ESTIMATED FATIGUE DAMAGE IN A

RAILWAY TRUSS BRIDGE:

AN ANALYTICAL AND EXPERIMENTAL EVALUATION

FRITZ ENGINEERING

&:iASORA TORY U8RARY

by

Antonello De Luca

A THESIS

Presented to the Graduate Committee

of Lehigh University

in Candidacy for the Degree of

Master of Science

in

Civil Engineering

Lehigh University

Bethlehem, Pennsylvania

October 1981

-r 0

ACKNOWLEDGMENTS

The research reported herein was conducted at Fritz Engineering

Laboratory, Lehigh University, Bethlehem, Pennsylvania. The Director

of Fritz Engineering Laboratory is Dr. Lynn S. Beedle, and the Chair­

man of the Department of Civil Egnineering is Dr. David A. VanHorn.

The help and guidance of Dr. J. W. Fisher, Project Director and

Thesis Supervisor, is greatly appreciated. Special thanks to the

Institute of International Education, Fulbright Coilliilission which

awarded the "Fulbright Scholarship" and to the American Embassy,

which awarded the "Italian Student Loan Corporation" loan.

The staff of Fritz Engineering Laboratory is acknowledged for its

support throughout this investigation. Thanks to Mr. R. N. Sopko, who

provided the pictures, to Mr. H. T. Sutherland and Mr. H. M. Hassan,

who superivised the field test. Very special thanks to Mrs. Ruth A.

Grimes and to Ms. Shirley M. Matlock, who typed the manuscript.

iii

TABLE OF CONTENTS

ABSTRACT

1. INTRODUCTION

1.1 History and Description of the Bridge

2. FIELD TEST OF THE BRIDGE

2.1 Purpose of Field Test

2.2 Test Setup

2.3 Test Procedure

2.4 Gage Locations

3. SUMMARY OF TEST RESULTS

3.1 Introduction

3.2 Stress Cycle Counting by Rainflow Method

3.3 Data Reduction

3~4 Impact Considerations

3.5 Stresses Measured at Different Locations

4. MATHEMATICAL MODEL

4.1 Analytical Models

4.1.1 Plane Simple Truss Model

4.1.2 Plane Frame Model

4.1.3 Three Dimensional Model

4.1.4 Three-Dimensional Model with Rail Acting Components with Stringers

4.1.5 Three-Dimensional Model with Ties

iv

Page

1

2

2

4

4

4

5

6

7

7

7

8

9

12

13

13

14

15

15

18 /

18

4.2 Pony Trusses: Summary of Previous Studies 19

4.3 Results of Analysis 20

4.4 Comparison Between Different Three-Dimensional Models 22

4. 5 Program LODPLO 23

5. COMPARISON BETWEEN FIELD TEST AND MATHEMATICAL MODEL

5.1 Plane Truss Model

5.2 Plane Frame Model

5.3 Three-Dimensional Models

5.4 Conclusions

6. ESTIMATED CUMULATIVE FATIGUE DAMAGE

6.1 Review of Basic Fracture Mechanics Concepts

25

25

26

26

29

31

31

6.1.1 Fracture Mechanics Approach to Crack Propagation 31

6 .1. 2 Fatigue Life Estimates 32

6.1.3 Variable Amplitude Load Spectrum 34

6.1.4 Stress Intensity Factor Estimates 35

6.1.5 Stress Intensity Factors for Riveted Stuctures 36

6.2 Fatigue Life of Riveted Joints

6.3 Predicted Stress Range from Train Traffic

6.4 Traffic Estimates Between 1905 and 2000

6.5 Estimated Histograms of Highest Stress Members

6.6 Estimated Fatigue Damage

7. CONCLUSIONS

TABLES

v

39

40

43

44

45

49

50

FIGURES 62

REFERENCES 98

APPENDIX Al: STATIC STRESS VERSUS TU1E RESPONSE 103 CAUSED BY THE PASSAGE OF TRAIN

APPENDIX A2: STATIC STRESS VERSUS TIME RESPONSE 110 OF CRITICAL MEMBERS CAUSED BY THE PASSAGE OF TRAIN

VITA 142

vi

Table

1

2

3

4

5

6

7

8

9

10

11

12

LIST OF TABLES

Section Properties

Gage Locations and Identification

Record of Test Runs

Summary of Maximum Measured Stresses

Comparison of Computed and Measured Stresses in the Floor System

Comparison of Maximum Computed and Measured Stresses in Truss Members

Comparison of Stresses in the Floor System by Different Models

Static Stress Cycles Caused by the Passage of One Car

Stress Cycles at Rivet Hole Caused by the Passage of One Car

Estimated Yearly Number of Cars and Locomotives which Crossed the Bridge

Estimated Total Number of Cars and Locomotives which Crossed the Bridge

Estimated Cumulative Fatigue Damage in 1981 and 2000

vii

50

51

52

53

54

55

56

57

58

59

60

61

LIST OF FIGURES

Figure

1.1 General View of the Kohr Mog Bridge 62

1:2 Plan and Elevation of the Bridge Structure 63

1.3 Cross-Section of Bridge 64

1.4 Built-Up Sections Used in the Bridge 65

1.5 Stringer-Tie-Rail Connection 66

2.1 View of Gages on Top-Chord and Bottom-Chord Members 67

2.2 View of Gages on Floor Beam and Diagonal 68

2.3 Gage Installed on a Rail 69

2.4 Instrumentation Used 70

2.5 Typical Analog Traces Recorded During Passage of Train 71

2.6 Geometric Properties of Locomotives and Cars 72.

2.7 Summary of Gage Locations on the Plan and Elevation 73

3.1 Response of Critical Members to Passage of Test Train 74

3.2 Stress Cycle Counting by Rainflow Method 75

4.1 Plane Truss Two Dimensional Finite Element Model 76

4.2 Three-Dimensional Finite Element Idealization 77

4.3 Complete Three-Dimensional Model with Ties and Stringers 78

4.4 Influence Lines for Plane Truss Model 79

4.5 Influence Lines for Plane Frame Model 8Q

4.6A Influence Lines for Three-Dimensional Hodel 3 81

4.6B Influence Lines of Floor System Members with 82 Different Models

viii

Figure

4.6C

4.7

4.8

4.9A

4.9B

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

Influence Lines of Floor System Members with Different Models

Influence Lines for Three-Dimensional Model 4

Influence Lines for Three-Dimensional Model 5

Deformation and Moment Diagram of Beam No. 8 Under Load No. 10 with Plate Elements

Deformation and Moment Diagram of Beam No. 8 Under Load No. 10 without Plate Elements

Assumed and Predicted Bridge Traffic

Stress Histograms at Floor Beam Center

Stress Histogram at Stringer Center

Stress Histogram at Diagonal u1L2

Stress Histogram at Diagonal u1

L1

Estimated Fatigue Life at Floor Beam Center

Estimated Fatigue Life at Stringer Center

Estimated Fatigue Life at Diagonal u1

L2

Estimated Fatigue Life at Hanger u1

L1

ix

83

84

85

86

87

88

89

90

91

92

93

94

95

96

APPENDIX Al

Figure Page

Al Bottom Chord 1414 - Test Train 103

A2 Top Chord u4u4 - Test Train 104

A3 Diagonal u112 - Test Train 105

A4 Hanger u111 - Test Train 106

AS Floor Beam Center - Test Train 107

A6 Stringer Center - Test Train 108

X

APPENDIX A2

Figure Page

A7 Floor Beam Center - One Diesel Locomotive and One Oil Car 110

AS

A9

AlO

Ml

Al2

M3

Al4

Al5

Al6

M7

Al8

Al9

A20

A21

A22

A23

A24

A25

A26

A27

A28

Stringer Center - One Diesel Locomotive and One Oil Car

Hanger u1L

1 - One Diesel Locomotive and One Oil Car

Diagonal u1

L2 - One Diesel Locomotive and One Oil Car

Floor Beam Center - One Steam Locomotive 200

Stringer Center - One Steam Locomotive 200

Hanger u1

L1

- One Steam Locomotive 200

Diagonal u1L2 - One Steam Locomotive 200

Floor Beam Center - One Steam Locomotive 220

Stringer Center- One Steam·Locomotive 220

Hanger u1L1

- One Steam Locomotive 220

Diagonal u1

L2

- One Steam Locomotive 220

Floor Beam Center - One Steam Locomotive 500

Stringer Center - One Steam Locomotive 500

Hanger u1

L1

- One Steam Locomotive 500

Diagonal u1

L2 - One Steam Locomotive 500

·Floor Beam Center - One Steam Locomotive 500

Stringer Center - 7en Freight Cars

Hanger u1

L1

- Ten Freight Cars

Diagonal u1

L2 - Ten Freight Cars

Floor Beam Center - Ten Oil Tanks Full

Stringei Center - Ten Oil Tanks Full

xi

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

Figure Page

A29 Hanger u111 - Ten Oil Tanks Full 132

A30 Diagonal u1

12 - Ten Oil Tanks Full 133

A3l Floor Beam Center - Ten Oil Tanks Empty 134

A32 Stringer Center - Ten Oil Tanks Empty 135

A33 Hanger u1 1

1 - Ten Oil Tanks Empty 136

A34 Diagonal u1

12 - Ten Oil Tanks Empty 137

A35 Floor Beam Center - Ten Passenger Cars 138

A36 Stringer Center - Ten Passenger Cars 139

A37 Hanger u111 - Ten Passenger Cars 140

A38 Diagonal u112 - Ten Passenger Cars 141

xii

I

ABSTRACT

An analytical and experimental analysis of the Kohr Mog Bridge,

at Port Sudan is presented herein.

Several finite element models were used to simulate the behavior

of the structure. The floor system restraint and the participation of

the ties and rail to the bending of stringers and floor beams were

among the variables.

The results of the analytical studies were compared with test

data in order to determine the applicable model. This permitted all

members in the bridge structure to be examined. The analytical model

was then used to develop the stress history that the structure had

experienced as a result of the various pieces of rolling stock that

had crossed the bridge. The resulting predicted stress history was

used to assess the possible fatigue damage by comparing the results

with test data on riveted connections.

This study indicated that if the riveted connections were good

with tight rivets, no significant damage would have developed as a

result of the passage of trains. If the rivets provided reduced

levels of clamping, some fatigue damage was predicted to have devel­

oped. Nonetheless, no appreciable indication of such damage would

become apparent until well into the next century.

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1. INTRODUCTION

1.1 History and Description of the Bridge

The bridge examined in this report is part of the Sudan Railway

System which was designed at the beginning of the century. This sys­

tem consists of a main north-to-south line, 925 km in length, extend­

ing from Wadi Halfa to Khartoum North, while an important branch line,

472 km in length, stretches across the country from Atbara Junction to

the Red Sea, providing a connection to Port Sudan. A

second branch extends from the main line at Abu Harned to Karirna for a

distance of 251 krn and provides the province Dongola with direct com­

munication with the Red Sea and Egypt.

The Khor Mog Bridge, at Port Sudan, is located on the branch

which connects Atbara to the Red Sea, at km 781.688 from Khartoum.

The bridge carries a single track 106.84 em gauge railway and con­

sists of nine simple 32.004 rn truss spans of the earliest design of

the Sudan Railways. The structures are half-through truss girder

spans.

Th~ steelwork was designed and fabricated by Sir William Arrol

and Company, Ltd., and the bridge was erected by the Sudan Railways in

1904/5. During the period 1935- 1937, the 32.004 rn spans were

strengthened to carry higher loads by welding a 381 x 12.7 rom plate to

the top chord over the central 19.6586 rn of each span.

In 1960, after a British consulting engineering firm was re­

tained to evaluate the bridge, bracing members were added which

-2-

connected the floor beams to the stringers. These members were bolted

to the bottom of the floor system.

A general view of the bridge is given in Fig. 1.1, while the plan

and the elevation are shown in Fig. 1.2. The cross-section of the

bridge is given in Fig. 1.3, while Fig. 1.4 shows the stringers, ties

and rails.

All members, except the bracing members and diagonals are

built-up members. The floor beams, stringers, and hangers consist of

four angles riveted to web plates (see Fig. 1.5A). The bottom chords

were built up by two webs and a flange plate to form two channel

sections (Fig. 1'.5B). Figure 1.5C shows the top chord section, which

consists of four angles riveted to two webs and to the upper flange

plate. The figure also shows the welding reinforcement plates. The

diagonals are two plates with no connection between them. All sec­

tion properties are given in Table 1.

The size of all rivets was 22.225 mm. A value of 25.4 mm was

therefore used to compute the net area of each member.

-3-

2. FIELD TEST OF THE BRIDGES

2.1 Purpose of Testing

In 1980 it was decided to carry out a general study on the Sudan

Railroad system in order to assess the conditions of the bridges and

evaluate any fatigue damage.

Fritz Engineering Laboratory was requested to carry out the study.

Five bridges were tested under the supervision of H. T. Sutherland in

January and February 1981 to provide the experimental basis of the

analysis. ·The structures examined were located at Atbara, Port Sudan,

over the Blue Nile at Khartoum, El Butana and over the White Nile at

Kosti.

The half-through truss girder span examined in this report is

used in several other bridges in Sudan. There are almost one hundred

such spans throughout the Sudan Railway System that are comparable to

the span which was tested at Port Sudan.

2.2 Test Setup

On January 24, 1981 the third span from Port Sudan of the Kohr

Mag Bridge was instrumented with electrical resistance strain gages.

These were placed at sixteen different locations on the structure.

The locations of gages on top and bottom chord are shown in Fig.

2.1. Figure 2.2 shows the gages on the floor beam and diagonal u2L3,

while Fig. 2.3 shows the gage on the rail.

-4-

In Fig. 2.2A it can be seen the plate of connection between the

hanger and the floor beam and the tapering of the floor beam at the

floor beam-to-hanger connection.

The surfaces were prepared by grinding them smooth, and the gages

were then attached to the surface with epoxy cement and covered with a

rubber strip of protective tape. The gage was then connected to

cables which hung from the bridge to the river bed, where the instru­

mentation was located (Figure 2.4) - Table 2 shows the correspondence

between the channels of ___ the amplifier, the cables and the location s

on the bridges.

The sixteen strain gages were connected to a recording oscillo­

graph with amplifiers which transmitted the signal to the analog

trace recorder. The records were then collected on UV sensitive

paper. Figure 2.5 shows typical time response conditions recorded on

the UV paper.

2.3 Test Procedure

Field testing was carried out on February 2, 1981. A train of

known weight crossed the bridge eleven times in the northbound and

southbound direction. Two of these runs were at 45 kph, while all

the others were made at 15 kph. Data are given in Table 3. Dis­

placements and strains under static load, with the center of gravity

of the train at midspan, were also taken. The train which made the

several test runs, consisted of two diesel locomotives, each weighing

986.38 N. Both units were GE engine class 1819. The dimensions of

-5-

each locomotive, their axle spacings and the axle loads are given in

Fig. 2. 6.

After the eleven runs with the test train were recorded, the

passage of a coach train was recorded in order to assist in evalu­

ating the.strain history experienced by the bridge.

2.4 Gage Locations

The location of all sixteen gages on the elevation and plan of

the bridge is shown in Fig. 2.7. One gage was placed on the top chord

at midspan 254 mm from the splice. Six gages were placed on bottom

chord members 1213, 1314, 1414. Two gages were placed on each bottom

chord member: one on the top of the lower flange plate, and another

on the bottom of the plate. One gage was placed on the bottom flange

angles of the stringer at the center of panel 111 2, and one on the

top flange angles 355.6 mm from the floor beam at panel 1 2 . The floor

beam at panel point 1 2 was gaged at its midspan and at the connection

to the hanger. Diagonals u1

12

and u21

3 were gaged at midlength. One

gage was installed on a rail and one on hanger u21 2 right above the

plate connecting the hanger to the floor beam.

-6-

3. SUMMARY OF TEST RESULTS

3.1 Introduction

Since no digital values for strain were obtained, the only pos-

sible method of data evaluation was manual measurement of the analog

trace. Some typical traces are shown in Figs. 2.5 and 3.1. By com-

parison of a trace of this type with a standard calibration record,

the stress change for each train or axle passage at a given detail

can be evaluated.

In order to count the stress cycles, a method had to be chosen.

In the United States most stress cycles have been counted using the

. (35) peak to peak method of cycle count1ng . In recent years, several

other methods have been examined including the peak count, the mean-

crossing peak count, the range count, the level crossing count and the

rainflow count methods. The rainflow method was selected for this

investigation. For a more detailed discussion on the difference

between the aforementioned procedures, see Woodward and Fisher's

reports( 35 ).

3.2 Stress Cycle Counting by Rainflow Method

The rainflow method can be easily understood using the analogy of

rain running down a series of pagoda roofs. The general rules for

counting are:

1. Rainflow begins at the beginning of the test and successively

at the inside of every peak.

-7-

2. Flow initiating at a maximum drips down until it comes

opposite a maximum more positive than the one from which

it started. It does similarly when it starts at a minimum.

3. Rain also stops when it meets rain from the roof above.

4. The beginning of the sequence is a minimum if the initial

straining is in tension.

5. The horizontal length of each rainflow is counted as a

half cycle at that strain range.

In Fig. 3.2 rain initiates at a, flows to b, drips to b', flows

to d and finally stops opposite e, because e is more negative than

Rain initiating at e stops at b' where it meets rain dripping from

Rain initiating at d flows to e and stops at the end of the record,

and flow initiating at e flows to f and stops at the end of the

record.

3.3 Data Reduction

a.

b.

All the data from the analog traces were then manually reduced

using the procedure previously outlined. The results of the strain

gage reduction are given in Table 4. A statistical average from the

nine runs at 15 kph was used for the subsequent computations.

-8-

The difference in the stress range when the train was headed in

the northbound or southbound direction was negligible, only an inver­

sion in the sequence of different stress ranges was observed.

The bottom and top chord only experienced one significant cycle

at each train pasage, aS' can be seen in Fig. 2.5, while the diagonals,

the floor beams, the stringers and the hanger experienced four stress

cycles at each test train passage. The coach train caused two,signifi­

cant stress cycles for each car. These cycles corresponded to each

set of axles of the locomotives and cars making up the train. The

gage located on the rail experienced a cycle corresponding to each

axle passage. As shown in Fig. 2.6, there are two sets of three axles

for the diesel engines.

It will be shown, in the chapter of evaluating fatigue damage

how many cycles and of which magnitudes are caused by different trains

at the critical locations.

3.4 Impact Considerations

Moving vehicles crossing a bridge structure result in a dynamic

response as a result of the interaction between the vehicle and the

bridge. The resulting dynamic increment is called the impact factor

and is a~tributed to three main reasons:

1. Speed effect

2. Roll effect: spring borne weight of the locomotive

oscillating about longitudinal axis.

-9-

3. Track and hammer blow effect: wheel and track irregu-

larities and unbalanced forces in the locomotive.

The specifications generally use an increment based on experi­

ence, theory and measurements. It is used as an upper bound to the

likely dynamic response. This is not always true for all members,

but most specifications use some function of velocity.

During the past two decades, considerable experimental and

theoretical research on highway bridge vibration has been made. A

general method of analysis regarding such vibration produced by a

single moving vehicle has been formulated by Velestos and Huang(32 ).

So far as an analytical solution of railway bridge vibrations is con­

cerned, no realistic model has been developed for a railway train

moving on a bridge. Chu, et al. (6 • 7) have developed a lumped mass

model of a railway train bridge in which they consider only the verti­

cal degree of freedom of each train joint. In their analysis the

vehicle system is idealized as a three degree of freedom model con-

sisting of the car body and wheel axle sets. This model allowed

Chu, et·al. to analyze the fatigue life of the critical member of a

seven panel railway truss bridge, by taking into account the impact

factor and the effect of different speeds( 33 ). In Ref. 34 Chu et al.

consider the..bridg~_impacts resulting from flat wheels and track

irregularities.

-10-

The use of such a model requires a knowledge of the spring and

dashpot constants of the locomotive and cars, and therefore, the

integration of the equations of motion of the bridge:

.. [M] [X) + [C) [X) + [K] [Z] = [F (x, t)] (3.1)

in which [F (x,t)] takes into account the external loads and the

interaction between the moving vehicles and the bridge.

The use of such a procedure is beyond the scope of this report.

It was decided to rely on the experimental data in order to evaluate

any impact factor.

All that can be concluded from the test runs is that for all

of the components measured, no major change was observed between

dynamic response at 15 kph and 45 kph, i.e.: no significant impact

was observed.

Several field tests on similar one track railway

b .d (11,28,29,30) h h h d . . . 1 r~ ges s ow t at t e measure ~mpact on cr~t~ca mem-

bers: hangers, stringers and floor beams, varied from a maximum of

11% in the Frazer, Ottawa and Rainy Bridges to 26% in the Assinbone

Bridge.

In our analysis, based on the test results, no impact was con-

sidered. If faster locomotives cross the Khor Mog Bridge, it is

recommended that a more realistic assessment of impact be made at

different speeds in order to better assess the dynamic changes that

are likely to develop in the various bridge components.

-11-

3.5 Stresses Measured at Different Locations

Based on the small variation in response of the ten runs at

15 kpm, that velocity will be assumed to a static load. The 15 kph

test data was used to determine the maximum static stresses on; the

structure.

The R.M.S. values of live load stress for the ten runs at sixteen

different locations are given in Table 5. The most highly stressed

members are:

The floor beam with 49.36 MPa

The diagonal u1

1 4 with 42.34 MPa

The top chord u4u4 with 36.48 MPa

The stringer with 39.99 MPa, and

The diagonal u21 3 33.37 MPa

All the other members experienced less than 27.58 HPa. The members

which are of major concern for the fatigue life estimation are the

floor beams, the stringers, and the diagonals.

The fact that the stress measured in hanger u111 was only

14.62 MPa can be explained by considering the behavior of a pony truss

The bending in the hanger is not significant, since the end connected

to the top chord is free to move in the out-of-plane direction. In

addition, diagonals framed into all hangers except at panel point 11 .

-12-

4. MATHEMATICAL MODEL

4.1 Analytical Models

Several stress analyses were made using five different analytical

models of the 105 ft (32.004 m) half-through truss girder span. The

stress analyses were performed using the SAP IV program (SAP IV is a

finite element program which can perform static and dynamic analysis

of linear systems).

The five analytical models were:

(1) a plane single truss model assuming all joints pinned,

(2) a plane frame model assuming all joints rigid,

(3) a three-dimensional model, which considered the plane

frame, floor beams, stringers and lateral bracing. It

was assumed that the frame joints, floor beam to truss

connection, and stringer-to-floor beam connections were

continuous (rigid). Four different sub-models were also

examined. In these models several of the joints were

released to simulate flexible joints in order to provide

lower bound restraint condition and evaluate its influence

on the stress resultants.

(4) a three-dimensional model which differed from the previous

one by taking into account the composite action of the

stringers and rail system. Figure 1.4 shows the physical

and analytical model used to simulate th~' participation of

the rail in the bending of the stringers.

-13-

(5) a three-dimensional model which simulates the actual floor system.

In this model the rail was assumed to behave as a composite beam

with the stringer (same as in model 4) , in addition all the ties

and timber beams were taken into account. Hence the floor system

behaved as an orthotropic deck.

The loading conditions included concentrated loads at lower

chord panel joints for modell and 2, and concentrated loads and moments

acting on the stringer to floor beam connection for models 3, 4

and 5. No out of plane loading due to wind or nosing was considered

in this analysis since the probability of occurrence of such a loading

condition is very rare and doesntt affect the fatigue life of the

bridge. The structure and the loading were therefore assumed synnnetric.

Reduction of nodal points due to symmetry consideration was taken into

account only for model 5 since the largest model between 1, 2, 3, and

4 (935 D.O.F., 209 bandwidth and 16 load cases) only required 137

system seconds to solve on the CYBER 720 at the Lehigh University

Computing Center.

The five models and relevant loading conditions are described

in more detail in the following sections:

(1) PLANE SIMPLE TRUSS MODEL

This is the usual model used for the linear elastic analysis of

a truss bridge span. It provides an upper bound to overall forces

and displacements in the truss. Figure 4.1 shows the 105 ft.

(32.004 m) half through span member and node numbering scheme used

-14-

L

in the stress analysis by SAP IV computer program. All joints were

assumed pin connected. The loading condition consisted of a single

vertical unit concentrated load at nodes 3, 5, 7, 9, 11, 13, 15, 17.

The gross area for all members was considered in this analysis

as well and in the other models. No reduction of the cross section

due to corrosion was observed in the field.

(2) PLANE FRAME MODEL

The plane frame model was identical to the plane truss model

shown in Fig. 4.1. All joints were assumed rigidly connected. As a

result of this assumption, moments can develop at the ends of the

frame members. Due to the very low inertia in the plane of the

bridge, all diagonal members behaved as truss members.

The loading conditions were the same used in the plane truss

model. Neither models 1 or 2 provided stresses in the floor system.

A separate analysis was required.

(3) THREE DIMENSIONAL MODEL

This model was developed to closely simulate the actual behavior

of the 105 ft. (32.004 m) bridge span. It also took into account

the floor system participation. Beam elements were used for all

members except for the bottom bracings. The bracing members' section

properties were negligible compared to the stringers and floor beams.

As can be seen in Figure 1.3a, floor beam to hanger connection

consist of a triangular plate riveted to the hanger and to the floor

-15-

beam by two angles. These plates were also taken into account in the

three-dimensional model. Two plate bending elements were used for

this purpose as shown in Figure 1.3b. Figure 4.2b shows the finite

element mesh used in this analysis, while Figure 4.2a shows the

three-dimensional model without connection plates between the

hangers and floor beams.

The model shown in Figure 4.2a consisted of 55 nodes, whereas the

model shown in Figure 4.2b consisted of 172 nodes. Six degrees of

freedom were allowed at all the nodes which were not supports.

Since beam members were used in this analysis and they were

assumed to be unidimensional elements in the "Saint-Venant ideali­

zation", it was necessary to establish the location of the elements

of the connection between the hanger floor beam and bottom chord.

Figure 1.3 shows that the centers of gravity of the three members do

not coincide. It was decided to assume that the three members were

connected at the center of gravity of the bottom chord. Hence the

location of the triangular plate had to be moved from its real

position. This resulted in a lowering of the intersection between

the hanger and the triangular plate (point Bin Figure 1.3), thus

providing less restraint to the hanger. To account for this diver­

gence between the physical and analytical model, an analysis was also

made using a larger triangular plate. No major difference was

observed in the stress resultants due to this change.

The loading condition consisted of two vertical unit loads

concentrated at the rail to floor beam connection which resulted

-16-

in an equivalent load which consisted of two unit loads and two

moments concentrated at the floor beam to stringer connection.

In this model the floor beam to truss connection was assumed

rigid (continuous) as were the stringer-to-floor beam connections.

Actually the stringer behavior can be expected to be bounded by

simple and continuous beams.

In order to bound the real behavior of the structure, four

sub-models having different floor beam to hanger and stringer to

floor beam connections were analyzed. They were:

MODEL 3A: The floor beam to hanger connections were assumed

rigid and a triangular plate element was used to

resist the bending moment. The stringer-to-floor

beam connections were assumed rigid.

MODEL 3B: Same as model A but with hinged connection between

floor beams and stringers.

MODEL 3C: Floor beams to hanger connections were rigid, but

the triangular plates were assumed to be loose.

No differential rotation was allowed between the

stringers and floor beams.

MODEL 3D: Same as model C with hinged connection between floor

beams and stringers.

The influence of the bracing system added in 1962 was also

studied by removing those elements and computing the difference in

stresses that resulted from that retrofit condition.

-17-

(4) THREE-DIMENSIONAL MODEL WITH RAILS ACTING COMPOSITE WITH STRINGERS

This model provided a better idealization of the actual behavior

of the stringers. It was observed in the field that the connection

between the ties and the rail and between the stringers and the

ties was not loose. This model simulated this condition.

The stringer to floor beam connection was considered rigid.

The spring connecting the rail to the stringer (see Figure 1.4)

was considered of infinite stiffness and fully effective. The section

properties for the composite section are assumed to be the new proper-

ties 6f the stringers.

The loading conditions were th~ same used in the previous model.

(5) THREE DIMENSIONAL MODEL WITH TIES

A new mesh, shown in Figure 4.3, had to be generated in order

to take into account the ties. Symmetry was considered in this case

since the number of nodes increased significantly. It was also

decided not to include the bracing system in this analysis since

its influence on the stress resultants had proved to be negligible.

Adding the bracing system would have resulted in a large bandwidth

(therefore high number of record blocks) which would have resulted

in prohibitive computing costs.

The stringer to floor beam connection was assumed rigid and the

section properties of the stringers were those defined in model 4.

The loading conditions were the same used in model 4.

-18-

4.2 Pony Trusses: Summary of Previous Studies

During the 1960's several studies were carried out on pony

trusses in order to investigate the behavior of this type of

structure. Of major concern was the distribution of stresses along

the top chord and the influence of the floor system.

The top chord of a pony truss is a compression member which is

restrained against lateral movement at the panel points by an elastic

transverse frame made up of the truss web members, the floor beams,

and their connections. The chord may buckle laterally over several

panels, as well as between two panel points carrying with it the

ends of the web members to which it is connected. A good approxi-

mation of the behavior of the pony truss top chord can be made by

analyzing it as a continuous beam on elastic support. (17)

Among the various secondary effects which made the analysis

more complicated, is the participation of the floor system which

reduces stresses in the bottom chord and also reduces the effective

depth of the truss thus increasing the top chord stresses.

(17) . . E. G. Holt developed one analyt1cal model for determ1ng

the buckling load of top chords of single-span pony truss bridges.

His studies led to the development of an "effective length factor" to

evaluate buckling of the top chord. His experimental results were

compared with buckling models developed by Engesser, Bleich and

Timoshenko.

-19-

The effect of floor systems, ignored in Holt's studies, is

taken into account by R. M. Barnoff(4 ) who conducted an experimental

and theoretical analysis which indicated an increase of 2% or 3%

in the chord stresses under normal working loads, due to the effect

of floor system participation.

Csagoly(9 ) has recently developed a method for investigating

the lateral stability of pony truss bridges. The method is an

iterative one which takes into account the secondary stresses caused

by change in geometry of the structure due to loading. It is, mainly

an extension of the concepts of a beam column elastically supported

at panel joint, which was introduced by Bleich(6a) in 1928.

4.3 Summary of Analytical Results

(1) PLANE TRUSS MODEL. The simple trusB model yielded axial forces

in all members. Figure 4.4 shows the influence lines for axial force

in the most critical members due to a unit train axle load moving on

the track. The stresses obtained in these members by loading the

influence lines with the train described in Figure 2.6 are shown in

Col. 2 Qf Table 6.

(2) PLANE FRAME MODEL. This model yielded axial forces and bending

moments (in the plane of the frame). Figure 4.5a shows the influence

lines for axial forces in the truss members. The stresses induced by

the moments were negligible. The stresses developed in the members

are given in Col. 3 of Table 6.

-20-

(3) THREE DIMENSIONAL MODEL "A". The three dimensional model (space

frame model) closely approximates the continuity conditions in the

actual bridge span. The model yielded axial forces as well as bending

moments in the hangers when the joints

previously described, four different sub-models were analyzed in

order to bound the real behavior of the bridge.

The influence lines are given in Figures 4.6A, B, C. As can be

seen, the difference between top u4u4 and bottom chord 141

4

influence lines are not significant. This is in good accord with

Barnoff~s study. The influence of floor system participation in

increasing top chord stresses was small. The stresses obtained in

the truss members and in the floor system are shown in Col. 4 of

Table 6.

The influence lines of axial force in the truss members do not

differ significantly from those obtained with the two-dimensional

models.

The influence of the stringer to floor beam connection and

floor beam to hanger connections did not significantly influence the

truss members. Only the floor system appears sensitive. When a

hinged connection is assumed between the floor beams and the stringers,

the stresses in the stringers and floor beams were maximum with no

redistribution outside the panel.

The triangular plates increased restraint at the floor-beam-

hanger connection.

-21-

Table 7 shows the stresses in the floor system computed by the

different sub-models. Figures 4. 9a and B show the deformed shape and the

bending moments that are developed in the floor beams and hangers.

Considerable out-of-plane displacements of the top chord can be

observed due to the bending of the floor beams.

(4) THREE DIMENSIONAL MODEL WITH RAILS ACTING COMPOSITE WITH THE

STRINGERS. Figure 4.7a shows the influence lines of this

model while the stresses in the members are given in Table 6.

(5) THREE DIMENSIONAL MODEL WITH TIES. Influence lines computed

by this model are summarized in Figure 4.8a. The st~esses

corresponding to the test train are given in Table 6.

The boundary conditions for all the models were: a roller on

the left side and a pinned end on the right side. To provide for

friction which inevitably exists in the roller giving more restraint

to the structure, a pinned-pinned condition was also examined for all

the models.

The bracing sy~tem added in 1962 proved to be negligible and

had no major effect on the stress resultants.

4.4 Comparison between Different (Three Dimensional) Models

The stresses in the truss members were not significantly affected

by changes in models 3A, 3B, 3C and 3D. This confirmed that the

restraint of the stringer floor beam connection and the presence of

the triangular plates connecting the floor beams to the hanger did not

-22-

influence the overall behavior of the truss.

Models 3A and 3C yielded about the same stresses in the floor

system as well as in the truss members. Models 3B and 3D, which

represent simple beam conditions for the stringers, also yielded the

same results. The reason why the plate had no effect on the moment

in the floor beams can be seen by examining the deformation of the

cross section shown in Figure 4.9. Since the top chord has no

restraint against out-of-plane movement, and the inertia of the top

chord members in the minor axis bending is very low, the plates

can only provide a rigid connection between the hangers and the floor

beam, but they can't avoid the rotation of the whole connection. The

plates will decrease the local stress introduced into the floor­

beam-hanger connection.

Table 7 shows that the model which best simulated the floor

system was model B. This model was therefore used to compare the

computed stresses with the measured ones.

4.5 Program LODPLO

A computer program was developed in order to load the influ­

ence lines with the train used in the tests. This program gave

the maximum stress in a member as well as the stress versus time

diagram at a given location due to the passage of the train. These

diagrams allowed a comparison of the mathematical model response to

the recorded strain data. The scale factor built into the program

reduces the length and the height of the diagrams to those obtained

. -23-

by the analog trace. The results are given in Chapter 5.

-24-

5. COMPARISON BETWEEN FIELD TEST AND MATHEMATICAL MODEL

Tables 5,6 summarize the predicted liveload stresses and the

measured stress ranges in several bridge members.

Both the maximum measured stresses and the root mean square

stress for the eleven test runs are summarized in Tables 5,6 for the

gages placed on hangers, chords, floor beam, stringer.

(1) PLANE TRUSS MODEL

The results of the analytical study are summarized in column 1

of table 6. The computed axial force was converted to average stress

using the gross section of the appropriate member.

This model provided the highest stresses in all members except

bottom chord member L2L3

, thus providing an upper bound value for

axial stresses. All calculated stresses were higher than the measured

stresses. The predicted stresses were about 23% greater than the

measured stresses ignoring any dynamic effects.

When a roller and a fixed end were used as boundary conditions,

the bottom chord stresses were overestimated by 56%. When both ends

were fixed against longitudinal displacements, the bottom chord stress­

es were underestimated by 61%.

It can be seen from the table that the support boundary conditions

only affected the bottom chord stresses. No change was observed in

all the other members. This behavior was common to all models studied.

-25-

The greatest deviation was observed in the hanger U2L2 which

was overestimated by 64%.

( 2) PLANE FRAME MODEL

The axial stresses predicted from this model were nearly

identical to those computed with plane truss. Hence about the same

deviations were observed between the predicted and measured stresses

as discussed in section 5.1.

(3) THREE-DIMENSIONAL MODELS

The model referred to in Column 3 of Table 6 is sub-model B.

Table 7 shows that the model which best fits the measured floor system

stresses is sub-model B. This suggests that according with Lewitt

(19) . and Chesson , the str~ngers behave as simple beams. The compari-

son in the various members are as follows:

(a) BOTTOM CHORD -

With the fixed-roller boundary condition the stresses were

overestimated by 46%. The fixed-fixed boundary condition under-

estimated the stresses by 65%. This comparison of the predicted

bottom chord stresses with the measured test train values suggests

that some fixity of the support existed. The measured bottom chord

stresses were bounded by the two extreme models of the support

conditions. This discrepancy was not of major concern since the

bottom chord members only experienced one stress cycle at each passage

of the train (see Fig. 2.5). The low magnitude of stress range and

its only occurrence one time during passage of a train indicates that

-26-

no fatigue problems should ever develop in the bottom chord since

even the fixed-roller condition results in a stress range below the

fatigue limit.

TOP CHORD

Model 4 and 5 underestimated the stresses by 0.6%. While model

3 overestimated the stresses by 0.6%. This agreement indicates that

either model provides a satisfactory prediction of the top chord

stress. The predicted values are all in good agreement with the

measured stresses shown in Table 6.

DIAGONALS

Model 3 overestimated the stresses in u11 2 by 8% and in u213

by

11.4%. Model 4 and 5 underestimated the stresses in u11

2 by 1.8%

and overestimated u213 stresses by 8.3%.

It will be discussed in section 5 why the diagonals are not

usually the most critical members.

HANGERS

All models overestimated the stresses in the hangers (see

Table 6).

The gage was placed on the interior side of the hanger in

which the bending moment causes tension. The difference between

the measured and computed stresses on this side was about 20%. The

value given in the table is the maximum stress which will be

experienced by the other side of the hanger.

-27-

It is also believed that the gage was installed in a region

influenced by the bracket plate. However this is not a fatigue

critical member since it is subjected to cyclic compression.

Hanger u1

,11

was not gaged. Considerations on its fatigue life

are given in paragraph 6.

FLOOR BEAMS

The measured floor beam stresses were overestimated by model

3B and 3D by avout 2.8%. Sub-models 3A and 3C underestimated the

floor beam stresses by 13%.

The floor beam end stringer stresses were in best agreement with

the measured stresses when the stringers were assumed to be simply

supported members. Models 4 and 5 which assumed the stringers-to­

floor beams connections to be continuous, underestimated the stresses

by 20 and 42%, respectively.

STRINGERS

Model 3B which gives the same results of a simple beam analysis

for each stringer, underestimated the stresses by 11%.

-28-

The other models had higher discrepancy (up to 65%). It has been

observed that this type of connection behaves closely to simple beam

condition. It should be noted from the analog trace (Figure 2.5)

that there was some degree of restraint in the stringers. Since no

allowance was provided for impact, it is believed that the real

behavior has some restraint. Therefore the impact should be about

15%. The value of impact suggested by Area Specifications which is

usually very conservative:

imp. % 100 + 40 s

312

1600

where S is the truss spacing and L the stringer span length, gives

a value of 45.6%.

5.4 Conclusions

The results of this study has indicated that the bridge structure

was best modeled as a three dimensional structure with relatively

little fixity in the floor beam stringer connections. The forces

in the chords diagonals and verticals were found to be in good

agreement with the measured response when the 3D structural model

was used.

The floorbeams and stringers are the structural members most

susceptible to impact forces. The differences between the

predicted and measured strains indicated that the stringers were

best modeled as simply supported components. The floor system

would be most susceptible to impact. The differences observed

-29-

indicated a dynamic factor of about 10% or 15% would provide

agreement for the simply supported systems (stringers). Another

value of impact was observed for the floor beams.

When continuous stringers were assumed the variation between

the predicted and measured stresses approached 45%. As can be seen

in Fig. 2.5 the measured strain response did not seem to indicate

such a large dynamic increment.

-30-

6. ESTIMATED CUMULATIVE FATIGUE DAMAGE

6.1 Review of Basic Fracture Mechanics Concepts

6.1.1 Fracture Mechanics Approach to Fatigue

Fracture mechanics was initially used in aerospace and initiat­

ing applications to analyze failures of rockets, airplanes, and ships

and other structures after catastrophic failures proved to be due to

small fatigue cracks existing in the structures.

Although brittle fractures had occurred in many types of struc­

tures, fracture mechanics was not considered a design tool until the

failure of Point Pleasant Bridge at Point Pleasant, West Virginia on

December 15, 1967, which resulted in the loss of 46 lives.

In spite of advances in materials, design, and fabrication in

steel industry, several steel bridges have experienced structural

failure after the Point Pleasant Bridge failure.

Every structure contains discontinuities where size and distribu­

tion are dependent upon the material and its processing. These dis­

continuities can grow and enlarge under cyclic loads and eventually

reach a critical size.

The three major factors that control brittle fracture can be

related by the equation:

K=a ~ (6.1)

which is valid for an infinite plate in uniform tension with a crack

of length 2a (MODE I).

-31-

In this equation K is the stress intensity factor, 2a is the flaw

size, and o is the applied stress. When the stress intensity factor

at the crack tip reaches a critical value of material toughness depen-

dent on temperature and strain rate, unstable fracture occurs, which

may result in complete failure of the member.

The critical stress intensity factor K , represents the terminal c

condition in the life of a structural component. The total useful

oife of the component is determined by the time necessary to initiate

and propagate a crack from the initial crack dimensions to the criti-

cal size a . Crack initiation and crack propagation may be caused by t

cyclic stress (fatigue)(23a).

6.1.2 Fatigue Life Estimates

f k · k h da · 11 The atigue crac propagat1on or crac growt rate, dn 1s usua y

related to the variation of stress intensity factor /J.K by the empiri­

cal Paris Power law< 22 ).

(6. 2)

where c·and n are material constants for the steels, n can be taken as

about 3 for most structural steels, and C is equal to 2 x 10- 10 for a

mean value and 3.6 x 10- 10 for an upper bound fit to available crack

growth data.

For simple through thickness cracks!

!J.K = !J. 0 (Tia)112 (6. 3)

-32-

and fatigue life is mainly affected by the stress range and the

initial flaw size. Stress range means that only live load stresses

need to be taken into account.

Equation 6.2 can be integrated:

Nf af

f dN J da (6 .4) =

Cl~Kn N. a. ~ ~

Usually, liK is a function of crack size and is difficult to inte-

grate directly. Therefore, numerically integration must often be per-

formed. For the case of an infinite plate subject to uniform tension,

Eq. 6.3 holds. In this case, integration of Eq. 6.4 leads to:

(6. 5)

It should be noted that Eq. 6.5 represents the fatigue curves

given in the Bridge Fatigue Guide which satisfies the relation:

liN = C S - 3 r

and have been shown to fit full scale tests well (l 4,l5).

(6.6)

Equation 6.5 also shows that when af >> ai, the initial crack

length is the dominant factor in fatigue life.

-33-

6.1.3 Variable Amplitude Load Spectrum

The testing which resulted in the curves given in the Bridge

Fatigue Guide, were developed from constant amplitude applied loads.

Extensive test results, by Schilling, et al.,( 2S) showed that vari-

able amplitude loading with a random sequence stress spectrum, simu-

lating actual bridges, can be conveniently represented by a single

constant amplitude effective stress range that would result in the

same fatigue life as the variable amplitude stress range spectrum.

The effective stress range is defined by:

B 1/B [ y. s . ] l. rl.

(6. 7)

. h . h s . h . th d . h f f 1.n w l.C . 1.n t e 1. stress range an y. l.n t e requency o occur-rl. l.

renee of the stress range(ZS). If B is taken as 2, Sr£ is equal to

the root mean square (R M S) of the stress ranges in the spectrum.

If B is taken as the reciprocal of the slope of the constant-amplitude

SN curve, the equation is equivalent to Miner's law:

n. l. N. l.

= 1 (6. 8)

The test conducted by Schilling,et al. showed that both the R M S

and Miner's effective stress ranges satisfactorily represented the

variable amplitude spectrum. The R M S method provided a slightly

better fit to the test data, and Miner's law was slightly more con-

servative.

-34-

In this study, both stress ranges were computed, but the

Sr(MINER) was used to estimate life, since it was more conservative.

6.1.4 Stress Intensity Factor Estimates

For most general conditions, K may be expressed as a central

through crack in an infinite plate under uniform stress modified by

several correction factors(l).

K = F e

F s F w F . o (rra)112

g (6.9)

The correc·tion factors modify K for the idealized case· to account for

effects of crack shape F , free surface F , finite width F , and non-e s w

uniform stress acting on the crack F . g

To evaluate fracture instability, the total sum of stresses due

to residual stresses, dead load and live load must be considered. For

cyclic fatigue, loading only the variation of stresses due to live

load have to be considered ~ORMS or ~OMINER'S'

When an exact solution of the stress intensity field for the

particular geometry and loading condition is not available, Eq. 6.9

has to be used. Analytical closed form solutions are also available

in the literature(26 • 27 •31 ) for certain well defined crack conditions.

A free surface correction factor of:

F = 1.211- 0.186 s

-35-

a b

(6.10)

is employed here on edge crack in semiinfinite plate subjected to

uniform stress.

For a central crack in a plate of finite width, the function

F w

Tia Sec Zb (6.11)

a . (17B) has an accuracy of 0.3 percent for an b rat1o less than 0.7 .

Integral transformation of a three-dimensional elliptical crack

shape has resulted in the elliptical correction factor Fe. For the

. (17C) point on the ellipse of maximum stress intensity, its value 1s :

F e = 1 E(k)

where E(k) is the complete integral of the second kind:

TI/2

E(k) = J 0

2 [1- (1 - ~ Sin2 8)] dB

2 c

h a . h . . . .d. . (10) w ere - 1s t e m1nor to maJor ax1s sem1 1ameter rat1o . c

integral is often referred to as the shape factor ~.

(6.12)

(6.13)

This

Expression for stress gradient correction factor F can be very g

complex. There are two methods of estimating this function. The

first procedure, which was suggested by Albrecht(l), requires the

determination of the stress field in the uncracked structure either

by a sttess analysis if possible, or using a finite element solution

for more complicated cases. The resulting stresses are then removed

-36-

from the crack surface by integration as a Green function. The

second procedure which is conceptually easier, requires the use of a

specialized finite element program, in which special elements, having

the crack singularity built in, are provided.

6.1.5 Stress Intensity factors for Riveted Structures

All riveted structures contain stress concentrations due to the

rivet holes. The majority of service cracks nucleate in the area of

stress concentration at the edge of a hole. For the application of

fracture mechanics principles to cracks emanating from holes, know-

ledge of stress intensity factor is a prerequisite.

By using a technique of conformal mapping, Bowie(6B) developed a

K solution for radial through cracks emanating from unloaded open

holes. The stress intensity factor is given by:

(6.14)

where the crack length is measured from the edge of the hole, and D is

the hole diameter. The function f (;) is given in tabular or graphi­

cal forni (6C).

Kobayashi(lS), developed estimates of the stress intensity factor

at holes using the Green's function procedure. The resulting expres-

sian for the stress intensity is given by:

-37-

'I

KI = a rnaj 1 + _l_ [ aR ~-20R4 + 9a2

R2

- 4a4

) 2TI 3

(R2 - a2)

+ R3 (iR4 + 5a2

R2 + 2a4 ) ( ; - . -1

*) l] (6.15) 3 s1.n

I R2

- 2 (R2 - a2) a

This expression is valid for a < R and gives results within 10%.

For the case where the crack is not small compared to the hole,

the effective crack size is then equal to the physical crack plus the

diameter of the hole. The stress intensity factor for the asymmetric

case with 2 aeff = D + a will be:

K = a In aeff = a ;.n:-a ( J2_ + l )1/2 2a 2

(6.16)

Expressions 6.14, 6.15 and 6.16 apply for through thickness

cracks. Often, fatigue cracks develop, as either corner cracks or

semielliptical shaped cracks in the hole. For the corner crack, the

shape factor( 6B) has to be taken into account. In Ref. 18,

Kobayashi gives an approximate solution to this problem as well as

to the crack emanating from a circular hole loaded with a concen-

trated load. Kobayashi in Ref. 18 also provides an approximate solu-

tion for cracks emanating from a circular hole loaded with a con-

centrated load.

Hsu (llA) took into account the clamping force as well as other

effects. He conducted an analytical and experimental investigation in

order to characterize the fracture and cyclic growth behavior of

-38-

cracks emanating from various types of fastener holes. He examined

tolerance, inteference-fit, and cold-worked fastener holes. He

developed analytical inelastic estimates of stress intensity factors

for through cracks and approximate stress intensity factors for

quarter-elliptical cracks emanating from a corner. Experimental

crack growth rates for holes, with residual strains (cold-worked or

interference-fit fasteners) were significantly lower than for

straight reamed holes without ariy conditioning, especially for small

initial cracks. This benefit decayed as the crack length increased.

Hsu also indicated the possibility that the beneficial residual

compressive strains induced by the cold-working operations were

relaxed during the subsequent applications of cyclic loads.

These mathematical models of crack growth can be used together

with test data on riveted structural joints in order to characterize

the fatigue and fracture behavior of riveted structures.

6.2 Fatigue Life of Riveted Joints

The present AREA Manual for Railway Engineering requires that

riveted connections that are subjected to fatigue or repeated loadings

must meet the requirements of Category D of Articles 1.3.13 and 2.3.1

if the engineer can verify that the rivets are tight and have

developed a normal level of clamping force, Category C may be used to

estimate the fatigue resistance.

Since Category D seemed conservative, a study ha.s been carried

b M (20 ) h .1 bl . . d . . out y arcotte . on t e ava1 a e r1vete J01nt test data.

-39-

The conclusions of this study are:

1. Members with riveted connections subjected to zero-to­

tension or half tension-to-tension load cycles shall meet

the requirements of Category C, except as noted below.

2. Members with riveted connections subjected to zero-to­

' tension or half tension-to-tension load cycles and with

severly reduced levels of clamping force hole meet the

requirements of Category D.

3. Members with riveted connections subjected to full several

load cycles shall, in all cases, meet the requirements of

Category D. The effective stress range shall be taken as~

the tensile portion of the stress cycle.

It should be noted that the above recommendations are based on the

assumption that the allowable bearing ratio is low which is usually

the case for older bridge structures.

6.3 Predicted Stress Range from Train Traffic

Program LODPLO was used in order to obtain the computed time

response at all the gage locations due to the passage of different

cars. Figures Al to A6 (see Appendix A) show the strain- time

relationship for bottom chord members L4L~, L3L4 , L2L3 , top chord

member u4u

4, diagonals u

1L2 and u2L

3, hangers u1L1 , u1L2; the floor

-40-

beams and stringers due to the test train. They all correspond

well with the experimental analog traces at the appropriate section.

The program was also used to compute the stress ranges at various

locations due to different- cars and locomotives known to use the

bridge structure.

As can be seen from Fig. Al to A6 and from the analog traces, the

only members which experienced more than one major stress cycle during

passage of the test train were the hanger u1

11

, the floor beams and

the stringers. The floor beams, the stringers and the diagonals are

the most highly stressed members. As a result of the frequency of

loading and the magnitude of the stress range, they are the most

critical members in the bridge structure for fatigue damage. Fatigue

failures have been observed in railway truss bridges, mainly in hanger

members, although a few stringers have also experienced cracking.

In this particular pony truss structure the hangers do not

experience the high stress cycles that are experienced by the floor

beams and stringers.

Figures A7 to A39 show the time response of the hanger u11

1,

stringer, floor beam, and the diagonal u11 2 due to the stress loco­

motive series (220, 200 and 500) and the diesel engine 1800 followed

by one oil car. Also shown is the response due to cars and passenger

cars. As can be seen in Figs. A7, A9, All and Al3, the hanger and the

floor beams experienced two cycles during the passage of each steam

locomotive and one cycle due to the passage of each car. The

-41-

stringers (see Figs. A8, Al2, Al6) experience two cycles of stress

range during the passage of each car and one cycle as a result of the

passage of each axle of stress locomotives. This can be attributed

to the behavior of the stringers as simple beams.

Figures A9, Al3 show that the diagonal u11 2 which is the highest

stressed truss member, only experienced one major stress cycle during

the passage of a train.

The steam locomotive 500 caused the highest stresses (56.54 MPa)

in the floor beams, while the empty oil tanker and passenger car only

caused 8.34 MPa and 11.58 MPa in the floor beams.

The Rainflow Counting Method (see Section 3.2) was used in order

to assess the stress range cycles.

Model 3B was used in all the computation of stress versus time

response (three dimensional model with stringers or single beams)

described in Section 4.1.

The net section was used to evaluate the fatigue resistance of

the riveted members. In these bridge members the difference between

the net and gross section stresses was small.

In the computation, the net section of the hanger u1

11

and of the

diagonal u11 2 was used. For the floor beam and the stringer the modi­

fied section modulus was taken into account, and the stress at the

rivet holes was used. This resulted in no increment due to the lack

of area at the rivet location. The local effect of stress concentra­

tion in proximity of the rivet should be taken into account if a

-42-

fracture mechanics model is used to compute the fatigue behavior of

riveted structure in bending, since the modified section modlus

doesn't reflect any local effect. However, since no study seems to

be available in this direction, the modified section modulus was used.

Table 8 summarizes the stress cycles experienced by hanger u1

11

,

the floor beams, the stringer and diagonal u211

as a result of passage

of different cars. Table 9, instead shows the stresses computed with

the net section. As it can be seen, only the first cycle is affected.

The floor beams and stringers of each panel point behave like

single beams (use influence lines in Fig. 4.7) and experienced the

same stress cycles.

Figures A23 to A39 shows that a consist of ten oil cars crossing

the bridge caused one main cycle in diagonal u1

1 2 equal to 35.85 MPa

and ten minor cycles equal to 2.76 MPa. Only the main cycle was

taken into account, since the amplitude of the other stress cycles

was negligible.

6.4 Traffic Estimates Between 1905 and 2000

A detailed survey of all the traffic crossing the bridge since

1905 was not available. However, from several meetings with Mr. H.

M. Hassan, Senior Engineer of the Sudan Railways, the following gen­

eral information was acquired and used in the analysis:

1. Prior to 1960 all engines used on the system were steam

locomotives. Before 1935.engine type 220 and 200 were used

-43-

and after 1935, engine type 200 and 500. It was assumed that

before 1935, 60% of the locomotives were engine type 200 and

after 1935, 70% of the engines were type 500.

After 1960, all locomotives were diesel engine 1800- 1819.

2. Prior to 1960 the trains crossing the Kohr Mog Bridge con­

sisted of 20 to 25 cars. After 1960, the diesel engine

pulled forty cars.

3. Forty percent of the freight cars were assumed to be oil

tankers. These oil tank cars are full when they cross the

bridge going to Khartoum, while they return empty. The

freight cars were always assumed to be full.

4. Since the Kohr Mag Bridge is located on a hill, two steam

engines were used to pull the cars prior to 1960. After hav­

ing pulled the train across the bridge, one engine returned

to Port Sudan. This means that in one complete trip (Port

Sudan- Khartoum- Port Sudan) two sets of cars crossed the

bridge ~and. four steam locomotives.

5. The distribution of freight trains and passenger trains per

month between 1905 and 1960 was assumed to be constant.

From 1960 to 1980 the number of trains has increased in a

linear manner. By the year 2000 it is assumed that the total

number of freight trains will double over the number crossing

the bridge in 1980.

-44-

Figure 6.1 shows the assumed traffic distribution. It should be

noted that running 614 trains per month in the year 2000 means 22

trains per day, which means an extremely high number of trains.

All axle loads and spacings are summarized in Fig. 2.6.

Table 10 shows the yearly number of cars which crossed the bridge

before 1960, in 1980 and in 2000. The total number of cars which

crossed the bridge in 1960, 1981 .and will cross in 2000 is given in

Table 11.

The total number of cycles, their magnitude and the histograms

of the critical members were evaluated by taking into account the

stress ranges caused by the passage of each car or locomotive crossing

the structure.

The cumulative fatigue damage estimated to occur by 1980 and

2000, as well as the residual fatigue resistance is given on Table 12.

Recomputation of the total number of cycles is recommended if

the past and future traffic data is proved to be different from the

one assumed herein.

6.5 Estimated Histograms of Highest Stress Members

A program was developed to use the procedure outlined in para­

graph 6.3. This program used as input the number of cycles and ranges

due to each car given in Fig. A7 to A37·and the total number of pas­

sages of the cars in the year in which the fatigue damage had to be

computed.

-45-

The output included the total number of cycles experienced by the

section considered, the stress histogram, the root mean stress range

Sr(RMS) and the Miner stress range Sr(MINER)"

The stress histograms of the critical section are shown in Figs.

6.2, 6.3, 6.4 and 6.5.

Up to 1981 the Sr(MINER) was estimated to be 17.44 MPa for the

hanger, 19.24 MPa for the stringer, and 28.89 MPa for the floor beam.

The total number of cycles experienced by the hanger was 5,768,000~ 9Y

the stringer was 11,179,160, and 5,768,100 for the floor beam. The

stringers experienced almost twice as many stress cycles as the floor

beams and hangers. However, the value of SrMINER and SrRMS is only

19.31 MPa. The diagonals have only experienced 444,150 cycles.

By the year 2000, SrMINER and SrRMS for the hanger, floor beams

and stringers changed very little. The total number of cycles experi­

enced by these members will about double if the conservative assump­

tions are satisfied.

Table 12 summarizes the SrRMS' SrMINER at the different loca­

tions and the total number of cycles in 1980 and 2000.

6.6 Estimated Fatigue Damage

Figures 6.6 to 6.9 show the stress distribution of the critical

members plotted on the S-N curve of riveted structures. It seems that

if no stress cycle in the variable amplitude loading spectrum is

higher than the endurance limit, then it can be assumed that the

-46-

.ll.h .f .. l.f(l2) structure w~ ave ~n ~n~te ~ e . The endurance limit for Cate-

gory C is 68.95 MPa, while it is 48.27 MPa for Category D.

It is apparent that no fatigue damage should develop in the

hangers. The floor beams, stringers and diagonal will not experience

fatigue damage if Category C applies. If the clamping force is not

high, and Category D is used to assess the fatigue resistance, then

the floor beam will exhibit fatigue cracks by the year 2100, the

diagonal by the year 2025, and the stri~gers will have infinite life.

Such crack formation should be easy to monitor and detect should it

develop. It should also be noted that fatigue cracking of one of the

multiple component members will not result in collapse of the member.

This should permit the inspection to detect any such cracks that form.

It should be noted from the analog trace (Fig. 2.5) that the

stringer didn't behave exactly as a simple beam, therefore experienced

some reversal in the stress cycle.

The residual fatigue life, after cracks were detected, could be

estimated with a fracture mechanics approach.

Charpy V-notch tests are currently being run at Fritz Engineer-

ing Laboratory, while this report was in preparation.

On the basis of the material properties, fracture toughness,

service temperature of the bridge, and considering that residual

stresses are low in this type of structure, brittle fracture is not

likely to occur. Fatigue cracking should be the only possible means

of developing cracks.

-47-

)

The total number of cycles to failure can be computed by inte­

grating formula (6.4), where ~K, for through cracks smaller than the

hole is given by Eq. 6.15. If cracks are larger than the hole

(25.4 mm) then formula 6.16 applies for the stress intenstiy factor

of a through crack.

-48-

7. CONCLUSIONS

Several analytical models of the railway truss bridge at Port

Sudan were studied. The three dimensional finite element analysis of

the bridge which assumed the stringer as simple beams provided the

bestagreement betweenthe field test and the analytical models. The

stresses measured in the top chord and diagonals were within 9% of

the predicted values. The hanger stress was overestimated by 18%.

The bottom chord was found to be bounded by the fixed-fixed bearing

and the fixed-roller models. The floor beam stresses were overesti­

mated by 3%, and the stringer stresses were underestimated by 11%

neglecting dynamic effects. Hence, an impact factor of about 15% was

assumed applicable to the stringers~

The fatigue of vertical members proved to be the floor beams and

the diagonal u112 if the worst condition is assumed at the riveted

joints. The first hanger, in the pony truss structure allows out-of­

plane displacements of the top chord which minimizes bending stress

and decreases the stress range.

Estimation of the condition at the rivet holes is critical, since

Category C connections will not experience crack growth and will have

infinite life. If Category D is applicable, the floor beams will

start to experience cracking in 2100 and the diagonal in 2025 under

the assumed stress spectrum.

It is recommended to verify by test, the magnitude of dynamic

response, the tightness of the rivets and the validity of the assump­

tion in past and future traffic.

-49-

TABLE 1: SECTION PROPERTIES

I (cm4) I (cm4

) X y

Members AREA (cm2) X 10 2

X 10 2

ulu2 216.13 811.31 . 413.88

Top Chords u2u3 264.52 875.89 502.57

u3u4 282.26 922.06 528.10

u4u4 282.26 922.06 528.10

LOLl 174.19 607.07 211.01

LlL2 174.19 607.07 211.01 Bottom

L2L3 174.19 607.07 211.01 Chords L3L4 245 .16 791.74 262.46

L4L4 245.16 791.74 262.46

LOUl 179.65 243.30 83.13

UlL2 106.45 82.41 0.14

Diagonals U2L3 88.71 4 7.68 0.11

U3L4 70.97 12.81 0.10

U4L3 41.51

U4L4 41.51

UlLl 92.66 131.40 15.50

Verticals U2L2 102.23 151.82 29.26

U3L3 92.66 131.40 15.50

U4L4 92.66 131.40 15.50

14.83 1.13 0. 77

Bracing 12.45 0.97 0.66

After 1960 20.97 1.50 1. 50

Floor 266.19 2138.18 103.64 Beams

Stringers 132.97 661.58 14.84

-so-

Channel

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

TABLE 2: GAGE LOCATIONS AND IDENTIFICATIONS

Table No.

Cl

Cl

C2

C2

C3

C3

C4

cs

C6

C7

C8

C9

ClO

Cll

Cl2

Cl3

-51-

Location

Bottom Chord L2L3

Bottom Chord L2L3

Bottom Chord L3L4

Bottom Chord L3L4

Bottom Chord L4L4

Bottom Chord L4L4

Vertical u2L2

Floor Beam-Top @ L2

· Stringer- Top @ L1

Diagonal u1

L2

Floor Beam- Center @ L2

Rail

Stringer Center @ L1L2

Top Chord u4u4

Bottom Chord L4L4 (on splice)

TABLE 3: RECORD OF TEST TRAIN RUNS

Date Test No. Velocity Direction

2/2/81 1 15 kph South

2/2/81 2 15 kph North

2/2/81 3 15 kph South

2/2/81 4 15 kph North

2/2/81 5 15 kph South

2/2/81 6 15 kph South

2/2/81 7 15 kph North

2/2/81 8 15 kph South

2/2/81 9 15 kph North

2/2/81 10 45 kph South

2/2/81 11 45 kph North

2/2/81 12 0 kph (static) Dead Weight

2/2/81 13 15 kph Coach Train

-52-

TABLE 4: SUMMARY OF MAXIMUM MEASURED STRESSES

AT 15 kph AND 45 kph

All Values in MPa

15 kph 15 kph 45 kph 45 kph Maximum R.M.S. Maximum R.M. S. Between Average Between Average

Member 9 Runs Between 9 Runs 2 Runs Between 2 Runs

1213 26.32 21.72 21.17 20.89

1314 27.03 24.68 25.86 27.03

1414 18.82 17.03 17.65 17.03

u4u4 -36.41 -34.96 -34.06 -34.06

Ul12 42.34 40.82 42.34 42.34

U213 33.37 31.03 30.55 29.99

U212 -14.14 -13.31 -14.14 -12.20

Floor Beam 49.37 46.33 47.02 45.30 Center

Floor Beam 18.75 46.33 47.02 45.30 Top

Stringer 39.99 38.47 39.99 39.99 Center

Stringer 8.96 7.79 8.21 7.65 Top

-53-

TABLE 5: COMPARISON OF MAXIMUM COMPUTED AND MEASURED STRESSES

IN THE FLOOR SYSTEM

Neasured

R.M.S. Model 5* Model 4 Model 3B Maximum Average

Floor Beam 29.44 34.68 50.82 49.37 46.33 Center (29.44)

Floor Beam 8.21 13.93 13.99 18.75 15.45 Top (8.23)

Stringer 27.44 27.58 36.19 39.99 38.47 Center (27.44)

Stringer 9.31 10.34 9.65 8.96 7.79 Top (9. 31)

* The values in parenthesis represent the fixed-fixed condition at the bearings

-54-

TABLE 6: COMPARISON OF MAXIMUM COMPUTED AND MEASURED STRESSES

IN TRUSS MEMBERS

Two Dimensional Models Three Dimensional Models Measured

Mem- Plane Plane R.M.S. ber Truss Frame Model 3 Model 4 Model 5* Maximum Average

1213 41.03 41.58 40.13 38.89 39.44 26.34 21.72 (16.41)

1314 39.37 39.03 37.65 36.20 37.03 27.03 24.68 (14.14)

1414 41.92 41.37 40.41 37.10 39.16 18.82 17.03 (8.41)

u4u4 -37.44 -36.82 -36.61 -36.20 -36.20 -36.41 -34.96 (-35.92)

UlL2 47.71 45.99 42.96 41.44 41.58 42.34 40.82 (40.89)

U2L3 41.09 39.72 37.16 36.13 36.13 33.37 31.03 (35.51)

U2L2 -23.72 -23.58 -27.00 -21.24 -21.24 -14.14** -13.31** (-20.82)

* The values in parenthesis represent the fixed-fixed condition at the bearings.

** N.B. These stresses were measured on the side on which bending moment caused tension stresses.

-55-

TABLE 7: COMPARISON OF STRESSES IN THE FLOOR SYSTEM

BY DIFFERENT MODELS

All Values in MPa

Measured Sim:ele Beam Continuous Beam Stresses

Member MPa Model 3B Model 3D Model 3A Model 3C

Floor Beam Center 49.37 50.82 51.00 44.06 44.50

Floor Beam Top 18.75 15.26 15.60 14.03 14.03

Stringer Center 39.99 36.19 36.19 19.94 20.00

Stringer Top 8.96 9.65 9.65 6.00 6.02

N.B. The location of the gage on the top of the stringer and on the

top of the floor beam was not exactly defined.

-56-

TABLE 8 STATIC STRESS CYCLES CAUSED BY THE PASSAGE OF ONE CAR*

All Values in MPa

Hanger Floor Beam Stringer Car Type UlLl Center Center Diagonal

Diesel Engine 22.41 50.20 35.85 41.44

1800 21.93 43.71 35.85

22.20 48.27 21.38 42.96

Steam Locomotive 1.31 2.34 21.72

200 8.00 14.41 22.82

12.48

16.82 40.54 29.51 37.16 Steam Locomotive 8.14 15.10 21.86

220 7.86

23.78 56.47 39.16 47.51

1.01 1.66 8.07 Steam Locomotive 11.92 21.24 7.72

500 24.27

12.14

Freight Car 15.44 29.65 24.55 32.89

3.93

16.13 30.61 27.58 35.58 Oil Car Empty 11.10

Oil Car Empty 4.41 8.34 7.52 9.65

2.97

Passenger'Car 5.31 11.58 10.00 9.38

10.00

* No dynamic Effect Included

-57-

TABLE 9: STATIC STRESS CYCLES AT RIVET HOLE (NET AREA)

CAUSED BY THE PASSAGE OF ONE CAR*

All Values in MPa

Hanger Floor Beam Stringer Car Type UlLl Center Center Diagonal

Diesel Engine 26.68 50.20 35.85 49.30

1800 21.93 43.71 35.85

24.06 48.27 21.38 51.02

Steam Locomotive 1.31 2.34 21.72

8.00 14.41 22.82

12.48

20.00 40.54 29.51 44.20

Steam Locomotive 8.14 15.10 !21. 86

7.86

28.34 46.47 39.16 56.54

1.01 1. 66 8.07

Steam Locomotive ll.97 21.24 7.72

24.27

12.14

15.44 29.65 24.55 32.88 Freight Car

3.93

16.13 30.61 27.58 35.58 Oil Car Full

11.10

Oil Car Empty 4.41 8.34 7.52 9.65

2.97

Passenger Car 5.31 11.58 10.00 9.38

10.00

* No Dynamic Effect Included

-58-

TABLE 10: ESTIMATED YEARLY NUMBER OF CARS AND

LOCOMOTIVES WHICH CROSSED THE BRIDGE

Before 1960 In 1980 In 2000

Oil Cars Full 9,000 27,000 54,000

Oil Cars Empty 9,000 27,000 54,000

Freight Cars 27,000 81,000 162,000

Passenger Cars 8,400 25,000 25,000

Steam Locomotives 4,272

Diesel Engine 4,000 7,400

-59-

TABLE 11: ESTIMATED TOTAL NUMBER OF CARS AND

LOCOMOTIVES WHICH CROSSED THE BRIDGE

Oil Cars Full 495,000 855,000 1,665,000

Oil Cars Empty 495,000 855,000 1,665,000

Freight Cars 1,485,000 2,575,000 5,005,000

Passenger Cars 462,000 796,000 1,296,000

Steam Locomotives 78,940 78,940 78,940 200

Steam Locomotives 51,260 51,260 51,260 220

Steam Locomotives 74,760 74 '760 74,760 500

Diesel Engine 61,400 175,100 1800

-60-

I

TABLE 12: ESTD1ATED CUMULATIVE FATIGUE DAMAGE IN 1981 AND 2000

All Values in MPa

8RMINER 8RRMS Total Number of Cycles

1981 2000 1981 2000 1981 2000

Diagonal VlL2 46.27 46.20 44.54 44.44 441,150 800,200

Hanger 17.44 16.41 5,768,100 Infinite

UlLl Life

Floor Beam 28.89 28.96 26.96 27.24 5,768,000 10,544,800 Center

Stringer 19.24 19.24 17.03 17.03 11,129,160 Infinite

Center Life

-61-

L

Fig. 1.1 General view of the Kohr Mog bridge

-62-

I 0\ w I

LO

Ul U2

Ll L2

U3 U4 U4 U3 U2 Ul

L3 L4 L4 L3 L2 Ll LO

9 @ 3.66m. = 32.92m

ELEVATION

fLAN AT BOTTOM LATERALS

Fig. 1.2 Plan and elevation of the bridge structure

HANGER

STRINGER

PHISICAL MODEL

three beam elements

FINITE ELEMENT MODEL

Fig. 1.3 Cross section of the bridge

-64-

1.22 2.63 3.32

~ (intersection with the bottom chord)

two isoparametric plate bending el~ents

H = .2 t=

H = .56m s

I R=0.20m

n

0.45

_f_

n

I ,A r r

I s

C.L.

1.07m

. 1.37m

1.97m

PHYSICAL MODEL

C.L.

I. ~I +I +A (H +H /2) 2 eq. s r r t s

ANALYTICAL MODEL Fig. 1.4 String~r-Tie-Rail connection

-65-

a

b

c

B

HANGERS, FLOOR BEAMS, STRINGERS

B

~ BOTTOM CHORDS L >

TOP CHORDS

Note: All Dimensions in inches 1 inch = 25.4mm

Hangers Floor Beams Stringers Bottom Cord Top Cord

4x3x3/8 5x3x3/8 6x4!rz_3/4 4x3~x7/16 3~x3~xl/2 3~x3~xl/2

n 314x3/8 11314x3/8 27 X 1/2 72 X 3/8 14 X 7/16 14 X 7/16

22 X 3/8 + 22x3/8 + (22xl/2) (15xl/2)

Fig. 1.5 Built Up Sections Used in the Bridge

-66-

rig. 2.1 View of gages on top chord and bottom chord members

-67-

L ------------~---------------------------~

Fig. 2.2 View of the gages on floor beam and diagonal

-68-

!ig. 2.3 eage installed on rail

-69-

- ~ .. r '""·

L __

~ig. 2.4 Instrumentation used

-70-

,Fig, 2.5 Typical analog traces recorded during passage of train

-71-

[o-o-o---o-o-c:l ·DIESEL

ENGINE CLASS 1800-1819

6 63

164 164 164 164 164 164

J ~ . :Ji FUEL OIL

0 0 o-o-o-o. 0 0 0 0 0

ENGINE CLASS 500

ENGINE CLASS 220

6.7 6.7 11.51 1.510. 12. 12.1 12.1

FUEL OIL ENGINE CLASS 200

0 0 0 0

L FREIGHT OJ l.Ot3 1.75 6.17 1.75 1.03 1.26 1.75 6.17 1.75 1.26 t ~ j t 1

+--r ;--J --t-1---t-J -tf---tt r12=2~12~2~------~1~2~2~1~2~2 _____ ~13~7 137 137 137

~(T-1J------------------~~ PASSENGER CAR

11.96 1.75 1.87 J t (

50 50 50 50

Fig. 2. 6 Geometric properties of locomotives and cars

-72-

I ....... w I

Cl2 C8

Cl

Fig. 2.7 Summary of gage locations on the bridge

C2 C3 C4 C7

C6 C9

\\

\\ "\ ~

\ \\

C5 Cll

... .,. ~

0.30 1.83

1

0.68

1.38

0.50 I

,..= ..... --...........

HANGER Ul'U

DIAGONAL UlL2

FLOOR BEAM CENTER

STRINGER CENTER

Fig. 3 .1 Response .of critical members to passage of test train

-74-

STRAIN

STRAIN

B

A

~

B'~ --- ~ ~

D

TYPICAL STRAIN TIME DIAGRAM

D

----- -·

~

~

F

'\%--- -- --· E

Fig. 3.2 Stress cycle counting by rainflow method

-75-

I '-l (j\

I

PLANE TRUSS ANALYSiS OF THE SUDAN BRIDGE BY SAP4

Fig. 4.1 Plane Truss two-dimensional finite element model

I -....) -....)

I

Fig. 4.2 Three-d' . l.IIlensional finite · element idealization

I "-1 00 I

· ~<OHR MOC BRIDGE. 3D-ANALYSIS WITH PLATE EL .. AND TIES

Fig. 4.3 Complete three-dimensional model with ties an~ stringers

U2 Ul

' ~ ........ ..::t ...-4 N N - ..::t ........ ...-4 M 1.1) ..::t ..::t 1.1) M ...-4

Ll"' 0 1.1) 0 0 1.1) 0 1.1) CHORD AXIAL

~1 N~ L4 L4

..::t co N co ..::t ...... co 0\ 0\ co co ...... M - ........ M M ...... ...-4 1.1) 0\ 0\ N ..0 . . . 0 0 ...-4 ...-4 - - ...-4 0 CHORD AXIAL

~I I~ L3 L4

...-4 ...-4 ...-4 N N M M ...... ("") ..0 0\ N Ll"' co ...-4 0 N ..::t ..0 0\ ...-4 M \D co . . . . . . . 0 0 0 0 ...-4 ...-4 ...-4 0 CHORD AXIAL

I

I ~ L2 L3

~ ~ CHORD AXIAL 1.1) 0 Ll"' - U4 U4 . . 0 . ...-4 - N N - ...-4 0 I I I I I I I I

0 0 0 0 0 0 0 ...-4 ..0 \D N co ..::t 0 ..0 N ...-4

...-4 M ..::t ..0 co 0\ ..... . . . . . . . 0 DIAGONAL AXIAL 0 0 0 0 0 0 ...-4 I

l7 ollii Ul L2

0 0 0 0 0 0 0 0 N ..0 ..0 N co ..::t 0 ..0 M ...-4

..... M ..::t ..0 co 0\ . . . . . . . . 0 0 0 0 0 0 0 0 /I- I DIAGONAL AXIAL

I v U2 L3

..... N ("") ..::t ..0 ......

...-4 N M ..::t 1.1) ..0 N .....

..... N M ..::t Ll"' ..0 N ...-4 . . . . . . N ...-4

0 0 0 0 0 0 . . I. I I I I I 0 0

HANGER AXIAL U2 L2

0 0 0 . ...-4 HANGER AXIAL

m:sJ/' Ul Ll Fig. 4.4 Influence ·lines for plane truss

-79-

"':! 1-'•

()Q . ~ -0.0003 ~-0.107 ·~ 0.155 ~ 0.157 -.05~ ~0.230 tl0.392 A515 .

028 ~ /IS 1.11

H ::s I-to -0.0005 -0.214 0.310 0.313 -1.0 0.460 0. 784 A~ ..... (:: ro ::s n

1-0.0007 H -o.321 H o.466 H o.47o ro -1.5f4-\ t-1 0.680 H1.175 t---+f546 tK 718 ..... 1-'• ::s ro

1-0.0017 H -o.427 H o.62o (I) H o.626 -2.0~9 l H o.926 1 p .569f 21025 ~ 71~·· I-to 0 fi

"d 1-0.0031 H -0.537 H o.782 H o.782 -2.oH H1.150H1.913Ho25 lK 7li! ..... I 1\l

00 ::s o ro I I-to

1-0.0032 r7 -0.631 \:i 0.910 H o.919 -1.5'r*-f H 1.381\ 11.891 H546 t:K: fi 18 ~ ro

I

~ ~ -0.267 1;---ll. 059 -Lob-r/ H 1.578 \--11.277 0.0125 0.237 ~ozs~s 0.962 \1 0.113 H -0.159 t\-0.134 -0.5P:-f1 \-1 0.804 \---10.636 515 c::

. ......

~ ~ tj tj

0 (') () g ()

~ ~ 5 ::c ::c 0

c~ 0

G) G) ~~ ~~ 5;:~ s~ s~ c::o c::o

N!Z:

~~ Ci:1 ~f< t-'~ ~iei Ciei 5;:iei 5;:~ w N

~ ~ ~ ~ ~ ~ !:1 ~ ~ ~

\0 II) ..;!" ...... II) ...... ,..., 00 ..;!" 0 0 .0 ...... ("'') 00 . . . . . N 0 0 0 0 0 ..;!" \0 ..;!"

FLOOR BEAM CT. SUB-MODEL 3d

\.0 ...... N ..;!" ..;!" ,..., 0 ("'') ...... ..;!" N ,..., 0'1 0 ...... N ..;!" 0 0 . . . . 0 0 0 0 ...... II)

FLOOR BEAM TOP SUB-MODEL 3a

..;!" 0 ("'') II) ("'') II) ("'') ...... ,..., ...... 0 \0 N ..;!" ...... 0 0 N ..;!" 00 \0 . ,...,

FLOOR BEAM TOP . . . . \0 0 0 0 0 0 C?

~ SUB-MODEL 3b

00 II) 0'1 II) 00 00 00 0'1 C? ...... N ("'') ("'') N \0 \0 0 ...... N ("'') 0'1 0 0 . . . . ("'') 0 0 0 0 0 II)

~ FLOOR BEAM TOP SUB-MODEL 3c

\0 ..;!" ...... ...... II) ,..., ..;!" N ...... ,..., 00 ("'') ...... N 0 ,..., 0 0 ...... ("'') ,..., ..;!" ..;!" . . . . . ,..., 0 0 0 0 0 ("'') ...... ("'')

~ FLOOR BEAM TOP

SUB-MODEL 3d Fig.4.6b Influence lines of floor system members with different models

-82-

LD 00

Ll ~ L2 ~ L3 ~ L4 ~ L4 ;;;; L3 ::::: L2 cr. Ll LO cg; 0 .-I .-I 0 U"\ CX) CX) . . . . . . . 0 0 0 0 0 .-I

0 ,..... \0 ·.-I \0 C"") \0 0 11"1 0 N 11"1 0 0 .-I .-I 0 \0 CX) . . . . . . . a a a a

:r 0'1 \0 C"") 0'1 CX) C"") ,..... N N \0 CX) 0 0 .-I N \0 11"1 \0 . . . . . N . 0 0 0 0 0 ~ .-I N·

A-:

"V

-::t CX) 11"1 CX) 11"1 0'1 C"") \0 .-I 0 .-I C"") C"") 11"1 0 0 .-I N .-I 0'1 . ,....... . . . . . N 0 0 0 0 0

~ .-I N

.e-

"J7"

STRINGER CT. SUB-MOD~L 3a

STRINGER CT. SUB-MODEL 3~1 4

STRINGER CT . SUB-MODEL 3c

STRINGER TOP SUB-MODEL 3a

STRINGER TOP SUB-MODEL 3};> 4

STRINGER TOP SUB-MODEL 3c

Fig. 4.6c Influence lines of floor system members with different models

-83-

"':: ..... 00 . ~ 0.0012 ~ -0.104 . ~ 0.151 ~0.155 -0.~ ~0.202 ~0.353 ~.469 . '-1 ~I /IS H ::l 0.0023 -0.208 0.302 0.309 -1. 30 0.414 o. 720 1-t! ..... H -- ~r 7IS c:! (1) ::l ()

1 o.oo32 H -0.313 H o.455 ~10.463 -l.r-o-\ H0.632 Hl.100 (1)

..... r--1 Cl\: 718 1-'· ::l (1)

1 0.0034 ~ -0.426 ~ 0.619 ~0.620 -2.109 \ /-10.856 1-----P. 486 I Ill

1-t! I ~i\: )1~ 0 li

rt -0.0024 -0.533 o. 776 . 0. 786 -2. 1.093 1. 787 .876 ::r' li ~'\./1~ I ro

(X) (1) ~ I I p. 0.0066 -0.464 0.679 0.941 -1. 60 1.296 1. 715 .464

~· \ t;l" 18 (1) ::l Ill A 0.192 I ~-0.054 t--1 0.834 -1.~ t-'• 0.074 Hl.344 \--11.207 0 H t;l" lS ::l Ill .....

8. ~ 0.623 I 0.130 ~~-0.165 ~ 0.100 -o.w \-i 0. 760 ~0.594 \f- --~I }8 (1) .....

~

~ ~ t:1 t:1 n g n n ~ ~ ::.t:l ::.t:l ::.t:l -0 0 0 0

(j) (j) G) (j) ~~ ~~ c~ s;:~ s~ C::tt:l c::o sg N:;d N:Z:

t::~ r;~ t-'~ t-'~ Z!~ t;~ ~~ ~~ w N~ ~ ~ ~ ~ ~ ~ ~ ~ ~

>'%j 1-'•

()Q . ~ I o.oo15 ft -0.103 II o.15o llo .154 -0.51~\ 110.203 . OJ

H ::I I 0.0030 H -0.207 H 0.308 H0.308 -1.03M ll0.410 fl0.410 t-4.983~ I-to ...... !:! (D

::I n I o.oo42 u -0.317 u 0.462 (D

..,:... 1-'· ::I (D

1 o.oo35 II -0.424 I I o.617 I 10.619 -2.01b. \ I _10.843 I __ 11. 493 (I)

I-to 0 li

rt I -o.oo25 u -0.533 u o. 777 I lo. 786 -2.011.1 I I h .on I h. 790 ::T li

I (D OJ (D

Y' ~ J 0.0044 U -0.462 LJ o.682 I lo. 943 -1.5fl0 I I 11.319 1 11.736

(D ::I (I)

::;· IJ 0.189 , 0.078 l -0.060

~ ......

I I 0.634 , 0.132 r0.168 ~0.097 -0.5

~~. . .. . . t-< 0

~ ~ t::l t::l (') (') (') g ~ ~

::X:: ::X:: ::X::

~~ t-<0 0 0 G"l G'1 G"l "'~ tEl ~El

8~ s~ s~ c:::o

;~ ~~ t;~ ~~ ~~ ~~ ~~ t-<~

w N~ ~ ~ ~ ~ ~ ~ ~ ~ ~

. \ 2.54mm

\

WITH PLATE ELEMENTS

/ ~

.. , ____ v

---FLOOR BEAM TO STRINGER CONNECTION CONTINUOUS

- -- FLGOR BEAM TO STRINGER CONNECTION HINGED

I I

~

/ DEFORMED SHAPE

MOMENT DIAGRAM

-Fig. 4.9a Deformation and moment diagram of beam No.8

under load No.lO. With plate elements

-86-

2.54mm

ELEMENTS

/ ____ /

--- FLOOR BEAM TO STRINGER CONNECTION CONTINUOUS

- - - - FLOOR BEAM TO STRINGER CONNECTION HINGED

2260 Nm v ~

/

/ /

/ / DEFORMED SHAPE

/ /

/ MOMENT DIAGRAM /

Fig. 4.9b Deformation and moment diagram of beam no.8 under load no. 10. Without plate elements

-87-

NUMBER OF TRAINS PER MONTH

600 (2000)

500

400

300

200 1905) (1960)

100

---------> --- --- _.,.. ---0,---------~------~--------~------~----~

0 25 50

FREIGHT TRAINS

----- PASSENGER TRAINS

75

Fig. 6.1 Assum~d and predicted bridge traffic

-88-

100 YEARS

60

50

40

30

20

10

Q

FREQUENCY (PERCENT)

0.0 1.0 2.0 3.0 (0.0)

FLOOR BEAM CENTER

Sr1-1INER =28. 89MPa

SrRMS =26.96MPa

4.0 5.0 6.0 (34.48)

7.0

STRESS RANGE NET SECTION

Fig. 6.2 Stress histogram at floor beam center

-89-

8.0 9.0 KSI (62.l)MPa

60

50

40

30

20

10

0

0.0 1.0 (0.0)

STRINGER CENTER

SrMINER=19.24 MPa

SrRMS =17.03 MPa

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 KSI (34 .48) (62 .1)MPa

STRESS RANGE NET SECTION

Fig. 6.3 Stress histogram at stringer center

-90-

60

50

40

30

20

10

0

FREQUENCY (PERCENT)

a.o 1.0 z.o 3.o 4.o (0.0)

DIAGONAL UlL2

SrMINER=46.27 MPa

SrRMS =44.54 MPa

5.0 6.0 (34.48)

7.0 8.0 9.0 KSI (62.2l)MPa

STRESS RANGE NET SECTION

Fig. 6.4 Stress histogram at diagonal u1L

2 -91-

li\

FREQUENCY KPERCENT)

60 -

50 -

40 -

30 -

20 - .

10 -!-

0 I

0.0 1.0 (0.0)

-

..i I I

2.0 3.0

HANGER UlLl

S =17.44 MPa rMINER

srRMs =16.41 MPa

-I I I

4.0 5.0 6.0 (34.48)

...... I I 1-

7.0 8.0 9.0 KSI (62.21) MPa

STRESS RANGE NET SECTION

Fig. 6.5 Stress histogram at hanger u1

L1

-92-

I \.()

w I

~ ( KSI) 1 KSI =6.895 MPa

NET SECT ION

-5

.2 I '

0

. '

'0 4!1~ Fa o

.,p. ' :: ~ A l •dJ

IPo QJ$ $

• •

FLOOR BEAM CENTER

c

D

--1981

Fig. 6.6 ESTIMATED FATIGUE LIFE AT FLOOR BEAM CENTER

-

N

I \0 .J:--1

~ ( K S I ) 1 KSI = 6. 895 MPa

NET SECTION 0

0 + 0 r!.P £DJ $0 liD 0 DO O 0 00 'l'tb------- X 8

• X 4 •& 0~~~~.

'b

.,p. ::~

0 Do 0

I I I I 7 7 2 7 2

Fig. 6.7

~o-,p 0

0 .l

A.J•Ifl • •

IP

.> 7 7 7 7 7 7

STRINGER CENTER

c

D .........

............. ........ ......... -....

...:..... .........

......... . .........

z ? ~7 ·1981

/ L. ·--

N

~ ( K S I ) l KS I = 6. 8 95 MP a

NET SECT ION D

DIAGONALUlL2 fb ££1 o--

Po D

A A. l •dJ .

• ¢1

c

D 198l 2000 --- --- -

N ,.__ _____ ,___ __ ,~-L--L-1'--L' _IL-J.I__l_l ------·-- I I I I

Fig. 6.8 ESTIMATED FATIGUE LIFE AT DIAGONAL UlL2

~6r<(KSI) 1 KSI =6.895 MPa

NET SECT ION 0 HANGER UlLl

0 + 0

110 CDJ tflo liDO 00 O 0 Do 'fldJ---- X • • X 4 • &

0 t!J-- 0 0 [h -----:_ ~ ~ ~-' co Q!)o-

Po (]

.,p. ! .. ~ A.J·~ e

•• • 0 Oo o QJ

c

D ...........

-- --

1981

N

Fig. 6.9 ESTIMATED FATIGUE LIFE AT HANGER UlLl

REFERENCES

1. Albrecht, P. and Yamada, K. RAPID CALCULATION OF STRESS INTENSITY FACTORS, Journal of St~uctural Division, ASCE, Vol. 103, No. ST2, Proceedings Paper 12742, February 1977.

2. AREA Specification, Chicago, 1980.

3. American Society for Testing Materials FATIGUE CRACK GROWTH UNDER SPECTRUM LOADS, STP 595, American Society of Testing Materials, Philadelphia, PA. 1976.

4. Barnoff, R. M. and Mooney, W. G. EFFECT OF FLOOR SYSTEM ON PONY TRUSS BRIDGE, ASCE Journal of the Structural Division, April 1960.

5. Bathe, K. J., Wilson, E. L. and Peterson, F. SAP IV - A STRUCTURAL ANALYSIS PROGRAM FOR STATIC AND DYNAMIC RESPONSE OF LINEAR SYSTEMS, EERC Report 73-11, University of California, Berkeley, CA, April 7, 1974.

6. Biggs, J. M. INTRODUCTION TO STRUCTURAL DYNAMICS, McGraw-Hill Book Co., Inc., New York, NY, 1964.

6a. Bleich, F. BUCKLING STRENGTH OF METAL STRUCTURES, McGraw Hill Book Company, New York, NY, 1952.

6b. Bowie, 0. L. ANALYSIS OF AN INFINITE PLATE CONTAINING RADIAL CRACKS ORIGINATING AT THE BOUNDARY OF AN INTERNAL CIRCULAR HOLE, Journal of Mathematics and Physics, 1956.

6c. Broek, D. ELEMENTARY ENGINEERING FRACTURE MECHANICS, Sijthoff and Noordhoff, The Netherlands, 1978.

7. Chu, K. H., Garg, V. K. and Dhar, C. L. RAILWAY-BRIDGE IMPACT SIMPLIFIED TRAIN AND BRIDGE MODEL, ASCE, Journal of the Structural Division, September 1979.

-97-

8. Clough, R. and Penzien, J. DYNAMICS OF STRUCTURES, McGraw Hill Book Company, Inc. New York, NY, 1970.

9. Csagoly, P., Bakht, B. and Ma, A. LATERAL BUCKLING OF PONY TRUSS BRIDGES, Hinistry of Transportation and Communication, Ontario Report RR 109, October 19 7 5 •

10. Edinger, J. A. I1~LUENCE OF INCREASED GROSS VEHICLE WEIGHT ON FATIGUE AND FRACTURE RESISTANCE OF STEEL BRIDGES, M.S. Thesis, Lehigh University, Bethlehem, PA, 1981.

11. Elkholy, I. A. S. ESTIMATED FATIGUE DAMAGE IN THE 336'-6" THROUGH TRUSS SWING SPAN OF THE RAINY RIVER BRIDGE, CN Rail, Montreal, Quebec, Canada, May 1981.

12. Fisher, J. W. BRIDGE FATIGUE GUIDE, DESIGN AND DETAILS, American Institute of Steel Construction, New York, NY, 1977.

13. Fisher, J. W. and Struik, J. H. A. GUIDE TO DESIGN CRITERIA FOR BOLTED AND RIVETED JOINTS, John Wiley & Sons, New York, NY, 1974.

14. Fisher, J. W., Albrecht, P. A., Yen, B. T., Klingerman, D. J. and McNamee, B. M.

FATIGUE STRENGTH OF STEEL BEA}IS, WITH TRANSVERSE STIFFENERS AND ATTACHMENTS, NCHRP Report 147, Highway Research Board, 1974.

15. Fisher, J. W., Frank, K. H., Hirt, M.A. and McNamee, B. M. EFFECT OF WELDMENTS ON THE FATIGUE STRENGTH OF STEEL BEAMS, NCHRP Report 102, Highway Research Board, 1970.

16. Hertzberg, R. W. DEFOR}~TION AND FRACTURE MECHANICS OF ENGINEERING MATERIALS, John \o/iley & Sons, New York, NY, 1976.

1 7 . . Ho 1 t , E . J . , J r . THE STABILITY OF BRIDGE CHORDS WITHOUT LATERAL BRACINGS, Department of Engineering Research, The Pennsylvania State University, Report No. 3, 1956.

-98-

17a. Hsu, M. A., McGee, W. M. and Aberson, EXTENDED STUDY OF FLAW GROWTH AT FASTENER HOLES, AFFDL, Report TR 77-83, Lockhead - Georgia Company, Marietta, GA, 1978.

17b. Irwin, G. R., Liebowitz, H. and Paris, P. D. A HISTORY OF FRACTURE MECHANICS, Engineering of Fracture Mechanics, Vol. 1, 1968.

17c. Irwin, G. R. FRACTURING AND FRACTURE MECHANICS, Theoretical and Applied Mechanics Reports 202, University of Illinois, Urbana, Illinois, October 1961.

18. Kobayashy, A. S. A SIMPLE PROCEDURE FOR ESTIMATING STRESS INTENSITY FACTOR IN A REGION OF HIGH STRESS GRADIENT, Proceedings of the Japan-U.S. Seminar, University of Tokyo Press, 1973.

19. Lewitt, C. W., Chesson, E., Jr. and Munse, W. H. RESTRAINT CHARACTERISTICS OF FLEXIBLE RIVETED AND BOLTED BEAM-TO-COLUMN CONNECTIONS, Department of Civil Engineering, University of Illinois., Urbana, Illinois, 1966.

20. Marcotte, D. J. on: FATIGUE OF RIVETED JOINTS, Unpublished M.S. Thesis, Lehigh University, Bethlehem, PA,' 1981.

21. Miner, M. A. CUMULATIVE DA}~GE IN FATIGUE, Journal of Applied Mechanics, September 1945.

2la. Palmgren, A. BERTSCHRIFT des VEREINES INGENIEURE, 58, 1924.

22. Paris, P. C. THE GROWTH OF FATIGUE CRACKS DUE TO VARIATIONS IN LOAD, Ph.D. Dissertation, Lehigh University, Bethlehem, PA, 1962.

23. Reemsnyder, H. S. FATIGUE LIFE EXTENSION OF RIVETED CONNECTIONS, ASCE, Journal of the Structural Division, December 1975.

23a. Roberts R., Barsom, J. M., Fisher, J. W., Rolfe, S. T. FRACTURE MECHANICS FOR BRIDGE DESIGN, Federal Highway Administration, U.S. Department of Transportation, P.O. 5-3-0209, July 1977.

-99-

24. Rolfe, S. T. and Barsom, J. M. FRACTURE AND FATIGUE CONTROL IN STRUCTURES, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1977.

25. Schilling, C. G., Klippstein, K. H., Barson, J. M. and Blake, G. T.

FATIGUE OF WELDED STEEL BRIDGE MEMBERS UNDER VARIABLE AMPLITUDE LOADINGS, NCHRP Report 188, Transportation Research Board, 1978.

26. Sih, G. C. HANDBOOK OF STRESS INTENSITY FACTORS, Institute of Fracture and Solid Mechanics, Lehigh University, 1973.

2 7 . S ih , G . C • MECHANICS OF FRACTURE I, METHODS OF ANALYSIS AND SOLUTIONS OF CRACK PROBLEMS, Woodhoff International, 1973.

28. Sweeney, R. A. P. and Elkholy, I. A. S. ESTIMATED FATIGUE DAMAGE IN THE 206'-3" THROUGH TRUSS SPAN OF THE OTTAWA RIVER BRIDGE STE. ANNE de BELLEVUE, P.Q. CN Rail, Montreal, Quebec, June 1980.

29. Sweeney, R. A. P. ESTIMATED FATIGUE DAMAGE IN THE 380' SWING SPAN AND THE 159'-3" FIXED SPANS OF THE FRASER RIVER BRIDGE, New Westminster, B. C., Final Report, CN Rail, Montreal, Quebec, Canada, April 1979.

30. Sweeney, R. A. P. and Elkholy, I. A. S. ESTIMATED FATIGUE DAMAGE IN THE 116'-10-1/2" THROUGH TRUSS SPAN OF THE ASSINBONE RIVER BRIDGE, Nattress, Manitoba, CN Rail, Montreal, Quebec,, August 1980.

31. Tada, H., Paris, P. C. and Irwin, G. R. THE STRESS ANALYSIS OF CRACKS HANDBOOK, Del Research Corporation, Hellertown, PA, 1973.

3la. Tada, M. and Irwin, G. R. K-VALUE ANALYSIS FOR CRACKS IN BRIDGE STRUCTURES, Fritz Engineering Laboratory, Report No. 399.1, Lehigh University, 1975.

32. Velestos, A. S. and Huang, T. ANALYSIS OF DYNAMIC RESPONSE OF HIGHWAY BRIDGES, ASCE, Journal of the Engineering Mechanics Division, Vol. 96, No. EMS, Proceedings Paper 7591, October 1970.

-100-

33. Wiriyachai, A., Chu, K. H., Garg, V. K. FATIGUE LIFE OF CRITICAL MEMBERS IN A RAILWAY TRUSS BRIDGE, Association of American Railroads, Report R.742, Chicago, Illinois, February 1981.

34. Wiriyachai, A., Chu, K. H. and Garg, V. K. RAILWAY BRIDGE IMPACTS RESULTING FROM FLAT WHEEL IRREGULARITIES, Association of American Railroads, Report R-475, Chicago, Illinois, March 1981.

35. Woodward, H. M. and Fisher, J. W . . PREDICTIONS OF FATIGUE FAILURE IN STEEL BRIDGES, Fritz Engineering Laboratory Report No. 381-12(80), Lehigh University, Bethlehem, PA, August 1980.

36. WEI• R. P. FRACTURE MECHANICS APPROACH TO FATIGUE ANALYSIS IN DESIGN, Journal of Engineering Materials and Technology, Vol. 100, April 1978.

-101-

APPENDIX Al: STATIC STRESS VERSUS TIME RESPONSE CAUSED

BY THE PASSAGE OF THE TEST TRAIN

-102-

....... ,.._ (/)

:::=:: -(/) (/)

w 0:::: ;-(/)

0 0 0 0

\!)

0 0 0 0

11')

0 0 0 0

"'Of"

0 0 0 0

!"")

0 0 0 0

N

0 0 0 0

0 0 0 0

0

o.

lKSI=6 .895 MPa

8. i 2.

TIME

Fig. Al Bottom Chord 1414 - Test Train

-103-

1 6.

-,..._ U)

~ -U) U)

w a:: 1-(f)

0 0 0 0

0

0 0 0 0

0 0 0 0

(\J

l

0 0 0 0

I"? J

0 0 0 0

..,. l

0 0 0 0

1./")

l

0 0 0 0

c.D

lKSI=6 .895 MPa

T l ME Fig. A2 Top Chord u4u

4 - Test Train

-104-

-....-U)

~ -U) U)

' w a:: 1--U)

I· #.

0 0 0 0

(\J

0 0 0 0

0

0 0 0 0

00

0 0 0 0

<.D

0 0 0 0

-.:-

0 0 0 0

(\J

0 0 0 0

0

l 1KSI=6. 895 MPa

i "l

I I I

1

i

l

o. 8. 24- 32.

TIME Fig. A3 Diagonal u

11

2 - Test Train

-105-

0 0 0 CCI

r"'? -, I

i

0 lKSI=6.895 M1ta

0 0 0

r"'? ~ I i

0 ! I

0 i 0 i (\l I

I (\l 1 -- !

U) 0 \

:X:: 0 \ 0

~ I U)

l U) I w I 0:: ! i 1-

I (/) 0

\ 0 0 c.o

I \ -, I I !

I l

0

\ 0 I 0

(\l

I

l

0 0 0 0

o. B. ·, 6. 24- 32.

TIME Fig. A4 Hanger u

11

1 - Test Train

-106-

-...-U)

::::::::

U) U)

LJJ a::: 1-U)

0 0 0 0

(\.!

0 0 0 0

"l l :

0-i i !

i g i 0 i 0 I

I ool

0 0 0 0

' tO l

I

i i 0 ' 0

0 0

~

~ 0 0 0 0

(\.!

l

0 0 0 0

0

1KSI=6. 895 Mlt '

\

0 .. B.

Fig. AS Floor

I I

\

i 6. 24. 32.

T·1 ME Beam Center - Test Train

-107-

-< ,. --:L -, . . ..,,. (./')

L.L: 0:: 1--(./.

0 0 0 0

oo-, ..:-

0 0 0 0

0 ~-

0 0 0 0

(\; -:"')

0 0 0 0

..;-· -~"\.:

0 0 0 0

<.0 -

0 0 0 0

0:

0 0 0 0

0

-

o.

r""1 ,.... ,..... ,....

...... ,_. L...l ~

I

I I I . ,. I 0 · 24 .

T}ME Fig. A6 Stringer Center - Test Train

-108-

I

32.

APPENDIX A2: STATIC STRESS VERSUS TIME RESPONSE OF CRITICAL MEMBERS

CAUSED BY THE DIFFERENT CARS AND LOCOMOTIVES

-109-

-<. CL :L -U) U)

~ 0:::: ,_.. (/)

0 0 0 0

0 (.;>

0 0 0 0

0 II')

0 0 0 0

0 -.:-

0 0 0 0

0 :"'">

0 0 0 0

0 0.)

0 0 0 0

0

0 0 0 0

0

a. 8. i 6. 24- 32.

T l ME

Fig. A7 Floor Beam Center - One Diesel Locomotive and One Oil Car

-110-

-< a.. ::L

:/)

(./)

w ~ ....... V':

0 0 0 0

00 v

0 0 0 0

0 v

0 0 0 0

N-:"')

0 0 0 0

-.:-_ ':'\!

0 0 0 0

!.0-

0 0 0 0

0:: -

0 0 0 0

0

~

I

8.

,...

~ -

I

T 1 ME

T

24. I

32.

Fig. AS Stringer Center - One Diesel Locomotive and One Oil Car

-111-

-<: c.. !:

C/) (/)

LlJ 0:: 1--

c.::

0 0 0 0

,.._ :\!

0 0 0 0

C\J -:"\!

0 0 0 0

,.._

0 0 0 0

C\J

0 0 0 0

:-._

0 0 0 0

0 0 0 0

o. i 6. 24. 32.

T j ME

Fig. A8 Stringer Center - One Diesel Locomotive and One Oil Car

-112-

-< Cl.. 2: -r..r; (./)

LLJ a::: 1--

c..r:

0 0 0 0

00 "<:"

0 0 0 0

0 "<:"

0 0 0 0

N :"")

0 0 0 0

"<:" ·~

0 0 0 0

r..O

0 0 0 0

0

o. 8. 24. 32.

T 1 ME

Fig. AlO Diagonal u1L2 - One Diesel Locomotive and One Oil Car

-113-

-< CL E -(./") C/) LL.j

cr. 1-cr.

0 0 0 0

0 <..?

0 0 0 0

0 \.1")

0 0 0 0

0 v

0 0 0 0

0 :"')

0 0 0 0

0 ~

0 0 0 0

0

0 0 0 0

0

TIME

-Fig. All Floor Beam Center - One Steam Locomotive 200

-114-

0 0 o-(')

00 -.:-

0 0 0 0

0 v

0 0 0 0

':'\! :""> -<

0.. 0 :L 0 0 0

v;, (/) -.:-w ·::".!

0:: t-(f: 0

0 0 C?

lO

I 0 0 0 (')

l ,..,.. "'•

0 0 0 0

0

o. ;-'

I 0 · 24. 32.

TIME

Fig. A12 Stringer Center - One Steam Locomotive 200

-115-

...-.. <: Cl. :L. .......,

c.r; (f) l..J..j

cr. 1-if.

0 0 0 0

<..0 ('\l

0 0 0 0

;.,~

0 0 0 0

\.0

0 0 0 0

0 0 0 0

<..0

0 0 0 a

0 0 0 0

o. 24. 32.

TIME

Fig. Al3 - One Steam Locomotive 200

-116-

-< 0.. L. -' ' Cf)

/ Cf)

w . •,/

-~---- a::: I-U)

0 0 0 0

00 v

0 0 0 0

0 v

0 0 0 0

(\!

~

0 0 0 0

'V (\!

0 0 0 0

(.£)

0 0 0 0

a;

0 0 0 0

0·-t--~-------.-----------.-----------.--~~----~~---------

o. I 6 . 24. . 32.

TIME

Fig. Al4 Diagonal u1L

2 - One Steam Locomotive 200

-117-

-< CL :L -CJ) CJ)

w 0::. 1-

cr.

0 0 0 0

00 -.::-

0 0 0 0

0 -.:-

0 0 0 0

N !"'?

0 0 0 0

-.:-N

0 0 0 0

t.D

0 0 0 0

0:

0 0 0 o.

0

Q. 24. 32.

TIME

Fig. AlS Floor Beam Center - One Steam Locomotive 220

-118-

-< Q.. ~

'-...,, '..!)

w 0:::: 1--u:

0 0 0 0

0 :"'")

0 0 0 0

Lf') (\J

0 0 0 0

0 (\J

0 0 0 0

Lf')

0 0 0 0

0

0 0 0 0

0 0 0 0

0

o. 8. j 6: 24. 32.

TiME Fig. A16 Stringer Center - One Steam Locomotive 220

-119-

-<. 0.. r..

(./., U)

w 0:: I-Cf:

0 0 0 0

0 (\!

0 0 0 0

(.!)

0 0 0 0

N

0 0 0 0

X)

0 0 0 0

~

0 0 0 0

0

0 0 0 0

o. 8. i 6. 24. 32.

TIME

Fig. Al7 Hanger u1

11

- One Steam Locomotive 220

-120-

........ < CL. L: -(/) (/) LJ...j

0::: I-cr.

0 0 0 0

00 v

0 0 0 0

0 v

0 0 0 0

C"~ :"?

0 0 0 0

v N

0 0 0 0

!.0

0 0 0 0

0 0 0 0

0

TIME Fig. Al8 Diagonal u11

2 - One Steam Locomotive 220

-121-

-< c... L: -Cf) (./)

w 0::: -U)

0 0 0 0

0 ~

0 0 0 0

0 L.i

0 0 0 0

0 ~

0 0 0 0

0 :"")

0 0 0 0

0 ('\!

0 0 0 0

0

0 0 0 0

0

TIME

Fig. Al9 Floor Beam Center - One Steam Locomotive 500

-122-

.......... <. 0.. L ........,

(f) (/)

w a::: 1--(f;

0 0 0 0

00 ...,..

0 0 0 0

0 ...,..

0 0 0 0

C"\: !"'?

0 0 0 0

...,.. C"\:

0 0 0 0

(,{)

0 0 0 0

0:

0 0 0 0

0

o. I 6·. 24- 32.

TIME Fig. A20 Stringer Center - One Steam Locomotive 500

-123-

0 0 0 0

(£)

"" 0

~ 0 0 0

0.:

r 1 0 0 0

( I 0

i.O

I - I < ~ c... 0

I L 0 ........ 0 0

Cf: c.;; I

. lJ.j

~~ 0::::. f-u; 0

0 0 0

<..:) I

I

i

0 0 0

I 0

r-·

0 \( 0 0 \1 0

"<r

o. 8. 16. 24. 32-.

T- E . . l M

Fig. A21 Hanger u111

- One Steam Locomotive 500

-124-

-< Cl. r.

U"; (./)

LL..i a:: 1--

cr.

0 0 0 0

00 -:::-

0 0 0 0

0 ~

0 0 0 0

(\; :"")

0 0 0 0

~

0J

0 0 0 0

<.0

0 0 0 0

0::·

0 0 0 0

0

r J

I I

\ I

I

\

o. 16. 24. - 32.

TIME

Fig. A22 Diagonal U1L2 - One Stearn Locomotive 500

-125-

-<. o_ L -U') (/')

w 0:: 1--c.r;

0 0 0 0

00 -.:-

0 0 0 0

0 ~

0 0 0 0

(\J

!"'?

0 0 0 0

-.:-(\J

0 0 0 0

1.0

0 0 0 0

00

0 0 0 0

0

I ~ I I I I I I

I I.

I tl I II l u u

o. 2- 4. 6. - 8.

T 1 ~E

Fig. A23 Floor Beam Center- Ten Freight Cars

-126-

-<. Q._

.:r: ........

:.I) (./)

w a:: 1--

c.r.

0 0 0 0

0 !"")

0 0 0 0

tJ":)

N

0 0 0 0

0 N

0 0 0 0

tJ":)

0 0 0 0

0

0 0 0 0

0 0 0 0

0

-

-

-

-

-

-

r n

r~ ~ l ~~ r u

I I '

I

I

I

. I I I

I \ I l I

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C). 2. 4 . 6. 8.

T1ME

Fig. A24 Stringer Center - Ten Freight Cars

-127-

......... < Cl.. L

(./) CJ)

w 0:::: 1--

v:

0 0 0 0

N

0 0 0 0

,..._

0 0 0 0

!"")

0 0 0 0

Oi

0 0 0 0

u;

0 0 0 0

0 0 0 0

!"?

~ ~ A

I 1\

I I I

I

8. 2. 6.

TIME

Fig. A25 Hanger u111

- Ten Freight Cars

-128-

-< 0.. ~ ._,

CJ)

CJ: LL.1 0:: ......... r..r.

0 0 0 0

co v

0 0 0 0

0 ~-

0 0 0 0

·:'..: :-')

0 0 0 0

v •:"\!

0 0 0 0

<.0

0 0 0 0

0:::

0 0 0 0

0

Q.

vw~

'

2 . 4. 6.

T1ME

Fig. A26 Diagonal u1L2 - Ten Freigh~ Cars

-129-

-< C:l._

k: -CJ"; U)

w 0:::. 1--

c.r:

0 0 0 0

c:x:l ~

0 0 0 0

0 ~

0 u ~ L 0 0 0

N n

0 0 0 0

~ (\.!

0 0 0 0

<.0

I 0 0

J 0 0

a:: i !,;

0 0 0 0

0

o. 2. 6.

TiME

Fig. A27 Floor Beam Center- Ten Oil Tanks-Full

-130-

8-

-<. c.. !:

U) (/)

w cr. 1--CF.

0 0 0 0

0~ ~

0 0 0 0

If") ('\.)

0 0 0 0

0 C\1

0 0 0 0 . If")

0 0 0 0

0

0 0 0 0

\..:-.

0 0 0 C?

c

-

-

-

-

-

-~-~-,.., V•

n~ :

II

:~ ~

I I

' I

I

i I

r ~~ ~~ ~ . I I I

I

I I ; I

I I '

I I I

~ ! u

I I

I I I I

I j I ; I i

I ' I

I ! I I l I I

2. 4. 6.

TIME

Fig. A28 Stringer Center - Ten Oil Tanks Full

-131-

I

8.

....... < Q.._ :L -(/) (/)

w ~ 1--

~

0 0 0 0

0 0 0 0

0 0 0 0

!""')

0 0 0 0

IJ)

0 0 0 0

Lf)

0 0 0 0

0 0 0 0

n '

o. 2 .

Fig. A29

4 .

TIME

6.

I v

Hanger U L - Ten Oil Tanks Full 1 1

-132-

,....., < CL E ..._,

C/) U) LJ..,; 0::: 1--

cr.

0 0 0 0

00 -.:-

0 0 0 0

0 -.:-

0 0 0 0

N :"")

0 0 0 0

-.:-C\!

0 0 0 0

c.o

0 0 0 0

0 0 0 0

0 \

Q. 2. 41 • 6.

TIME Fig. A30 Diagonal u112 - Ten Oil Tanks full

-133-

.-.. < Cl.. :L

(f)

(/')

w 0:: 1--r..J;

0 0 0 0

0 0 0 0

0

0 0 0 0

0:::

0 0 0 0

t;)

0 0 0 0

-.:"

0 0 0 0

':"'\!

0 0 0 0

0

L~ ~ I i

.~

~ .I v

o. 2. 6. g.

-TiME

Fig. A31 Floor Beam Center - Ten Oil Tanks Empty

-134-

0 0 0 ('\!

~l I ;

0 :

g i 1 ~H~ 11 11 n ~~ ~~ lH~ ~ -l I i i I! I I ! I I I 1

11 :II lll\11 I

g i I \' ;

1

i I i i I i1

' 1 I 1~ 1' I i 1i I :

1

: I ! 0

i l ! ! II l 'I 'II ; !I l

~ ro ~'!I; :.1\1 !\It~~!!~~~ 111 ~ : i I: ',II :! \; II ! . I i 'I I :s 25 ·I~:~! l\l 1

1 1! ll. 1! ll i ~ ! I I ' l !I ! I I I

~ 1 I I I I . I i ! I i ~ gl \ l I I I I : I II

... ~ I I l ! I I I I i I ' I ! I i i I! ; I ! ; ! I I I ; I I

g : I : ! \ I I I I i I

N l ! ! ! i ll I I I I g ; I \ i I I I I ~ ' ! II ' I i : I i ' I i

I I I

o. 2 . 4. 6. 8.

·riME

Fig. A32 Stringer Center - Ten Oil Tanks Empty

-135-

·-< c... ::L.

Cf) Cf)

LLJ a::: ,....... (/')

0 0 0 0

0 0 0 0

~

0 0 0 0

!""')

0 0 0 0

N

0 0 0 0

0 0 0 0

0

0 0 0 0

~ I ) (

l v

o. 2. 4 . .- 8. 0·

TiME

Fig. A33 Hanger U1L1 - Ten Oil Tanks Empty

-136-

-< Q._

:L: ..._..

(./) (./)

l.J..1 a::: f-c.r.

0 0 0 0

0 0 0 0

0

0 0 0 0

00

0 0 0 0

lD

0 0 0 0

"o::t"

0 0 0 0

0 0 0 0

0

vvwwv

\

\

o. 2. 4. 6.

liME

Fig. A34 Diagonal u1

12 - Ten Oil Tanks Empty

-137-

...., < a.. !:

(f) (/)

w 0:: 1--

c.r:

0 0 0 0

(\J

0 0 0 0

0

0 0 0 0

00

0 0 0 0

r.D

0 0 0 0

~

0 0 0 0

l

I I I

11

d.

~ I

8.

T. Ml= . l ~

i 6.

Fig. A35 Floor Beam Center - Ten Passenger Cars

-138-

0 0 0 -...,.

C\J-,

! ' I

o I 0 I

g I ~l o I 0 1

~ I ~..,

:: . I . I Cf) 0

~ 0

- 0 C\J I I I I

Cf) '1 Cf) -w 0::: 1-­

I I I :

Cf) 0 0 0 co

0 0 0 ...,.

0 0 0 0

0

. I ! I I . . I

-i' I I i I ! \ ,. , I ! I I I ~ i

ll I

o.

I

I I \I

II ,I I

I I' I I I I

I I

I

I I ! I I

\I II I l II ! i

i I , I II I I' I I I

I I I i I;

I i I I I I

I l I\

I I I I I I,

I I :I

II II

I \

I!

I I

I

I I I i

I \ I I

I

4 .

I I I'

I I , I I I

i

I I. i

l i \ II I \ I ! I I 1 I

II I I I I \! II I. I I i I. II I

I

I I I I I I :I ll

I

l I . I II \I

II 'I I

I \.\

I

I I I

I I I I I I

I\ II I i ! !

I I i . I ; ! ; I i

I I'

I I I I I I I

i I I

I I

I I I I I I I I

l I

I l I I I I

t I 1 I I -8 .. i 2.

T 1 ME Fig. A36 Stringer Center - Ten Passenger Cars

-139-

I

16.

......... < c... ~

(,./) (./)

w 0::::. ...... ~

0 0 0 0

0 0 0 0

o;

0 0 0 0

r-.._

0 0 0 0

If':·

0 0 0 0

~

0 0 0 0

0 0 0 0

-

-

-

.

-

~ ! ~

I

i I -

-I

I I I

I ~

I I I 1-I I I II ~ I

~ II ! ! I

~ v I

! ' I r' I 1/ f I

I I I I o. 4. . i 6.

TiME

Fig. A37 Hanger u1L1

- Ten Passenger Cars

-140-

-< Q.. ~

,,.... v. U)

w 0::: 1--

cr:

0 0 0 0

(\J

0 0 0 0

0

0 0 0 0

co

0 0 0 0

t.D

0 0 0 0

-.:-

0 0 0 0

("J

0 0 0 0

0

J

o.

r r I

I I (

~

d. 8.

~ I

\ \

T"ME . l .

i

\

\ \ I I I

\

I 2 ·

Fig. A38 Diagonal u112

- Ten Passenger Cars

-141-

.I 0 ·

•• VITA

The author was born in Naples, Italy, on January 31, 1957. He

is the second of three offspring of Mr. and Mrs. Luigi DeLuca.

The author received his high school education at the

"Classical Lyceum Umberto I" in Naples. He graduated with honors from

the Technical University of Naples, Italy in January 1980.

In September 1980 the author entered Lehigh University on a half­

time research assistantship in the Fatigue and Fracture Division at

Fritz Engineering Research Laboratory. Since that time he has worked

on several projects. These projects include the "Blue Route Bridge

Defects and Structural Response," "Improved Quality Control Procedures

for Bridge Steels," and "Field Studies of Sudan Railroad Bridges." The

exposure to structural fatigue in general has provided the basis for

his thesis.

The author was awarded a full "Fulbright Scholarship" from the

Institute of International Education and an "Italian Loan fund Cor-­

poration" loan in 1980.

-142-