..

•

)' Technical ~eport Documentation Page / I. Report No. 2. Government At:cession No. 3. Recipient's Cotolog No.

DOT-FH-11-8198.6

Title and Subtitle:

FATIGUE OF CURVED STEEL BRIDGE ELEMENTS

EFFECT OF INTERNAL DIAPHRAGMS ON FATIGUE STRENGTH OF CURVED BOX GIRDERS

5. Report Dote

February 1978 6. Performing Orgoni zotion Code

1-=:---:--:-7"7-----------"-"7--:---~----------! 8. P edorming Orgoni zotion Report No. 7. Author1sl . · 1. . .i Fritz Engineering Laboratory

D. Abrah&n,,.B .• T. Yen, J. Hartley Daniels 9. Performing Organization Name and Address

Fritz Engineering Laboratory, Bldg. No. 13 Lehigh University

Report No. 398~6

10. Work Unit No. (TRAIS)

DOT-FH-11-8198 11. Contract or Grant No.

Bethlehem, PA 18015 · , 13. Type of Report ond Period Cov.,red

~------------------------------------~------~ 12. Sponsoring Agency Name and Address

U.S. Department Federal Highway Washington, DC

15. Supplementary Notes

16. Abstract

of Transportation Administration 20590

Interim Sept. 1976·-; Feb. 1978

14. Sponsoring Agenc:y Code

Research on the fatigue behavior of horizontally curved, steel bridge elements is underway at Lehigh University under the sponsorship of the Federal Highway Administration (FHWA) of the U.S. Department of Transportation. The investigation is centered on the effect of welded details on curved girder fatigue strength. Fatigue tests of five full-scale curved plate girder assemblies and three full-scale curved box girders are part of the investigation.

This report examines analytically the effects of spacing of rigid interior diaphragms on the stresses and deflections of .curved box girders. Available computer programs are employed and existing results are utilized with little emphasis on the procedure of computation. The objecti~e is to assess the quali­tative relationship between stresses and the coupling influence of diaphragm spacing and curvature, so as to gain insigh~ to the fatigue behavior of box girders.

Results of the analyses indicate that decreasing of diaphragm spacing effectively controls the torsional stresses. The ratio of diaphragm spacing to radius of curved box girders is introduced as a parameter for monitoring stress ranges. It appears that the relationship between stress range and the spacing­to-radius ratio is practically linear for a given geometry of curved box girder. More,study is recommended to explore further this ratio as a parameter for con-

~ trolling the ~agnitude of st~ess range.

17, Key Words

Bridges (structures), box girders, design, fatigue, girder bridges, struc­tural engineering, testing, torsion, welding

18. Distribution Statement

Document is available to the public through the National Technical Information Service, Springfield, Virginia 22151

19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price

Unclassified Unclassified

Form DOT F 1700.7 (B-72l Reproduction of completed poge authorized

I I I

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i I ,. !

I

I

I I

FATIGUE OF CURVED STEEL BRIDGE ELEMENTS

EFFECT OF INTERNAL DIAPHRAGMS ON FATIGUE STRENGTH OF CURVED BOX GIRDERS

Submitted by

D. Abraham

B. T. Yen

J. Hartley Daniels

miTl ENGINEERING LABORATORY L16RM'l..

"Prepared for the Department of Transportation, Federal Highway Administration under Contract Number DOT-FH-11-8198.

The opinions, findings and conclusions expressed in this publication are those of the authors and not necessarily those of the Federal Highway Administration."

LEHIGH UNIVERSITY

Fritz Engineering Laboratory

Bethlehem, Pennsylvania

February 1918

Fritz Engineering Laboratory Report No. 398.6

ABSTRACT

Research on the fatigue behavior of horizontally curved, steel

bridge elements is underway at Lehigh University under the sponsorship

of the Federal Highway Administration (FHWA) of the U.S. Department of

Transportation. The investigation is centered on the effect of welded

details on curved girder fatigue strength. Fatigue tests of five full­

scale curved plate girder assemblies and three full-scale curved box

girders are part of the investigation.

This report examines analytically the effects of spacing of rigid

interior diaphragms on the stresses and deflections of curved box

girders. Available computer programs are employed and existing results

are utilized with little emphasis on the procedure of computation. The

objective is to assess the qualitative relationship between stresses and

the coupling influence of diaphragm spacing and curvature, so as to gain

insight to the fatigue behavior of box girders.

Results of the analyses indicate that decreasing of diaphragm

spacing effectively controls the torsional stresses. The ratio of

diaphragm spacing to radius of curved box girders is introduced as a

parameter for monitoring stress ranges. It appears that the relation­

ship between stress range and the spacing-to-radius ratio is practically

linear for a given geometry of curved box girder. More study is recom­

mended to explore further this ratio as a parameter for controlling the

magnitude of stress range.

TABLE OF CONTENTS

ABSTRACT

LIST OF TABLES

LIST OF FIGURES

LIST OF ABBREVIATIONS AND SYMBOLS

1. INTRODUCTION

1. 1 Background

1.2 Objectives and Scope

1.3 Research Approach

2. INFLUENCE OF DIAPHRAGM SPACING

2.1 Selection of Curved Box Girder Details

2.2 Results of the Analyses

3. INFLUENCE OF DIAPHRAGM SPACING AND RADIUS

4. FATIGUE CONSIDERATIONS

5 • CONCLUSIONS

6. TABLES

7. FIGURES

8. APPEND ICES

APPENDIX A: STATEMENT OF WORK

APPENDIX B: LIST OF REPORTS PRODUCED UNDER DOT-FH-11. 8198

9. ACKNOWLEDGMENTS

10. REFERENCES

1

1

2

2

3

3

4

7

9

11

12

15

LIST OF TABLES

Table 1 Results of Survey of Six Box Girder Bridges and Values Selected for Analysis

Table 2 Comparison of Deflections

Table 3 Comparison of Distortions

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Fig. 12

Fig. 13

Fig. 14

Fig. 15

Fig. 16

Fig. 17

Fig. 18

Fig. 19

Fig. 20

Fig. 21

Fig. 22

Fig. 23

LIST OF FIGURES

Schematic of Horizontally Curved Box Girder

Definition of Parameters for Existing Curved Box Girder Bridges (see Table 1)

Dimensions of Curved Box Girder Used in the Analysis (Fig. 20 of Ref. 14)

Finite Element Discretization for Analysis of Curved Box Girder by SAP IV

Torsional Loading Components

Concentrated Loads and NQ~ber of Diaphragms (ND)

Total Longitudinal Normal Stress at Bottom of Inner Web

Total Longitudinal Normal Stresses at Bottom of Outer Web

Stress Gradient across the Bottom Flange

Vertical Deflection of Inner Web

Distortion of Cross Section (Angle y is Defined in Fig. 4)

Transverse Plate Bending Stresses at Bottom of Inner Web

Transverse Plate Bending Stresses at Bottom of Outer Web

Total Shear Stress at Bottom of Inner Web

Total Longitudinal Normal Stress at Bottom of Inner Web

Total Longitudinal Normal Stress at Bottom of Outer Web

Distortion Plus Warping Stress at Bottom of Inner Web

Distortion Plus Warping Stress at Bottom of Outer Web

Dimensions of Curved Box Girder for Analysis by CURDI

Finite Strip Discretization for Analysis of Curved Box Girder by CURDI

Stress Range, S , vs. Diaphragm Spacing to Radius, a/R, r Relationship

Box Girder Geometries from Ref. 12

Influence of Number of Diaphragms on Maximum Distortion Stress

LIST OF FIGURES (Continued)

Fig. 24 Influence of NQmber of Diaphragms on Maximum Distortion Stress

Fig. 25 Stress Range, s r' vs. Diaphragm Spacing to Radius, a/R, Relationship

Fig. 26 Stress Range, s r' vs. Diaphragm Spacing to Radius, a/R, Relationship

Fig. 27 Stress Range, s r' vs. Diaphragm Spacing to Radius, a/R, Relationship

Fig. 28 Stress Range, s r' vs. Diaphragm Spacing to Radius, a/R, Relationship

Fig. 29 Stress Range, s r' vs. Diaphragm Spacing to Radius, a/R, Relationship

Fig. 30 Stress Range, s r' vs. Diaphragm Spacing to Radius, a/R, Relationship

Fig. 31 Stress Range, s r' vs. Cycle Life, N, Relationship

Fig. 32 Stress Range, s r' vs. Diaphragm Spacing to Radius, a/R, Relationship

,,

LIST OF ABBREVIATIONS AND SYMBOLS

a = internal diaphragm spacing along centerline of curved box girder

b width of bottom flange of curved box girder center to center of webs

d = distance from center of top flange to center of bottom flange

e = distance top flange overhangs from the center of the web

tb = bottom flange thickness

tt top flange thickness

t = web thickness w

L = span length along centerline

M = bending moment or torque

ND = number of internal diaphragms

p == concentrated load

R = radius of curved box girder

X = coordinate axis

y = coordinate axis

z == coordinate axis

of curved box girder

1. INTRODUCTION

1.1 Background

The research reported herein is part of a multiphase investiga­

tion of curved girder fatigue at Lehigh University entitled "Fatigue of

Curved Steel Bridge Elements", sponsored by the FHWA. This investiga­

tion is broken down into five tasks as shown in Appendix A. In Task 1

the analysis and design of large scale horizontally curved plate and

box girder test assemblies are performed, including bridge classifica­

tion and selection of welded details for study. Task 2 concerns special

studies on stress range gradients, heat curving residual stresses, web

slenderness ratios, and diaphragm spacing as related to fatigue perfor­

mance. Fatigue tests of large scale plate and box girder test assem­

blies are performed in Task 3. Ultimate strength tests of the modified

test assemblies are performed in Task 4. Design recommendations for

fatigue are prepared in Task 5, based on the work of Tasks 1, 2 and 3.

Horizontally curved steel box girders with a composite concrete

deck are often used as bridge members on highways, particularly at the

entrances and exists of modern expressways. The ability of the closed

box shape to distribute vehicular loads in the transverse direction of

the bridge is the main advantage of box girders. Because of the curva­

ture, loads on a curved bridge generate torsional stresses in addition

to the flexural stresses in the bridge. The box-shaped cross section

enables distribution of the torsional stresses among its component parts

better than the distribution among parallel plate girders of deck-and­

girder type bridges.

Torsional stresses include the St. Venant shearing stresses in

the plane of a cross section and the warping normal and warping shearing

stresses resulting from warping of the plane cross section. In addition,

distortional normal and shearing stresses arise due to deformation of

the cross section. There are a number of methods for evaluating these

stresses, such as the theory of thin-walled elastic beams,(l, 2 , 3) the

theory of folded plates, (4) the method of beam-an-elastic-foundation

(BEF) analogy, (5 , 6) and various numerical procedures including the method

of finite element. ( 7,B) Numerous summaries have been made on the

application of these methods to the analysis and design of steel or

1 b . d (9,10,11) composite stee -concrete ox g1r ers.

The results of these studies have clearly indicated the necessity

for rigid transverse interior diaphragms in box girders. Figure 1 shows

a schematic view of a horizontally curved box girder of radius R. Suffi­

ciently rigid diaphragms at the supports and between the supports as

shown in the figure are required to control distortion thus maintaining

the cross-sectional shape of the box girder and reducing the distortional

stresses. ( 6, 12 , 13) The spacing of rigid transverse interior diaphragms

along the length of a box girder is an important factor in this regard.

1.2 Objectives and Scope

This report presents the results of an analytical study of the

influence of the spacing of rigid interior disphragms on the fatigue

strength of curved steel box girders. This study was undertaken as

part of Task 2 described in Appendix A. A listing of all project re­

ports is given in Appendix B.

The objectives of the research reported herein are: (1) to

examine analytically the effect of the spacing of rigid interior dia­

phragms on stresses in curved steel box girders, and (2) to evaluate the

influence of diaphragm spacing on fatigue strength of box girders.

1.3 Research Approach

One of the curved steel box girders designed and fabricated for

fatigue testing in Task 3 of the FHWA investigation (Appendix A) and re­

ported in Ref. 14 was selected for the theoretical study. An existing

finite element method of analysis was employed. The number of interior

diaphragms was varied from 0 to 5. Flexure, rotational and distortional

stresses were considered.

-2-

2. INFLUENCE OF DIAPHRAGM SPACING

2.1 Selection of Curved Box Girder Details

Besides the spacing of rigid interior diaphragms, the factors

influencing the stresses and deflections of horizontally curved steel

box girders include (1) the cross-sectional geometry and dimensions,

(2) the length and radius of the centroidal axis, (3) the sup-

porting conditions, and (4) the location and magnitude of the loading.

A random survey of six curved steel box girder bridges with rectangular

. ( 14) . h f d . 1 1 cross sect1ons gave t e ranges o non 1mensiona geometrica para-

meters as shown in Table 1. The various dimensions are defined in Fig.

2.

The influence of diaphragm spacing was examined through the results

of analyses of a curved steel box girder representative of the girders in

the random survey. The box girder selected was the same as those de­

signed for fatigue testing and reported in Ref. 14. (In particular,

curved box girder number 3 was selected.) Table 1 shows the values of

:the various dimensionless parameters for the girder. Figure 3 shows the

dimensionsof the curved box girder used in the analysis.

The finite element method was selected for the analysis, using

the SAP IV computer program. (l5) Figure 4 shows the discretization of

the box girder-for use in the program. The material properties were

assumed to be the following: a yield point of 36 ·ksi, a modulus of

elasticity of 30,000 ksi, and a Poisson's ratio of 0.3. Two concentrated

loads of 100 kips each were placed directly over the inner web of the

box girder at the quarter points of the simple span·as shown in Figs.

3 and ·4. The loads cause bending, rotation, and distortion of the box

girder cross section. In mathematical evaluations, the torsional moment

M is pure torsion and distorttonal components as shown in Fig. 5. The

finite element analysis of the box girder provides the total stresses

and deflections without separating them into bending, rotational and

distortional components. '•• -'.

·;···

-3-

The number of interior diaphragms (ND) was varied from 0 to 5 as

shown in Fig. 6. The diaphragms were placed symmetrically in the span.

For ND = 0, 1 and 5 the loads did not coincide with diaphragms whereas

for ND = 3 the loads occurred directly over diaphragms, as shown in the

figure.

2.2 Results of the Analyses

The results of the SAP IV analyses are given in Figs. 7 to 14.

In the figures, stress, deflection or distortion are plotted vertically

with respect to position along the half span of the girder. The loads

are over the inner web at the quarter points. The length-to-radius ratio

of the girder is L/R = 0.3.

Figures 7 and 8 show the total longitudinal normal stresses at

the bottom of the inner and the outer webs, respectively. Also shown

dashed in the figures are the corresponding bending stresses in a

straight girder of the same cross-sectional geometry and length, L. In

the vicinity of the loads, the normal stresses are reduced in the inner

web as the number of interior diaphragms increase. The corresponding

stresses in the outer web increase. For both webs an increase in the

number of diaphragms results in the total normal stresses approaching

the bending stress in the center half of the span. This indicates that

the stress gradient across the bottom flange reduces to a very small

value as the number of diaphragms increases. Figure 9 shows that with

five interior diaphragms, the stress gradient is indeed very small,

except near the load points. Thus, with a sufficient number of interior

diaphragms, longitudinal stresses due to torsion are present only locally

near the loads. '

The effects of~ the number of interior diaphragms on vertical

deflections and distortions are shown in Figs. 10 and 11. Figure 10

shows the reduction in deflection of the inner web as the number of

diaphragms is increased. The cross-sectional distortion (the angle y in

Fig. 5) is plotted with respect to position along the half span in Fig.

11. Distortion effects reduce quite rapidly with an increase in the

numb~r of diaphragms. -4-

The transverse plate bending stresses due to cross-sectional

distortion can also be expected to reduce as distortion reduces. These

stresses are plotted in Figs. 12 and 13 for the bottom of the inner and.

outer webs, respectively. The plate bending stresses reduced signi­

ficantly as the number of diaphragms was increased. For the case of

three interior diaphragms, the loads were at the quarter point dia­

phragms and the transverse flexural stresses in the webs near these

diaphragms were directly effected by the loads.

Figure 14 illustrates the effects of diaphragm spacing on the

total shearing stress. It appears that only small changes of shearing

stress take place when diaphragm spacing is decreased.

For comparison, the stresses and deflections were computed for a

straight box girder having the same loading condition, cross-sectional

geometry and centroidal length of the curved girder, except that for

the straight girder L/R = 0. The total longitudinal normal stresses

have the same pattern of variation along the span as the curved box

girders. This can be seen by comparing the stress profile plots in

Figs. 15 and 16 with corresponding plots in Figs. 7 and 8 for the curved

box girder. Because the bending normal stresseg are identical for the

straight and the curved box girder, a direct comparison of the torsional

normal stresses can be made. The comparison is made in Figs. 17 and

18 which shows that, regardless of diaphragm spacing, the torsional

stresses (distortion plus warping) were lower for the straight box

girder. The increase in number of interior diaphragms from 1 to 3,

however, had a stronger effect on the torsional stresses in the curved

girder.

Results of deflections and distortions for the straight and

curved box girders are compared in Tables 2 and 3. The influence of

curvature (L/R) and of diaphragm spacing on total deflections were not

significant as is indicated by the very small changes in values in Table

2. The effects of curvature and diaphragm spacing on distortion were

more important, however. As shown in Table 3, the distortion at the

load points (L/4) was reduced to approximately one-half by adding interior

-5-

-- ·•-•---~- ·-··----... ----·----··---~------.··--.·~--~·-~----"~-··~ -·--~ -----·~----..--~--~--·• .. • -· ••-- -~-- "'•-·•-·•r..___ ____ _

diaphragms at the quarter. points in both the straight and curved box

girders. The variation in distortion at the load points with respect

to L/R was of about the same order of magnitude. Thus, in order to

reduce distortional stresses in curved box girders, the required number

of interior diaphragms must be determined considering the curvature

(L/R).

-6-

3. INFLUENCE OF DIAPHRAGM SPACING AND RADIUS

To explore further the influence of diaphragm spacing and

curvature on the stresses in curved box girders, an analysis was made

of an arbitrary curved box girder with a prismatic cross s~ction, but

variable span length and radius. The dimensions of the box girder are

given in Fig. 19. Rigid interior diaphragms 9.5 mm (3/8 in.) thick

were placed at a spacing of L/2 to L/12. The span length-to-radius

ratio varied from 1/24 to 0.6. The computer program CURDI was used

for the analysis. (8 ) This program employs the finite strip method and

provides rapid solutions to theelement forces and displacements for

curved box girders. The discretization of the box girder for analysis

is shown in Fig. 20.

The computed stresses and deflections showed the same trend as

depicted by Figs. 7 to 18; an increase of the number of diaphragms in a

given girder decreased the torsional and distortional stresses. To

incorporate the effect of curvature, the maximum normal stress range,

S , corresponding to the range of applied load is plotted in Fig. 21 as r

a function of a/R, the ratio of diaphragm spacing to box girder radius.

It is apparent from this plot that the S versus a/R relationship is r

practically linear for any ratio of L/R. This relationship could

simplify the determination of the necessary spacing of diaphragms in

curved box girders with respect to fatigue criteria.

Because of the apparent importance of the S versus a/R relation-r

ship, information from the literature was used to examine the relation-

ship further. There are only limited results of analyses in the liter­

ature with regard to the influence of diaphragm spacing on stresses (or

stress ranges) in curved box girders. Heins(12

•16

) used the partial

differential equation of distortion developed by Dabrowski. (2) A para­

metric study was conducted by Heins from which an empirical formula was

established for estimating distortional stresses in curved box girders.

The results from the parametric study are used here in a further examina­

tion of the influence of diaphragm spacing and curvature on stresses.

-7-

Two box girder cross sections taken from Ref. 12 are shown in

Fig. 22. The girders have the same width but different depth. The

number of interior diaphragms varies from one to nineteen. Three

different lengths of span were examined. The computed maximum distor­

tional stresses from Ref. 12 are plotted in Figs. 23 and 24 as a

function of the number of diaphragms. Regardless of the L/R ratio, the

maximum distortional stress decreases with increasing number of dia­

phragms. Box girders with higher L/R values, that is, with longer

spans or sharper curvature, have higher distortional stresses.

The relationship between stress range and spacing-to-radius

ratios are depicted in Figs. 25 to 30 for the two box girders with three

different span lengths. The relationship is nearly linear for all values

of L/R in the practical range. For box girders with high L/R values,

the stress range decreases with decreasing of diaphragm spacing at a

rate slightly different from constant. For comparison, straight box

girder stress ranges are also included in the figures. For straight box

girders, L/R and a/R are zero, the ratio of a/L is used as the abscissa.

The resulting S -a/L lines are straight. r

Since the curves in Figs. 25 to 30 are concaved upward, straight

line approximations connecting the terminal points of the individual

curves are on the conservative side. Such straight lines could be used

in determining the spacing of interior diaphragms to ensure that stress

ranges are within permitted values.

-8-

4. FATIGUE CONSIDERATIONS

The generation of torsional and distortional stresses in curved

box girders under live load results in high ranges of longitudinal

stresses. Stress range controls the fatigue life of bridge compo­

nents. (ll) Thus, torsional and distortional stresses might reduce the

fatigue life of curved box girders if such stresses are npt controlled.

While studies are in progress to investigate the effects of

stress range gradient on the fatigue strength of curved plate girders . (14)

and box girders, the early stage of fatigue crack propagation has

been well described. (l8 ) Design stress range curves for various cate­

gories of details have been specified by AASHTO, AISC and AREA. (l9 , 20 ' 21,22,23) Figure 31 depicts the design stress ranges for different

categories of structural details. It is of paramount importance that the

live load stress ranges do not exceed the specified values for the gover­

ning categories at the desired life of the structure.

For given geometry, loading, and support conditions of a curved

box girder, the bending and torsional stresses are invariant. Only the

distortional stresses can be reduced by adding interior diaphragms. From

results such as reported herein it is obvious that curved box girders

with longer spans or sharper curvature have higher distortional stresses,

thus require more interior diaphragms. Plots of stress range versus

a/R ratio, similar to Figs. 25 to 30, could be used in conjunction with

the specification stress range curves to determine the diaphragm spacing,

for a given life.

As an example, consider a box girder with the cross section of

Girder No. 1 in Figure 22 where L/R = 0.5, and L = 15 in. The stress

range versus a/R relationship is given in Fig. 25 and is replotted in

Fig. 32. If a design life of two million cycles is specified and a

category E structural detail must be used, the design stress range is

8 ksi from Fig. 31. With one interior diaphragm, the a/R ratio is 0.25

and the stress range is approximately 10 ksi, being higher than the

design stress range of 8 ksi. Three rigid interior diaphragms are needed

-9-

to reduce the stress range below the design value, as is depicted in

Fig. 32.

Most known fatigue failures occur at structural details when the

applied stress ranges are high but are not considered in the design

process. <23 , 24) Many of these stresses are induced by the performance

of the structure under load. The distortional component of torsion of

box girders causes transverse flexural bending of the web plates (as

shown in Figs. 5, 12 and 13). These performance-induced stresses may

cause fatigue cracking at the junction of box girder components. For­

tunately the addition of transverse interior diaphragms reduces effec­

tively the transverse bending stresses along the box girder. However,

at diaphragms where a load is applied, local stresses may be significant

(see Fig. 13). Appropriate structural details must be used at the dia­

phragms to avoid high secondary stresses. <25 )

-10-

...

5 . CONCLUSIONS

This report analytically investigated the effect of the spacing

of rigid interior diaphragms on the longitudinal normal stresses

(bending, torsional and distortional) and transverse web bending

stresses in horizontally curved steel box girders. The study was con­

ducted using a number of box girders, including one of the large scale

box girders designed for fatigue testing. (l4)

The following conclusions can be made on the basis of the inves­

tigation:

1. Curved box girders are subjected to higher stresses compared to

straight box girders because of the curvature.

2. Increasing the number of diaphragms and reducing the diaphragm

spacing effectively control the longitudinal normal stresses

as well as the transverse bending stresses.

3. The overall deflection of curved box girders is not significantly

effected by diaphragm spacing but the distortion is effected.

4. The plots of stress range S versus spacing-to-radius ratio (a/R) r

are nearly linear for all values of span length to radius ratio

(L/R) in the practical range.

5. A conservative straight line approximating the S to a/R relation-r

ship for any curved box girder can be used with the specification

fatigue design stress range curves to determine the required dia­

phragm spacing for a given life.

In summary, it must be emphasized that the cross-sectional geome­

try, span length, curvature, supporting conditions and loading control the

primary stresses in curved box girders. Only distortional stress com­

ponents are controlled by the prese~ce of rigid interior diaphragms. The

use of interior diaphragms directly reduces distortional normal and dis­

tortional plate bending stresses, thus increasing the fatigue strength of

curved steel box girders. The stress range versus a/R relationship ap­

pears to be a practical guide for diaphragm spacing. Additional work in

in this direction seems beneficial, and is reco~~ended. -11-

·--~----~--~.-------·--···-·-·-·---------~---·---------·------------·

6. TABLES

..

-12-

TABLE 1 RESULTS OF SURVEY OF SIX BOX GIRDER BRIDGES AND VALUES SELECTED FOR ANALYSIS

Dimens ion1ess Range of Values Parameter Parameters for Selected for

(see Fig. 2) Existing Bridges Analysis

b 0. 96-1.0 1.0 d

e 0.1-0.6 0.33 b' '

L 16.0-37.0 12.0 d

d 88.0-168.0 96.0 t

w

b tb

24.0-330.0 96.0

tb 0. 3-1.0 0.33

tt

L 0. 06-1.09 0.3 R

-13-

TAB IE 2 COMPARISON OF DEFLECTIONS

Location along Span

L/R ND L/8 L/4 3/8 L L/2

0.3 5.59 9.91 10.92 11.18 1

0 5.16 9.27 10.54 10.79

0.3 5.08 8.89 10.16 10.41 3

0 4.85 8.79 10.29 10.72

TABLE 3 COMPARISON OF DISTORTIONS

Location along Span

L/R ND L/8 L/4 3/8 L L/2

0.3 0.00202 0.00267 0.00167 0.001 1

0 0.000312 0.00144 . 0.00089 0.00053

0.3 0.00081 0.0015 0.001 0.0007 3

0 -0.00026 0.0007 0.0004 0.0002

-14-

7. FIGURES

-15-

I I

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I

I I

I

I

/

~··. ~ :.

Rigid Interior

Diaphragm

End Diaphram

·-____ --=-..:.....;;. __________ . -· -· --.--·- ___________ ... ____ _..:. ____ __,_ _____ 1

( Fig. 1 Schematic of Horizontally Curved Box Girder i . -·- _________ !

. -:-~•··r· --~--~---· . •w· ·-:·-:~--'T~.-::::-:·?:.":"-':_-~~-:'-·,-_.-.-- ..... ,:-~ - ~:----;-:--- -_-c_-;;--=.-;:;..=-:=-·-- ·--

--- -···---------~--- ··- ... -··-·-

- : .. <.:: z~~=~~~ ... : :--:- .. _ .... -~ ~----::--:.---::::::- ~- .. -... . ,.:-_~ ~. --~----- ·- "'-':"--=-.-=::-;z· . . -

I I ,

I I

e e

• ftt

tw~ ~

.tb

t b

I i ~----L------'-1

Fig. 2 nefir.i.it:ton. -or -fiaramefers-"rar- Exl-sHng----~

Curved Box Girder Bridges (see Table 1) \ I

·--- ------, 17 I

~ ----- • I

d

.. ·t.~~~ .••..

' .

-·· .·. ·---

' i.

1.14 11

+

.._....,..... 3/all

t3/all

I ..

1 ..

36 11

36 1

.I P= lOOK

9'

_, Fig. 3 Dimensions of Curved Box Girder Used

i~ t:!t.~. A!J:alys i? ~Fig. ?O _o~ Re_!:. 14)

R= 1201

12 11 --· . . -- .. - "'-~ . ..

36 11

. .

IL/R = 0.31

. --- -- -,

"' ;-): ·.- _. __ ..

· .. };·:·~.x~·{r~ .'·. ·

Plate Element { typ.)

I• -

• -- -36 11 - 0

Symm.

Plan

6011

- -4 Spaces

@ 911

- - -Section A-A

--

Load at Quarter Point

Radius= 120'

Span= 36 I

4 Spaces@ 9'_' _

'F:lg·~-1;-· :Fi-niEe-Eleffient: niscre·i:-i.zatio_n __ [or-·-xn-af.YSis _______ l - _ ----~~ ~~~:.'::_~---~~-~-G~:-~::. ~~--?~~I~ _____ --··_ .-· l

-------1 .. 19

···--··-- .....

M

f) d

1- b .. , M II .M ~- b 4. I I I I I M

I t I I

~bJ r2Mb +L I

-M ...._:../ M .,1: - y 2b 2b 2

Torsional Pure Torsion Distorsiona I

Fig. 5 Torsional Loading Components

• -- :·-- --- ---1,-

20

0.5 L

(a)

ND=O '. •.

1- L -I P = 100 kips p

~ ... -~.··:."'•" ------- - l-.. ' .. -·· :

(b)

Interior Diaphragms

p ,_

(c)

ND = 3

(d) I I .

.. - ''·:,. .. :... :

. .. ~ .. -.- ·-. . . . . . ~· _. ~- ~ ...

. :.;: _ _':.-)~~}~~'

Concentrated Loads and N~~~~~---~~-~~~~~~~~~s--(ND) ____ l_- -- ---~-~."-------·-----,---~-- .....

- Fig. 6

N N

0

10

20

30

40

STRESS (ksi)

IL/R = 0.3, '

Bending ..... 5'....;.

' 3

0.25 X/L

i p

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

:: . Fig.' 7 Total Longitudinal Normal Stress at Bottom of Inner Web,::' -.: .:: _, .. " Lt

~ ~~"'·If~ I"" .. .. • '

), · .. . . ~·· . '' .. ·, ' ' . ~ . ·: ., ' .. .

·!!

l

. .

5

ND=O

p

Inner Web

0.5

ct Symm.

(Midspan)

. /" ._, ..

, .. r-·j

-10

10

20

30

STRESS (ksi)

..

...._ """- ..... ....... -...._....... 5

-...._......._ ............ ...._,

..... ,

p

-------Bending

Outer Web

.Fig. 8 Total Longitudinal Normal Stress at Bottom of Outer Heb ' ' :

! '·'

! : ..

' ·' ), ''

ND=O

p I

I i

Symm. (Midspan)

4

'3

2

N +:-

0

- I

-2

-3

-4

,.

,_ ... _ ....... - ""' ..... ~f."·"l""'--·"· ·~

STRESS GRADIENT ( k/ in 3 )

,.

j L/R = o.:3J

.. - ·-.

' ;·: .: ' ' .

··;

,, ;i'.":·· ...

Fig. 9

p

~ ND=O

I

I X/L 0.5

0.25

p

, . I , ,

·'

Sign Convention c:::::::::::iJ ·for Stress [ o;;p

Gradient

Stress Gradient across the Bottom Flange

5

Inner Web

ct Symm.

(Midspan)

0

0.5

1.0

1.5

..

DEFLECTIONS (in.)

I I -I . , .

r 0.25

5

3

Fig, 10 Vertical Deflection of Inner Web

X/L

p

Inner Web

I , . l

1

0.5

Cf_ Symm.

(Midspan)

0

10

20

30

. l

'· ·'" ........... ~, ... - - .

DISTORTION Y (Rad.) x 10-3

p

' 0.25 X/L

·'--. -. -.1 ---:---;:----~---------·--·--·-:----~--·--------------.----------·---~-. -.

, .. , ; : ' ... -;- -~ . :

• • J : ..

• , t· .· .. Y\}!:J~:$';i :·.;

.1:_:

0.5

I I

I

I I

I ct.

Symm. (Midspan)

4

0

STRESSES (ksi)

p

t

5

0.25 X/L

Fig. 12 Transverse Plate Bending Stress at Bottom of Inner Web

p

Inner Web

0.5

Cl Symm.

(Midspan)

N '(X)

2

-I

STRESSES (ksi)

NO= I

3

i

p

~ NO= I

X/L

Outer Web

Fig. 13 Transverse .Plate Bending Stresses at Bottom of Outer Web

•• 4 • __ ;: ... ~ '

. ""-;.

p

. ~ i

0.5 I I

I I

I <t

Symm. {Midspan)

-6

-4

-2

2

..

SHEAR STRESSES

( ksi)

p

~

I L/R = 0.3,

Fig. 14 · Total Shear Stresses at Bottom of Inner Web .. ..

:~'~·;:.'j, ·.·.··•· . ' . .. '

. ' . ~ r. .

! i)::" /~;;-:")<~ •·· .

Inner Web ct

Symm. (Midspan)

w a·

0

5

10

15

20

0.25. X/L

' ................. , tp ' '

TOTAL.NORMAL STRESSES

(ksi)

' ' ' ' ' ' ' ' . ', Bending

'---- -- -·-

NO= I

0.5

-------~

p

Inner Web

I

I I

I I ct

Symm. (Midspan)

; Fig. 15 Total Longitudinal Normal Stress at Bottom of Inner Web

. ;'\' .

i"'~ . -.. . ; .

/j -': ' '

0

5

10

15

20

STRESSES (ksi)

0.25 X/L

Outer Web

Fig. 16 Total Longitudinal Normal Stresses at Bottom of Outer Web

' .... _,~ ..

. i .. ··! . .I .. . . I

l

0.5

Symm. (Midspan)

w N

-10 p

~ -5

X/L o~----------_.------------~----_.------~--------~_.------------~

5

10

STRESSES (ksi)

0.25

Fig. 17 Distortion Plus Warping Stress at Bottom of Inner Web

, . . . .. :·. :. ; - · ....

:V" ... . '

. • .

p

. .# .•. ·

Inner Web

0.5

I I

I I

I I

I <t_

Symm. (Midspan)

-10

-5

; ;

5

STRESSES (ksi)

. ;

p

~

Fig .. 18 Distortion Plus Warping Stress at Bottom of Outer Web

p

Inner Web

0.5

I I

I <l

Symm. (Midspan)

,. ~ R= 59' to 236'

P = 1 ooK

. · ... ·.··

0. 1211 1211

0. 12"

1 .. 1811 ..I

l 1 ..

--- -------- . -1 --- I -L= 9.8- to 72

.. ·. :-:: ·· ..

Section At Which Internal Forces 8 Displacements Are Computed ( typ.)

X

Axis Of Symm.

Aj!

Plan

, ... 2211

-

~ 1211 • z I

Section A-A

.. I -

Radius= 60' to240 1

Span= I 0' to 72'

.,.,_y

. - - - - .. - - - --- - .. --- - . --. -- - - -Fig. 20 Finite Strip Discretization for Analysis of

Curved Box G~.r.~~~---by ~l!RDI .. _

20

:; 15 ~ -... ... (/)

.. w (!)

ZIO <( a:: (/) (/) w a:: ..... (/)

5

0

/L/R=I/s

z/R= l/12 .P L/R= 1/24 .

L/R = 0.3

0.15 OfR

L/R= 0.6

0.3

---F'Tg-:---2T-- st:ress-:Ranie-~- --s -~ -·:v-s-:-niai)l1raiffi. -------------- 1

______ Spacing to Radi~s, a/R, _ _l:{elaticmship .. \

36

.. ~~ :.' .

25.7511

48.0 11

48.011

.. -

0.94 II

I

0.3911 --+------I

0.51 111

96.0"

I 92.011

Girder No. I

1.10 II

I '

--+--~-. --'

0.6311 I

0.98 11

96.011

192.011

Girder No. 2

48.0 11

-

I

,_,....

5.95 11

9.8011 - -

I

-.-

29.31 II

1---I-

80.4811

52. 1711

48.011

.. -

. l

Fig •. 2~ .. -~~~---~~~~-~~ .G~~~~tries from Ref. 12 . \ . ~ ., .·- .

. -....... _.1 37 I

______ I

-C/) ~

7

-s .. -b 5 .. en en ~4 f-en z3 0

~2 0 f­en 0

0

Girder No. I

I L = 590.611 I

L/R =0.5

3 7 9 NUMBER OF DIAPHRAGMS, ND

··-·----------------· -· -·---

• . 'X

19

Fig. 23 Influence of Number of Diaphragms on Maximum Distortion Stress

.... -~- ~",-:''~"~ .......

•

. -:--· ..

l ~ . r ~

I !

r

I

8

:::"7 (/) ~ -;6 b

u)5 en w ~4 en

53 1--

~2 l­en Ol

0

L/R = 0.75

L/R=0.5

--- L/R =0.3

3 7 9

Girder No. 2

IL= 1771.711

1

NUMBER OF DIAPHRJ:\GMS, ND 19

F-~;-:-;4---;~~~~~~-ce -~-;-~u~~;;~;~iaph~~gm~- on ;~xim~ ;~~t~-rti~~-Str~ss--l - - - - - -- - - -- -- -- --- - - - . . -- . . - . - - . -- -- -·- -- .. - .

-~ -~~---- --- ~ \

39 \

' ' '

'i -~ !

J a I :II ! ' \ , I

STRESS RANGE

(Srl ksi

13

12

II

10

9

8

7

6

0 0.1

.-----·- .. --- ·---- ... -- - ··- . - -·-

Girder No. I

IL=590.611

1

L/R=0.5

0.2

L/R = 0.625

0.3 0/R

0.4

! Fig. 25 Stress Range, S , vs. Diaphragm ·, '-·----·-·--~Sp?_c_ing .. t9 .. :R:ac:I_:i~-~-' .. a/~, .. -~e_l_ati()n~h~p ___ _

L/R = 1.0

0.5

::.

. : -,i

• ··~· -~ '; ' I'

' . ' ~ '

; t:: :.: t:·: :~.,.·.. i

i '

..

16

15

STRESS 14 RANGE (Sr) ksi

II

0

--------

0.05 0.10

-- -­.-- ----Girder No. I

I L=ll81.111 I

0.20

--

--F{g~--26-·sfress Range, s , vs. Diaphragm _______ -------~p::_c_irl_g;_ to_ R:ad ~&s , ____ a/R, _ Re lationsh_ip ___ ·

..

LtR = 1.0

LtR=O ...---

0.25

'i '' . ··j ' ' .

.p.. N

28

27

26

25 STRESS RANGE

24 (Sr) ksi

23

22

21

0

'; ; .. :. r~ ..• ;

0. I

Girder No. I

I L = l 771.711 I

0.2 0.3 0/R

--­..-- -----­..-

0.4

1 Fig. 27 Stress Range, S , vs. Diaphragm · : . r . .. '. ___________ §p ac::_;_ ng ___ t:() __ R_~-~ t_l1_~L _C}/_R_, _ _gt::la t;_~o_t:!_~-~J:p ___ !

L/R::; 1.0

..-- L/R = 0

0.5

~ . ' .: } ;~ ' : ~

.. · i ;;:}{;, ::~,;;\>;/:,··

Girder No. 2

3 : I L = 590.611 I

..

. 0.5, 0.65, 1.0 [L/R = 0.25,

---.:====11==:!:==11===~=========::: L/R = 0

0.2 0.3 OfR

0.4 0.5

Fig. 28 Stress Range, Sr' vs. Diaphragm Spacing to Radius, a/R, Relationship

I ;j· .

..... :·. 1. ,..

.J::'

.J::'

4 STRESS RANGE

(Sr) ksi

2

0 0.1

Girder No.2

I L= 1181.111

1

L/R=0.5 L/R= 0.625

0.2 0.3

OfR

0.4 0.5

Fig. 29 Stress Range, S , vs. Diaphragm Spacing to Radius, a/R, Relationship r

' ! i

I .. -

l _. .. ·

'il'~;k<

· ... . ;, ... .

.... ~·- - i ' ... ,. 'I'

•

1

: +"' i: v.i

. r i I

L-·-~~'

STRESS RANGE

(Sr) ksi

' •

·•

Girder No. 2

I L = 1771.711 I

3 --------------L/R=O

2

o~cfz-~~__L_ 0. I 0.2 0.30/R 0.4 0.5 0.6

Fig.--~-~~··- - __ ress Range S ---·----- SpB:c~ng to Ra'd.r' vs. Diaphragm --- ---- _~us a/R ---' -- -- -' Relationshi - -- p ____ _

._:1 .. :.-· i

. i 1 i .. · ... •.

_ .. ·

'' ~~

. ·'

' ,,

'

' . ,

' ' .•... ·•.

., ', I

.: i ·I , I I·

1 .. · •.

'· ·,.

'I' I . ' " .. ' . ~

., ..

I i i''

' . I.

~ [: I

' ')

.!'

:,•

i

.!:l I'

I

... I

. :l ,, I I' 1: ,;t

:!1

'I ' !

t.

'

·i ,f .

; ~ I .l ... ' .. ~

' "

. ,' ,. ·.:. ·. j i ' ! ! ~-

'I

I I'

·' I i

., I

I I ,. ! I

; I I

I I I

, :I .. 1:'

'.

,, ,.

•_.·jj. · .• ':t' ··j

! '

,, J. ·"

l'

5xlQ~-·.:y.: ,J-.•. _;:.:, · .• · .... ; ; 2' i09 ·. ,_ I • r-· .Ji.bw· i.i.Wf!tdi.t.;,:,.J ••,i,'t'¥'iool'•*'"->''•j ''I~ ' • ... • •.

\Fig~ 3: Stress Range, Sr' vs. Cycle Life,

I ''

; ... ··.il .· ''

I .• · ..

'i I

,'I''

··.,:·,'/'I.· .,,

·I

'. '

Category E ·

i ·' ·'

,.

! .. I ·I .•' I ·I 1:··

I ·.r' I . ·' ' j

.. ! ' .. 1 , .. ' I '

· .. · CY9LE .LIFE . r· . '

-~~--·~--------"""'"""'--.""'""'""~l, •' ~ •I . . :

,_.:. :,;i.·. !"().·t·t~:..!~·>: .

;~r~ •. '!·~ .. ,

:···

'!,) . . r:

.i . ; ..

. ·-

'., . 8

• ·J·" ·. r··l·

' ,. -~· ~

i ' . 'I I I

·: t: i

13

12

II

10 STRESS RANGE

,.p. (Sr)

I -...) ksi ! ' 8 L_-·

7

6

0

..

Girder No. I

I L = 590. 6 II I

LtR = 0.5

0.1 0.2 0.4

·Fig-.- j2 - stress- Ra"i:ige·;--s·---,---v:s- ~ -Dia-phra-gm

------------~-~ac_~-~~-~_c>_R:~~i~s '--~{~-'--~=-~-ti~~~~~---

...

•

0.5

8. APPENDICES APPENDIX A: STATEMENT OF WORK

"Fatigue of Curved Steel Bridge Elements"

OBJECTIVE

The objectives of this investigation are: (1) to establish the

fatigue behavior of horizontally curved steel plate and box girder high­

way bridges, (2) to develop fatigue design guides in the form of simpli­

fied equations or charts suitable for inclusion in the AASHTO Bridge

Specifications, and (3) to establish the ultimate strength behavior of

curved steel plate and box girder highway bridges.

DELINEATION OF TASKS

Task 1 - Analysis and Design of Large Scale Plate Girder and Box Girder Test Assemblies

Horizontally curves steel plate and box girder bridge designs will

be· classified on the basis of geometry (radius of curvature, span length,

number of span, girders per span, diaphragm spacing, types of stiffener

details, type of diaphragm, web slenderness ratios and loading conditions).

This will be accomplished through available information from existing

literature and other sources, as required.

Current research on the fatigue strength of straight girders has

identified and classified those welded details susceptible to fatigue

crack growth. This classification shall be extended to include criti­

cal welded details peculiar to curved open and closed girder bridges.

These welded details shall be examined with respect to their suscepti­

bility to fatigue crack growth and analyses shall be made to estimate ::

the conditions for fatigue crack growth.

Based on the analyses described above, a selected number of

representative open and closed section curved bridge girders shall be

defined for purposes of performing in-depth analyses, designs, and

laboratory fatigue tests of large scale<: test assemblies. These girders

shall be typical and will characterize commonly used girders, to include

the use of welded details. The assemblies shall be analyzed and designed

using currently available design guides, methods, and/or computer

-48-

programs. Each test assembly shall be designed to incorporate the maxi­

mum number of welded details susceptible to fatigue crack growth. Stress­

es in all components of the cross section shall be examined so that the

significance of each stress condition can be evaluated. An assessment

of the significance of flexural stress, principal stress, stress range

and stress range gradient shall be determined at each welded detail.

The significance of curved boundaries on the stresses shall be exa~ined.

Stress states in welded details equivalent to those used in straight

girders shall be examined.

Curved plate and box girder test assemblies shall be designed so

that ultimate strength tests can be carried out following the planned

fatigue tests, with a minimum of modification.

Task 2 - Special Studies

In addition to but independent of the analyses and designs

described in Task 1, certain other special studies shall be performed.

These special studies are specifically directed towards those problems

peculiar to curved girder bridges, as follows: (1) the significance of

a fatigue crack growing across the width of a flange in the presence of

a stress range gradient shall be studied, (2) the effect of heat curving

on the residual stresses and fatigue strength of welded details shall be

examined, (3) newly suggested web slenderness ratios for curved girder

webs reduce present slenderness ratios of unstiffened webs. These

slenderness ratios shall be examined in terms of fatigue performance of

curved webs, and (4) the effect of internal diaphragms in box beam

structures will be examined with regard to fatigue behavior.

Task 3 - Fatigue Tests of Curved Plate Girder and Box Girder Test Assemblies

The plate and box girder test assemblies designed in Task 1 shall

be tested in fatigue. Emphasis shall be placed on simulating full-scale

test conditions. The test results shall be correlated with the analyses

made in Task 1 and the results of the special studies performed in Task

2.

-49-

•

Task 4 - Ultimate Load Tests of Curved Plate and Box Girder Assemblies

Following the fatigue tests of Task 3, each plate and box girder

test assembly shall be tested statically to determine its ultimate

strength and mode of behavior. Fatigue cracks shall be repaired, where

necessary, prior to the static tests. Consideration shall be given to

providing a composite reinforced concrete slab on each test girder prior

to the static tests.

Task 5 - Design Recommendations

Design recommendations for fatigue based on the analytical and

experimental work shall be formulated in a manner consistent with that

for straight girders. Specification provisions shall be formulated for

presentation to the AASHTO Bridge Committee.

-50-

•

APPENDIX B: LIST OF REPORTS PRODUCED UNDER DOT-FH-11.8198

"Fatigue of Curved Steel Bridge Elements"

Daniels, J. H., Zettlemoyer, N., Abraham, D., and Batcheler, R. P. ANALYSIS AND DESIGN OF PLATE GIRDER AND BOX GIRDER TEST ASSEM­BLIES, DOT-FH-11.8198.1, September 1976.

Zettlemoyer, N. and Fisher, J. W. STRESS CONCENTRATION, STRESS RANGE GRADIENT AND PRINCIPAL STRESS EFFECTS ON FATIGUE LIFE, DOT-FH-11.8198.2, June 1977.

Herbein, W. C. and Daniels, J. H. FATIGUE TESTS OF CURVED PLATE GIRDER ASSEMBLIES, DOT-FH-11.8198.3, May 1977.

Batcheler, R. P. and Daniels, J. H. FATIGUE TESTS OF CURVED BOX GIRDERS, DOT-FH-11.8198.4, January 1978.

Batcheler, R. P. and Daniels, J. H. EFFECT OF HEAT CURVING ON THE FATIGUE STRENGTH OF PLATE GIRDERS, DOT-FH-11.8198.5, August 1977.

Abraham, D., Yen, B. T., and Daniels, J. H. EFFECT OF INTERNAL DIAPHRAGMS ON THE FATIGUE STRENGTH OF CURVED BOX GIRDERS, DOT-FH-11.8198.6, February 1978.

Daniels, J. H., Batcheler, R. P., and Fisher, J. W. ULTIMATE STRDNGTH TESTS OF CURVED PLATE GIRDER AND BOX GIRDER ASSEMBLIES, DOT-FH-11.8198.7 (in preparation).

Daniels, J. H., Fisher, J. W., and Yen, B. T. DESIGN RECOMMENDATIONS FOR FATIGUE OF CURVED PLATE GIRDER AND BOX GIRDER BRIDGES, DOT-FH-11.8198.8 (in preparation).

-51-

• '

9 • ACKNOWLEDGMENTS

This study was conducted in the Department of Civil Engineering

and Fritz Engineering Laboratory, Lehigh University, under the auspices

of the Lehigh University Office of Research, as a part of a research

investigation sponsored by the Federal Highway Administration (FHWA) of

the United States Department of Transportation. Dr. D. A. VanHorn is

Chairman of the Department of Civil Engineering and Dr. L. S. Beedle is

Director of Fritz Engineering Laboratory.

The FHWA Project Manager is Mr. Jerar Nishanian. The contribu­

tions of the Lehigh Project Advisory Panel are gratefully acknowledged.

The Advisory Panel members are: Messers. A. P. Cole, C. G. Culver, R. S.

Fountain, G. F. Fox, T. V. Galambos, A. Lally, F. D. Sears and I. M.

Viest.

Dr. J. W. Fisher, Dr. R. G. Slutter and R. P. Batcheler,·faculty

and staff members of the Department and Laboratory, contributed to the

conduct of this work.

The manuscript was typed by Ms. Shirley Matlock. The figures

were prepared under the direction of John M. Gera.

-52-

)

10. REFERENCES

1. Vlasov, V. Z. THIN-WALLED ELASTIC BE~~S, TT61-11400, Office of Technical Ser­vices, U. S. Department of Commerce, Israel Program for Scienti­fic Translations, ·psT. CAT. No. 428, Washington, D. c., 1961.

2. Dabrowski, R. CURVED THIN-WALLED GIRDERS, Translation No. 144, Cement and Concrete Association, London, 1968.

3. Kollbrunner, C. G. and Basler, K. TORSION IN STRUCTURES--AN ENGINEERING APPROACH, Springer-Verlag, New York, 1969.

4. Goldberg, J. E. and Leve, H. L. THEORY OF PRISMATIC FOLDED PLATE STRUCTURES, Meinoires, Interna­tional Association for Bridge and Structural Engineering, Vol. 17, pp. 59-86, 1957.

5. Wright, R. N., Abdel-Samed, S. R. and Robinson, A. R. BEF ANALOGY FOR ANALYSIS OF BOX GIRDERS, Journal of the Struc­tural Division, ASCE, Vol. 94, No. ST7, Proc. Paper 6025, July 1968.

6. Abdel-Samed, S. R., Wright, R.N. and Robinson, A. R. ANALYSIS OF BOX GIRDERS WITH DIAPHRAGMS, Journal of the Struc­tural Division, ASCE, Vol. 94, No. STlO, Proc. Paper 6153, October 1968.

7. Meyer, C. and Scordelis, A. C. ANALYSIS OF CURVED FOLDED PLATE STRUCTURES, Journal of the Struc­tural Division, ASCE, Vol. 97, No. STlO, Proc. Paper 8434, October 1971.

8. Scordelis, A. C. ANALYSIS OF CONTINUOUS BOX GIRDER BRIDGES, Report No. SESM-67-25, Department of Civil Engineering, University of California, Berke­ley, California, November 1967.

9. Subcommittee on Box Girder Bridges of the ASCE-AASHTO Co~mittee on Flexural Members, TRENDS IN THE DESIGN OF BOX GIRDER BRIDGES, Journal of the Structural Division, ASCE, Vol. 93, No. ST3, Proc. Paper 5278, June 1967.

10. Subcommittee on Box Girder Bridges of the ASCE-AASHTO Committee on Flexural Members, PROGRESS REPORT ON STEEL BOX GIRDER BRIDGES, Journal of the Structural Division, ASCE, Vol. 97, No. ST4, Proc. Paper 8068, April 1971.

-53-

11. Petzold, E. H. and Galambos, T. V. BEHAVIOR AND DESIGN OF LARGE STEEL BOX GIRDER BRIDGES, Civil and Environmental Engineering Department Research Report No. 26, Washington University, St. Louis, Missouri, December 1973.

12. Oleinik, J. C. and Heins, C. P. DIAPHRAGMS FOR CURVED BOX BEAM BRIDGES, Journal of the Structural Division, ASCE, Vol. 101, No. STlO, Proc. Paper 11634, October 1975.

13. Yen, B. T., Hall, J. and Chen, Y. S. AFFECT OF DIAPHRAGM SPACING ON BOX GIRDERS, Fritz Engineering Laboratory Report No. 380.13 (in preparation), Lehigh University, Bethlehem, Pa.

14. Daniels, J. H., Zettlemoyer, N., Abraham, D. and Batcheler, R. P. ANALYSIS AND DESIGN OF PLATE GIRDER AND BOX GIRDER TEST ASSEM­BLIES, Fritz Engineering Laboratory Report No. 398.1, Lehigh University, Bethlehem, Pa., October 1976.

15. Bathe, K. J., Wilson, E. L. and Peterson, F. E. SAP IV, A STRUCTURAL ANALYSIS PROGRAM FOR STATIC AND DYNAMIC RESPONSE OF LINEAR SYSTEMS, Earthquake Engineering Research Center, Report No. EERC 73-11, University of California, Berkeley, California, June 1973.

16. Heins, C. P. BENDING AND TORSIONAL DESIGN IN STRUCTURAL MEMBERS, Lexington Books, D. C. Heath and Company, 1975.

17. Fisher, J. W. and Yen, B. T. DESIGN, STRUCTURAL DETAILS, AND DISCONTINUITIES IN STEEL, Proceed­ings, ASCE Specialty Conference on the Safety and Reliability of Metal Structures, November 1972.

18. Hirt, M. A. and Fisher, J. W. FATIGUE CRACK GROWTH IN WELDED BEAMS, Journal of Engineering Fracture Mechanics, Vol. 5, 1973.

19. Association of State Highway Officials STANDARD SPECIFICATIONS FOR HIGHWAY BRIDGES, Tenth Edition, AASHO.

20. American Association of State Highway and Transportation Officials INTERIM SPECIFICATIONS - BRIDGES - 1974, AASHTO, Washington, D.C., 1974.

21. American Institute of Steel Construction SPECIFICATION FOR THE DESIGN. FABRICATION AND ERECTION OF STRUC­TURAL STEEL FOR BUILDINGS, AISC, New York, N. Y., 1969.

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22. American Railway Engineering Association MANUAL FOR RAILWAY ENGINEERING - Chapter 15, Steel Structures, Specifications for Steel Railway Bridges, 1973.

23. Fisher, J. W. BRIDGE FATIGUE GUIDE - Design and Details, American Institute of Steel Construction, New York, N.Y., 1977.

24. Fisher, J. W., Yen, B. T. and Daniels, J. H. FATIGUE DAMAGE IN THE LEHIGH CANAL BRIDGE FROM DISPLACEMENT INDUCED SECONDARY STRESSES, Transportation Research Record 607, TRB, 1977.

25. Inukai, G. J., Yen, B. T. and Fisher, J. W. STRESS HISTORY OF A CURVED BOX GIRDER, Fritz Engineering Labora­tory Report No. 386.8, Lehigh University, Bethlehem, Pa., 1977.

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