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UILU-ENG-73-2006

e • .l" CIVIL ENGINEERING STUDIES STRUCTURAL RESEARCH SERIES NO. 391 Illinois Cooperative Highway Research Program

Series No. 138

EFFECTS OF DIAPHRAGMS IN CONTINUOUS SLAB AND GIRDER HIGHWAY BRIDGES

Katz-Referenoe Ro Civil Engineering Bl0'6 C. E. Buildi .. Univarsity of 11 __ ~~_,~ Urb&~a~~ Illino1a~

By A. Y. C. WONG

W. L. GAMBLE

Issued as a Documentation Report on

The Field Investigation of Prestressed

Reinforced Concrete Highway Bridges

Project IHR-93

Phase 1

Illinois Cooperative Highway Research Program

Conducted by

THE STRUCTURAL RESEARCH LABORATORY

DEPARTMENT OF CIVIL ENGINEERING

ENGINEERING EXPERIMENT STATION

UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN

in cooperation with the

STATE OF ILLINOIS

DEPARTMENT OF TRANSPORTATION

and the

U.S. DEPARTMENT OF TRANSPORTATION

FEDERAL HIGHWAY ADMINISTRATION

UNIVERSITY OF ILLINOIS

URBANA, ILLINOIS

MAY 1973

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1. Repor' No. '2. Go .... rnm.nt Acce •• lon No.

UI LU-ENG .... 73-2006 4. Title and Subtitle

Effects of Diaphragms in Continuous Slab and Girder Highway Bridges

7. Author( 5)

A. Y. C. Wong and W. L. Gamble

9. Performing Organixation Name and Address

Department of Civil Engineering University of Illinois 2209 Civil Engineering Building Urbana, Illinois 61801

12. Sponsoring Agoncy Name and Addro.---------1

Illinois Department of Transportation 2300 South 31st Street Springfield, Illinois 62764

15. Supplementary Not ••

TECHNICAL REPORT STANDARD TITLE PAGE

3. Roclplent'. Catalog No.

s. Report D.'. May 1973

6. P.rfo""lng Organization Coo.

8. P.rforming Organixation Report No.

SRS 391 ICHRP No. 138

10. Work Unit No.

11. Contract or Grant No.

IHR-93 13. Typ. of R.port and Period Covered

Interim Progress 14. Spon.oring Agency Cod.

Prepared in Cooperation with the U. S. Department of Transportation, Federal Highway Administration

16. Abstract

The results of a study of the effects of diaphragms on load distribution of continuous, straight, right slab and girder highway bridges are presented and di scussed. The parameters for the study are·: the re lati ve gi rder stiffness, H; the ratio of the girder spacing to span, b/a; the beam spacing, b; and the position and the stiffness of the diaphragms. Load distribution characteristics of bridges subjected to a single load, 4-wheel loadings aligned in the transverse direction and two truck loadings were studied.

In general, the criterion for comparison is the maximum moments, both positive and negative, in the beams, due to 4-wheel loadings. The variations of the maximum moments with diaphragm stiffness were studied, and the conclusions of this investigation were based on the analyses of a three-span continuous bridge and various two-span continuous bridges.

Although diaphragms may improve the load distribution characteristics of some bridges which have a large bja ratio, the usefulness of diaphragms is minimal and they are harmful in most cases. On the ground of cost effectiveness, it is recommended that diaphragms not be installed in highway bridges. The comparison of the design moment coefficients as obtained in this investigation and the AASHO recommended design values shows that the AASHO design coefficients are unconservative in many cases and unduly con~ervative in others.

11. OJ.trl""tlon Statemont

Highway bridges, Analysis, Influence lines, Beams, Moments, Diaphragms, 4-Wheel loadings, Truck loadings, AASHO ~pecifications

Release Unlimited

19. Securl ty Ciani f. (of thi. roport) 20. Socurlty Ciani f. (of thl a p ... ) 21. No. of Pogo.

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ACKNOWLEDGLEMENTS

This report was prepared as a part of the Illinois Cooperative

Highway Research Program, Project IHR-93, "Field Investigation of Pre

stressed Reinforced Concrete Highway Bridges," by the Department of Civil

Engineering, in the Engineering Experiment Station, University of Illinois

at Urbana-Champaign, in cooperation with the Illinois Department of Trans

portation and the U. S. Department of Transportation, Federal Highway Ad-

mi ni strati on.

The contents of this report reflect the views of the authors who

are responsible for the facts and the accuracy of the data presented herein.

The contents do not necessarily reflect the official views or policies of

the Illinois Department of Transportation or the Federal Highway Administra

tion. This report does not constitute a standard, specification, or regulation.

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Chapter

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TABLE OF CONTENTS

INTRODUCTION .

1 . 1 Genera 1 . 1.2 Previous Studies. . 1.3 Object and Scope of Investigation 1 .4 No ta t ion.

STUDY OF PARAMETERS AND IDEALIZATION OF THE BRIDGE

2.1 Idealization of the Bridge and Its Components

2.2 Study of Parameters·.

METHOD OF ANALYSIS OF BRIDGE.

3.1 Introduction . 3.2 Basic Assumptions. 3.3 Method of Analysis 3.4 Computer Program Information.

PRESENTATION AND DISCUSSION OF RESULTS .

4.1 General Discussion 4.2 Effects of Diaphragm Stiffness and

Location on Distribution of a Point Load in Continuous Bridges.

4.3 Effects of Diaphragm Stiffness on Load Distribution of 4W Loading .

4.4 Effects of Bridge Parameters on Moment Distribution.

4.5 Comparison of the Effects of Diaphragms on Simply Supported and Continuous Bridges

4.6 Effects of Diaphragms on Moments from Truck Loadings.

4.7 Effects of Diaphragms on Three Span Bridge

COMPARISON OF AASHO DESIGN MOMENT COEFFICIENTS WITH THEORETICAL DESIGN MOMENT COEFFICIENTS .

CONCLUSIONS AND RECOMMENDATIONS.

SUMMARY.

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15 15 16 23

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4.1

4.2

4.3

4.4

4.5

4.6

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LIST OF TABLES

Page

COMBINATIONS OF PARAMETERS FOR THE STUDIES OF EFFECTS OF DIAPHRAGMS IN TWO SPAN CONTINUOUS BRIDGES . 54

MAXIMUM POSITIVE MOMENT COEFFICIENTS DUE TO 4W LOADING 55

MAXIMUM NEGATIVE MOMENT COEFFICIENTS DUE TO 4W LOADING 56

EFFECTS OF CONTINUITY ON THE REDUCTION OF MAXIMUM POSITIVE MOMENT COEFFICIENTS IN PERCENT IN BEAMS RELATIVE TO MOMENTS IN SIMPLY SUPPORTED BRIDGES. 57

RATIO OF MAXIMUM POSITIVE MOMENT COEFFICIENTS TO MAXIMUM NEGATIVE MOMENT COEFFICIENTS, 4W LOADING. 58

COMPARISON OF THE EFFECTS OF DIAPHRAGMS ON SIMPLY SUPPORTED BRIDGE AND CONTINUOUS BRIDGES SUBJECTED TO 4W LOADING. 59

THREE SPAN CONTINUOUS BRIDGE H = 20, b/a = 0.1, b = 7 FT MAXIMUM MOMENTS COEFFICIENTS DUE TO 4W LOADING . 60

vii

LIST OF FIGURES

FIGURE Page

2. 1 LAYOUT OF THE BRIDGE . 61

2.2 IDEALIZATION OF THE GIRDER CROSS SECTION 62

2.3 POSITION OF LOADS REPRESENTING TRUCKS 63

3. 1 FORCES AND COUPLES ACTING ON DIAPHRAGM . 64

3.2 FORCES AND DISPLACEMENTS AT CROSS SECTION CONTAINING DIAPHRAGM

~ 65

3.3 LOCATIONS OF SECTIONS CONSIDERED FOR INCLUENCE COEFFICIENTS . . . . 66

3.4 GRAPHS TO SHOW THE CURVE FITTING OF INFLUENCE COORDINATES AT A, B, C, D, E . 68

3.5 EFFECTS OF NUMBER OF HARMONICS ON THE ACCURACY OF THE SOLUTION, BEAM A LOADED 69

3.6 EFFECTS OF NUMBER OF HARMONICS ON THE ACCURACY OF THE SOLUTION, BEAM B LOADED . . 70

3.7 EFFECTS OF NUMBER OF HARMONICS ON THE ACCURACY OF THE SOLUTION, BEAM C LOADED . . 71

4.1 POSITIVE MOMENT ENVELOPES OF GIRDERS DUE TO A 10 KIP CONCENTRATED LOAD MOVING' ALONG THE BRIDGE, ONE DIAPHRAGM AT MIDSPAN OF EACH SPAN: H = 20, b/a = 0.1, b = 8 FT .. . . 72

4.2 POSITIVE MOMENT ENVELOPES OF GIRDERS DUE TO A 10 KIP CONCENTRATED LOAD MOVING ALONG THE BRIDGE, ONE DIAPHRAGM AT 4/10 POINT OF EACH SPAN; H = 20, b/a - 0.1, b = 8 FT . 73

4.3 POSITIVE MOMENT ENVELOPES OF GIRDERS DUE TO A 10 KIP CONCENTRATED LOAD MOVING ALONG THE BRIDGE, TWO DIAPHRAGMS AT 1/3 POINT OF EACH SPAN OF THE BRIDGE; H = 20, b/a = 0.1, b = 8 FT. ' . 74

FIGURE

4.4

4.5

4.7

4.8

4.9

4.10

4. 11

4.12

4013

4. 14

viii

INFLUENCE LINES FOR NEGATIVE MOMENT AT SUPPORT OF GIRDERS DUE TO A 10 KIP CONCENTRATED LOAD MOVING ALONG THE BRIDGE, ONE DIAPHRAGM AT MIDSPAN OF EACH SPAN; H = 20, b/a = 0.1, b = 8 FT .

INFLUENCE LINES FOR NEGATIVE MOMENT AT SUPPORT OF GIRDERS DUE TO A 10 KIP CONCENTRATED LOAD MOVING ALONG THE BRIDGE, ONE DIAPHRAGM AT 4/10 POINT OF EACH SPAN; H = 20, b/a = 0.1, b = 8 FT .

INFLUENCE LINES FOR NEGATIVE MOMENT AT SUPPORT OF GIRDERS DUE TO A 10 KIP CONCENTRATED LOAD MOVING ALONG THE BRIDGE, TWO DIAPHRAGMS AT 1/3 POINT OF EACH SPAN; H = 20, b/a = 0.1, b = 8 FT .

MAXIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 20~ bla = 0.1, b = 8 FT.

MAXIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 5, b/a ~ 0.05 "

MAXIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 5, bl a :::: 0, 1

MAXIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 5, b/a = 0.2

MAXIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 10, b/a = 0.05.

MAXIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 10, b/a = 0.1 .

MAXIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 10, b/a = 0.2 .

MAXIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 20, b/a = 0.05.

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FIGURE

4.15

4.16

4.17

4.18

4.19

4.20

4.21

4.22

4.23

4.24

t~XIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM SITFFNESS; H = 20~ b/a = 0.1

MAXIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 20, b;a = 0.2 .

MAXIMUM NEGATIVE MOMENT COEFFICIENTS AT MIDSPAN OF UNLOADED SPAN DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS

INFLUENCE LINES FOR MOMENTS AT VARIOUS SECTIONS ALONG TWO SPAN CO~ITINUOUS BRIDGE DUE TO 4W LOADING; H = 20, b I a ;:. O. 1, b = 7 FT

POSITIVE MOMENT ENVELOPE AND INFLUENCE LINES FOR A TWO SPAN CONTINUOUS BEAM DUE TO A CONCENTRATED LOAD .

RELATIONSHIPS BETWEEN MAXIMUM POSITIVE MOMENTS IN GIRDERS AND RELATIVE GIRDER STIFFNESS~ H; NO DIAPHRAGM, 4W LOADING, b = 5 FT .

RELATIONSHIPS BET~JEEN MAXIMUM POSITIVE MOMENTS IN GIRDERS AND RELATIVE GIRDER STIFFNESS, H; NO DIAPHRAGM, 4W LOADING, b = 7 FT .

RELATIONSHIPS BETWEEN MAXIMUM POSITIVE MOMENTS IN GIRDERS AND RELATIVE GIRDER STIFFNESS s H; NO DIAPHRAGM, 4W LOADING, b = 9 FT .

RELATIONSHIPS BETWEEN MAXIMUM NEGATIVE MOMENTS IN GIRDERS AND RELATIVE GIRDER S11 FFNESS ~ H; NO DIAPHRAGfV!, 411J LOADI NG , b = 5 FT .

RELATIONSHIPS BETWEEN MAXIMUM NEGATIVE MOMENTS IN GIRDERS AND RELATIVE GIRDER STIFFNESS, H; NO DIAPHRAGM, 4~J LOADING, b = 7 FT .

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104

105

106

107

108

109

FIGURE

4.25

4.26

4.27

4.28

4.29

4.30

5. 1

5.2

5.3

5.4

x

RELATIONSHIPS BETWEEN MAXIMUM NEGATIVE MOMENTS IN GIRDERS AND RELATIVE GIRDER STIFFNESS, H; NO DIAPHRAGM, 4tN LOADING, b = 9 FT. c

MAXIMUM MOMENTS DUE TO TWO-TRUCK LOADING VERSUS DIAPHRAGM SITFFNESS; H = 20, b/a = 0.2.

MAXIMUM MOMENTS IN GIRDERS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; THREE SPAN CONTINUOUS BRIDGE, H = 20, b/a = 0.1, b = 7 FT .

INFLUENCE LINES FOR NEGATIVE MOMENTS AT INTERIOR SUPPORT OF THREE SPAN CONTINUOUS BRIDGE DUE TO 4W LOADING; H = 20, b/a = 0.1, b = 7 FT.

INFLUENCE LINES FOR MIDSPAN POSITIVE MOMENTS IN GIRDERS OF THREE SPAN CONTINUOUS BRIDGE DUE TO 4W LOADING; H = 20, b/a = 0~1, b = 7 FT. 0

POSITIVE MOMENT ENVELOPES AND NEGATIVE MOMENT INFLUENCE LINES FOR THREE SPAN CONTINUOUS BRIDGES WITH VARIOUS DIAPHRAGM STIFFNESS

MAXIMUM POSITIVE MOMENTS DUE TO 4W LOADING COMPARED WITH AASHO DESIGN VALUES; BEAM A .

MAXIMUM POSITIVE MOMENTS DUE TO 4W LOADING COMPARED WITH AASHO DESIGN VALUES; INTERIOR BEAMS ~

MAXIMUM NEGATIVE MOMENTS DUE TO 4W LOADING COMPARED WITH AASHO DESIGN VALUES; BEAM A .

MAXIMUM NEGATIVE MOMENTS DUE TO 4W LOADING COMPARED WITH AASHO DESIGN VALUES; INTERIOR BEAMS .

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Chapter 1

INTRODUCTION

1 . 1 Genera 1

In Illinois, the, most commonly,built highway bridges are slab and

girder bridges. The girders are either steel girders or prestressed concrete

girders. The slabs are cast-in-place reinforced concrete slabs. Composite

action between the slabs and girders is assured by the shear connectors on

the top of the girders.

During the last two decades, new techniques and methods have been

proposed and developed to study the load distribution characteristics in

these bridges. With the aid of the electronic computer, more realistic

approximations for the analysis of slab and girder bridges are now practi-

cal. Although many studies on the load distribution of slab and girder

bridges have been published, at the present time, the 1969 AASHO specifi

cations (1) for bridge design still do not reflect the state of the art of

the increasingly advanced structural analysis. Especially, the guidelines

for the installation of diaphragms in bridges as recommended in the AASHO

specifications are quite arbitrary and the problem of where and what kind of

diaphragm that should be installed in a bridge is mostly up to the dfscretion

of the engineers. Furthermore, the effects of the diaphragms are not accounted

for in the proportioning of the girders. Thus, the question ~hether, in all

senses of practi cal i ty, di aphragms are needed in bri dges' needs further i n

depth study.

1.2 Previous Studies

Various analytical methods have been used to analyze load

2

distributions in highway bridges. Assumptions have been made to simplify

the problem in order to get a more manageable solution. The slab and girder

bridges can be looked upon as a plate structure stiffened by the girders.

Along this line of thought, orthotropic plate theory is used to solve this

class of problem. The work was initiated by Guyon (2) and subsequently

improved and modified by Massonnet (3) to include all ranges of torsional

stiffness of the girders. Both Guyon and Massonnet assumed that the Poisson1s

ratio for the equivalent plate material to be zero. Rowe (4) further improved

the analytical technique to allow for any value of Poisson1s ratio. The

idealization of a slab and girder bridge as an orthotropic plate can only be

justified if the girders are placed close together. For most of the highway

bridges, the girder spacings are not close enough for the bridge to be satis

factorily considered as an orthotropic plate. In order to find a closed. form

solution for the slab-girder-diaphragm system, Dean (5) proposed an exact

macro-discrete field approach to solve this class of problem, but his method

is too complicated to use.

In the area of numerical approximations, Newmark's moment distribu-

tion' procedure (6) is readily adaptable to slab and girder bridges. The

who 1 e slab 'of the bri dge is cons i dered. to be an i sotropi c p 1 a te and is i nde-

pendent of the supporting beams, and the beams are treated as interior. sup~

ports. Knowing the boundary conditions of the plate and the loading condi

tion, the resultant stresses.in.-the plate are solved for by the Levy solution.

Then a slab strip is considered. to. be a continuous beam over' flexible supports

and a moment distribution.is.carried out for each harmonic to determine the

moments in the beams. Thetotal:of.moments in the slab and girders is the

sum of the moments due to each. harmonic. In this method, the in-plane forces

in the slabs and the contribution of the slab to the stiffness of the T-beam

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girders are neglected, but torsional stiffness of the girders can be taken

into account in this method. In the early forties, Newmark and Siess (7)

analyzed slab and girder bridges using this method and the results of their

analyses were one of the bases of the AASHO specifications on load distri

bution in highway bridges.

The finite element method (8) is a powerful tool for the analysis

of slab and girder bridges. Analyses of slab and girder bridges with the

finite element method have been carried out by Gustafson (9), using plate

elements which include both flexural and in-plane forces. The only disad

vantage of this method is that a large number of simultaneous equations

have to be solved in order to get an accurate answer. For a long bridge,

this method may become uneconomical.

Lazarides (10) idealized the slab and girder bridge as an equiva

lent grid system and solved the grid work problem by determining the deflec

tion compatibility equations at each beam intersection. This method also

leads to a large system of simultaneous equations and furthermore, the

effects of the twisting moments in the slab cannot be incorporated in an

equivalent grid system.

By combining the Goldberg and Leve folded plate theory and the

two-dimensional theory of elasticity, Van Horn and Oaryoush (11) included

the effects of in-plane forces and T-beam action in the analysis of slab

and girder bridges. Sithichaikasem (12) went one step further by including

the torsional and warping stiffness of the girders in the analysis. The

model of Van Horn, Oaryoush, and Sithichaikasem is much more realistic

but they needed a large, fast electronic computer to make their solution

process feasible.

4

The effects of diaphragms on load distribution of slab and girder

bridges were studied by B.C.F. Wei (13), Siess and Ve1etsos (14), and

Sithichaikasem (12). The studies of Wei, Siess and Ve1etsos neglected the

effects of torsional stiffness of the girders and they all used Newmark's

moment distribution procedure. The investigation by Sithichaikasem in-

c1uded the effects of the torsional sitffness and warping stiffness of the

girders and the effects of in-plane forces in the slab.

The analyses as mentioned above were all performed on simply sup

ported bridges. Much. of the present design criteria on load distribution are

based on the results of the analyses of simply supported bridges and pro-

vision for the design of the negative moment regions is inferred from the

behavior of the positive moments. Since most highway bridges are continuous

bridges, analysis on the effects of diaphragms on continuous bridges will·

undoubtedly provide new data and supplement the data on the design of slab

and girder bridges.

1.3 Object and Scope of Investigation

The object of this investigation is to study the effects of dia-

phragms on load distribution characteristics on continuous slab and girder

highway bridges. The variation of maximum positive and negative moments

as a function of diaphragm stiffness and location will be studied. The

effects of continuity will be investigated. The range of bridge parameters

under inves.tigation will be such that they will adequately cover the range

of the highway bridges built. The results of the analyses will be com

pared to that of the recommended design procedure as stated in 1969 AASHO

specifications, Finally, recommendations on the use of diaphragms will be

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made, based on the results of this investigation. The scope of the studies

can be stated as follows:

1 0 To study the effect of number of harmonics used as to the

accur~cy of the solution.

2. To find the best locations of diaphragms that will improve

load distribution for a point load.

3. To study the effects of diaphragm stiffness on load distri

bution-of point load.-

4. For various bridge parameters (i.e., H, bfa, b) and diaphragm

stiffness, k, the effects of diaphragm stiffness on load

distribution of 4 wheel loads at various positions along the

bridges will be studied.

5. To study the vari a ti on of the i nfl uence 1 i nes due to 4 wheel

load as a function of the diaphragm stiffness.

6, To compare the load distribution character.istics of simply

supported bridges and continuous bridges.

7. To compare the present AASHO recommended design moment coef

ficients with the moment coefficients as calculated in this

investigation.

8. To study the effects of diaphragms on moments from truck

loadings in continuous slab and girder bridges.

9. Recommendations on the use of diaphragms on highway bridges

will be made based on the results of the analyses.

6

It was recommended in Ref. 12 and will be recommended in this

report that interior diaphragms be eliminated from most prestressed I-beam

bridges unless they are required for erection purposes. One of the

practical arguments that has been raised whenever this change has been

proposed in the past is the feeling that diaphragms help limit damage

to an overpass structure which is struck transversely from below by

an oversized load.

There appears to be conflicting evidence as to whether the

diaphragms are damage limiting or damage spreading members, and the

only comment the authors would make at this time is that the diaphragms

currently being used in bridges are probably the wrong shape and size,

and are usually in the wrong locations, if one of their valid functions

is the reduction of damage to the structure due to horizontal impact

on the side of the bridge. The analyses reported here are not relevant

to this particular question.

1 . 4 No ta t ion

The following notation is used throughout this study:

a

b

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d

h

length of one span of the bridge

beam spacing

aspect ratio

di stance between mi d-depths of top and bottom fl anges (see Fig. 2.2)

thickness of the slab

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ratio of the stiffness of the diaphragm to that of the girder

constants for calculating J

thickness of the bottom flange, top flange and web of the idealized cross section of the girder, respectively

Poisson1s ratio

warping constant of the girder

moment coefficient = Mowent a

stiffness of an element of the slab

elastic modulus of the material of the diaphragm

elastic modulus of the material of the girder

elastic modulus of the material of the slab

relative flexibility and absolute flexibility.matrices for the b·ri dges

flexibility matrices for the plate, beam, diaphragm and an element X

shear modulus of the material of the girder

ratio of the stiffness of the girder to that of the slab

moment of inertia of the girder with a composite slab

moment of inertia per unit width of the cross section of the slab

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moments, of inertia of top and bottom flanges, respectlvely

torsional constant of the modified cross section of girder, analogous to the polar moment of inertia of a circular cross section

single wheel load or a single concentrated load

ratio of warping stiffness to torsional stiffness of

the girder

vector of internal forces at the junction of one diaphragm and girder, or the interior supports reactions

vector of internal forces at· section i of the primary structure due to loads at locations j

vector of internal forces at section i of the primary

structure due to unit loads at sections n

ratio of torsional stiffness of the girder to the

flexural stiffness of the girder

vector of relative displacements at the diaphragm locations of absolute displacements at the location

of interior supports

C - C mk=0.05 mk=O.O x 100, percentage change in moment

Cmk=O.O coefficients due to addition of diaphragms with k = 0.05

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Chapter 2

STUDY OF PARAMETERS AND IDEALIZATION OF THE BRIDGE

2.1 Idealization of the Bridge and Its Components

In the analysis, the bridge is idealized as shown in Fig. 2.1.

The prestressed girders are idealized as being made up of rectangular sec

tions, as shown in Fig. 2.2. The dimensions, which are the same as the real

girder, are: the width of the top flange, the thickness of the slab, the

depth of the girder, the moment inertia of the girder, and the centroid of

the section. The sections used are based on the set of standard prestressed

concrete girders developed by the Bureau of Public Roads. The properties

of these girders are described in a Portland Cement Association Bulletin,

"Concrete I nforma ti on II (15).

202 Study of Parameters

The parameters that may affect the load distribution of a bridge

can be divided into five main categories:

1. Material properties of the slabs and girders;

2. Relative dimensions of the girders and slabs;

3. Geometry of the bridge;

4. Type of loading on the bridge;

5. Location and stiffness of diaphragms and the location of

supports.

2.20' Material Properties

In the design of highway bridges, the elastic modulus of the slab

10

concrete is usually taken as somewhat less than that of the girder concrete,

since the slab concrete design strength is usually less than that of the

precast girders. The test bridge at Tuscola, Illinois (16), had shown

that the elastic modulus of the deck concrete was approximately the same as

that of the girder concrete. Because of the uncertainty in concrete prop

erties, the elastic modulus of both the slab and girders is taken as

4,000,000 lb/in. 2 in the analysis. Poisson's ratio is also assumed to be

0.15 for both the slab and girders.

2.2.2 Geometry of the Bridge

In this analysis, only straight, right, continuous bridges are

considered. The symbols are defined when they are first used and in Sec-

tion 1.4. The parameter b/a is defined as the girder spacing, b, divided

by the length of one span, a, as .shown in Fig. 2.1. Unless explicitly

specified otherwise, the bridge is a five beam, two span continuous bridge

with equal spans, The range of b/a varies from 0.05 to 0.2, and this will

adequately cover the range of highway bridges that are usually built. A

small b/a ratio means a relatively long bridge and a large bja ratio means

a relatively short one. The range of beam spacing, b, varies from 5 to

9 ft. This also is within the practical range of highway bridge design.

The range of the parameters is shown in Table 2.1.

2.2.3 Properties of the Girders and Slabs

The parameters that are under investigation are H, T, and Q. They

are defined as follows:

r---' j -

r

[

I' t. __

I .. [

[

(.-

,.-----I !

r- -t ;

1 -1

L_

d

h

v

c =

E g

G

E I H = --.9-.9. aO

11

length of one span of the bridge

di stance betweer. mi d-depths of top and bottom fl anges

(see Fig. 2.2)

thickness of the slab

constants for calculating J

thickness of the bottom flange, top flange and web of the idealized cross section of the girder, res

pectively

Poisson's ratio

warping constant of the girder

stiffness of an element of the slab

elastic modulus of the material of the girder

elastic modulus of the material of the slab

Shear modulus of the material of the girder

ratio of the stiffness of the girder to that of the slab

moments of inertia of top and bottom flanges, re

spectively

moment of inertia of the girder with a composite slab

h3 I =s 12

J

7T2E C

Q = ----==--9_ 2G -a J

12

moment of inertia per unit width of the cross section

of the slab

torsional constant of the modified cross section of

girder, analogous to the polar moment of inertia of a

circular cross section

ratio of warping stiffness to torsional stiffness of

the girder

ratio of torsional stiffness of the girder to the flexural stiffness of the girder

The effects of T and Q on load distribution of a simply supported

bridge was studied by Sithichaikasem (12) and Van Horn (11), and the results

of the studies showed that the parameter Q has no influence on the load dis

tribution characteristics. of a bridge. The study also showed that for

typical values of T for prestressed concrete I-section girders, the.effect.

is small unless the value of T approaches that of a box girder section.

Thus, in this investigation, Q is taken to be zero and T is the value cal-

culated for the idealized section of the girder. The idealized section is

shown on Fig" 2.2

H is a measure of relative stiffness of the slab and girder; the

larger the value of H, the greater is the girder stiffness relative to that

of the slab. The range ofH is varied from 5 to 20. The details of the

range of parameters are shown in Table 2.1.

2.2.4 Types of Loading on the Bridge

A wheel loading on the bridge is idealized as a point load. In

the analysis, the point loading is represented by a series, and because of

f" ".,

~

i .

r [

"

[""

rr.

[ r:f L

[ ,'

"

[

L. r r -i L ...

L

13

the behavior of the Fourier series, the "point load" will not be exactly

a point load, but rather is a concentrated load slightly spread out about.

the point of application. This will in a way compensate for the ideali

zation of a wheel load as a point load. The effect of multiple loading

can be analyzed by the principle of superposition since only the.e1astic

behavior of the. structure is considered. The analysis is conducted by

first analyzing to find the effects of a point load, and then the effects

of four wheel loads aligned across the bridge in the transverse direction,

with the spacing as shown in Fig. 2.3a, are investigated. The effects of

two trucks running in. the same direction~ representing AASHO HS loadings,

are also studied. The details of the loading and the wheel spacing of the

loads are shown in Fig. 2.3bn

2.2.5 Location and Stiffness of Diaphragms and Location of Supports

The location, stiffness, and number of diaphragms that may im

prove the load distribution within a bridge will be studied. Although it

has been shown that a relatively flexible diaphragm at midspan of a simply

supported bridge may improve the load distribution to some extent (12),

the effects of diaphragms in continuous bridges have not previously been

studied. In this report, the effects of diaphragms at the 5/10 point,

4/10 point, and two diaphragms at the 1/3 point of the spans, are con

sidered. The relative stiffnesses of the diaphragm to girder considered

will be zero, 0.05, 0.10, 0.20, 0040, and 1000.0. From the study of the

effects of the location of the diaphragms, an optimum location is selected.

The effects of this optimally located diaphragm on load distribution for

various bridges are studied in greater detail.

14

In.order. to verify-that the results obtained by analyzing a two

span bridge can also be applied to a mu1tispan bridge, a three span con~

tinuous bridge with H= 20, b/a = O~l, and b = 7 ft will be.analyzed.

A comparison of the results of the analyses of both simply supported and

continuous bridges will be made.

r L.

[

r [

U t;;.

[

[ .. t·'.·

r .... f ! L.

i ! L

15

Chapter 3

METHOD OF ANALYSIS OF BRIDGE

3. 1 I ntroducti on

A slab-girder bridge can be idealized as an assemblage of plate

and girder elements interconnected at longitudinal joints. Various methods

.have been developed to analyze this kind of system, as mentioned in Chapter 1.

Of all the methods of analysis, Fourier harmonic analysis is the best for

this kind of structure in terms of precision and computational effort. By

representing the loading and the internal forces in the structure by series

in one direction, it has the advantage of transforming a two-dimensional

problem into a one-dimensional one~ which greatly reduces the size.of the

problem. The only limitation of. this method is that the structure must be

straight and the ends must be simply supported. Fortunately, the above

conditions are satisfied by most highway bridges.

3.2 Basic Assumptions

The basic assumptions of the analysis are those of the ordinary

theory of plate flexure and two-dimensional theory of elasticity and the

following:

1. Diaphragms are installed at all supports. The support dia

phragms are perfectly rigid in their own planes but are free

to rotate in the direction normal to its plane.

2. Adequate shear connectors are provided to insure full com-

posite action of the girder and slab.

16

3. Spacing of the girders are the same for all girders in a

bri dge. '

4, Shear deformations of the diaphragms and girders are neg-

1 ected. '

5. Internal supports are nonyie1ding.

3.3 Method of Analysis

The method of analysis is essentially a force method of analysis

based on the Goldberg and Leve (17) prismatic folded plate theory and har-

monic analysis. The external load, displacements and internal forces are

all expressed in Fourier series components and the structure is solved

for the internal forces due to the applied external load. Once the ana1~

ysis of a loading has been developed for a particular harmonic, the total

effects of the load can be obtained by summing up all the harmonics con-

sidered. The number of harmonics needed to obtain an acceptable solution

will be discussed later~

The problem of load distribution of a multispan bridge subjected

to mu1tiwheel loadings can be solved in four steps.

3.3.1 Analysis of the Primary Structure

The flexibility matrix of the plate and the beam elements are

derived from the theory of elasticity and classical plate flexure theory.

The detailed equations for the plate and beam flexibility matrix were

reported in Refs. 11 and l2~ By considering the compatibility at each

joint, we can solve for the internal forces acting at each joint as follows:

is the flexibility of plate element

r-~'"

I i-. ~

r I "

1 I

l..

[

I r it '

.'-::'

17

FG is the flexibility of the beam element

Lxt ' Lxr are the displacement vectors at the left and right

edges, respectively, of element x due to applied

,external load on that element

are the vectors. of internal forces at the left and

right edges of element x

The flexibility matrix can be partitioned such that for an element x

F = x

F F xtt xtr F F xrt xrr

At joint N, the right edge of the plate element is connected to the left

edge of the beam element, the displacement of the right edge (i.e., r edge)

of plate element j is

and the displacement of the left edge of beam element i is

For compatibility of joint N, the displacement at the 'right edge of element

i must equal the displacement at the left edge of element j. Observing

we can write

and that

18

where PN is the internal force vector at joint N.

By applying the above condition for all the joints we have

* * * F p = L

where

* F is the assembled flexibility matrix of the structure

* p is the force vector at all joints

* L is the applied load flexibility vector

Solving the above matrix equations, the internal forces at the joints are

found. Internal forces in the slabs and girders can be obtained by sub

stituting the joint forces into the. equilibrium equations of the individual

element. In this analysis~ there is no way to calculate the effective

width of the I-section girder~ In order to get the composite moment of the

girder, the condition that there is no axial force in a composite beam

under pure bending is used (i.e., J adA = 0). The moment in the girder

which satisfies the above condition is taken as the composite moment of

the girder.

3.3e2 Analysis of the Effects of Diaphragms and Interior Supports

The effects of diaphragms and interior supports are such that

vertical reactive force and twisting moment are developed at the junctions

of the girders and diaphragms. The diaphragms are idealized as being

cross beams simply supported by the exterior girders. The flexibility of

r I l i ..

f;.~ (2

... , .... :

[ " [ .,

-'

19

the diaphragm is obtained by the conjugate beam method. Unit loads are

placed at B, C, 0; unit couples are applied at A, B, C, 0, and E (see

Fig. 3.1); and the flexibility coefficients are the displacements at points

A, B, C, D, and E due to the unit loads. Two equations of equilibrium

are also needed, i.e.,

}F = 0 ~ z

IM = 0 x

to account for the missing unit load at A and E. The flexibility matrix

of the primary structure at the location of the diaphragms or the interior

supports is obtained from solutions of unit loads applied on the primary

structure. To calculate the relative flexibility of the structure at the

location of the diaphragms and the absolute flexibility at the interior

support, unit loads and couples are placed at A, B, C, 0, and E consecu~

tively at each diaphragm and interior support location. The relative dis-

placements due to unit loads and couples are calculated with respect to

the line joining ~he deflected points of A and E (see Fig. 3.2). When the

section considered is an interior support, the absolute displacements of

the structure at the interior support section are used to generate the

flexibility coefficients.

The compatibility equations for the diaphragms and supports are

as follows:

At the sections where the diaphragms join the girders:

20

At the interior supports:

where

* FB R + 6. = 0

FO is the flexibility matrix of the diaphragm

FB is the relative flexibility of the structure at the dia

phragm locations

* Fg is the absolute flexibility of the structure at the interior

support locations

6. is the relative displacement vector at the sections of dia

phragms or the absolute displacement vector at the interior

support section due to applied external loads

R is the reactive force vector at the junction of one diaphragm

or interior support and the girders

The solution to the above equations will give the reactive forces at the

diaphragm and support locations.

If S". is the i nter'na 1 force vector at secti on i of the primary lJ

structure due to loads at location j, and S. is the internal force vector ln of section i of the primary structure dut to unit loads at sections n, then

the internal forces at section i of the continuous bridge with diaphragms

* wi 11 be S .. , where lJ

* S .. lJ

K = S.O + I s. R

1 J n=l 1 n n

and

K = sum of the number of sections with diaphragms and interior

supports

r-j

... , i 1·;

r L.

r~'

[

I, ~,,: .,

1-

1 .~ -. ...

21

For a two span continuous bridge with five girders, fifteen unit

load loadings and fifteen unit couple loadings are needed to generate the

flexibility matrix of the structure if there is one diaphragm in each

span. If symmetry is considered, only six unit load loadings and six unit

couple loadings are needed.

3.3.3 Determination of the Influence Lines for Four Wheel Loadings

Concentrated loads are placed at various locations on the beams

of the bridges as shown in Fig, 3.3, and an influence surface could be

generated. Instead of using the influence surfaces to find the critical

moments in the beams due to a system of applied loads, influence lines

for the moments in the beams due to a four wheel loading moving along the

bridge are used. The choice of the four wheel loading is to simulate the

loading of single axles of two trucks going in the same direction along

the bridge. A previous study (12) has shown that two lane loadings always

produce maximum moments in girders, if the reduction factors for multilane

loadings given in the AASHO specifications are used. The axles are aligned

in the transverse direction, The spacing of the wheels is in accord with

the AASHO recommendatinns (see Fig. 2.3)0

Then the problem is to locate the position of the wheels in the

transverse direction which will give a maximum moment for a beam at a

section. The maximum moment in each beam at section i due to a four wheel

loading (4W loading) at section j can be found by searching for the maxi

mum moment induced in a beam as the 4W loading moves from one edge to the

other edge of the bridge. Moments at section in each beam due to unit

loads applied at A, B, C, D, and E at section j are first calculated and

22

a curve is fitted through the five data points, giving the influence lines

for the moment in a beam at section i due to a unit load moving trans-

versely at section jo The effect of a 4W loading is the sum of the co

ordinates of the influence line at the positions of the loads multiplied

by a scalar constant (i.e., the magnitude of the load).

After some trial and error, it was found that the best fit curve

for the influence line for beam A is either a fifth degree polynomial or

three piecewise smooth parabolas connecting A, B, C; B, C, D; and C, D, E.

The three piecewise smooth parabolas were used with the average value ·of

the two curves being used between B, C, and C, D. For beam B, the best

fit curve is three piecewise smooth parabolas as in beam A. For beam C,

a seventh degree polynomial is used. The results of the curve fitting are

shown on Fig. 3.4. The points .at the beams, A through E, are the given

data points, and the intermediate points are those calculated in the curve

fitting process, The curves show good agreement with the value when the

load is applied on the slaboNote that there is a kink in the curve for

beam C between A Band D E when k = 0.0. This kink may give a small error

for the total moment due to a 4W loading, but the error is small because

of the large influence coefficient at C and the wheels will be in the

regions between Band D at maximum moment because of symmetry 0

30304 Determination of Effects of Truck Loadings

To get the absolute maximum moment in the beams, theoretically,

the trucks have to wiggle along the bridge. However, it has been assumed

that the trucks will move along straight lines parallel to the girders in

the bridge. The result of preliminary analyses showed that the absolute

....... r. i .

I .~

r

I?~

L

(.:.

I: [

[~.

L r- -1

L r r' r

L ..

23

maximum moment in a beam.due-to a.4W loading will be when the 4Wloading

is at or near,the.4/l0point.of-onespan.from the simply supported.end •.

For the purpose.of.finding,the.effect of truck loadings,.the transverse

position of the:wheel .ts .. assumed .. to be:that which will induce .amaximum.

positive moment at.the~4!10~point~-With the distance of the wheel from

the edge beam .. thus-.fixed, we can proceed to find the maximum.moment~i·n

the beams due to-the~two.truGk-loading. Again, having the influence co

ordinates for.a.beam.at a.particular section, a curve is fitted to pass

through all the points .. The.Lagrangian interpolation function (18) is

used to fit the data.points.and.the moment due to a two truck loading is

the sum of the influence coordinates at the position of the wheel multi-

plied by the respective fractions of-weight in each line of wheel. For

example, for a HS2Q-loading.the-front axles will h~ve a scalar factor of

0.25 and the real axles will have a scalar factor of 1.0.

3.4 Computer Program Information

3.4. 1 General Programming Considerations

Four computer.prQg~ams.were.developed to analyze the ,bridgeo --The·

first program,which ,handles the analysis,of the primary structure was-a

modification of the-program.developed.by.Sithichaikasem (12). Extensive

use of secondary. storage. l.S needed. to store the data generated during the

analysis of the prima~y .structure.. The second program is used to add in

the effect of supports and-diaphragms ,and-to do the back substitution. The

third program takes.care of the curve fitting process and finds the loca

tions of the wheels to .give the·maximum moments for 4W loadings and two·

truck loadings. The input of the third program is the output of the second

24

program and no rearrangement of data is needed for the input of the third

program. The fourth program plots the influence lines for sections 2

through 9 (designated in Figo 303) and punches the influence coefficients

on da ta ca r'ds.

304.2 Convergence Behavior of the Method of Computation

In order to find the effect of the number of harmonics on the

accuracy of the solutions, a simply supported bridge with H = 20, bla =

0.05, and b = 8 ft which was subjected to three loading cases was studied.

The loadings were

10 A 10 kip concentrated load at midspan of beam A

2. A 10 kip concentrated load at midspan of beam B

30 A 10 kip concentrated load at midspan of beam C

The results of the analysis are shown in Figs, 3.5 to 3.7. The total com-

posite moment of all beams is compared with the total static moment as

calculated from the elementary beam theory. It 1S noted that by using just

one harmonic the total moment at midspan is already 82 percent of the

static moment, and by using 5 harmonics 93 percent of the total static

moment is obtained. The rate of convergence is much smaller after using

5 harmonics, and at 35 harmonics 97 percent of the total static moment

is distributed among all the beams 0 The figures show that regardless of

the loading condition, the unloaded beams always converge to a constant

moment after a few harmonics but the moments in the loaded beam continue

to increaseo Of the 3 percent of the total static moment that is unac

counted for at 35 harmonics, 1 to 2 percent of the total moment may be

the slab moment (7) and the other 1 percent may p~obably belong to the

moment of the loaded beamo

. .. ~. [-,.

r r L ...

[

[

[ r~ 7 I

'" ! :I. iL ....

25

Chapter 4

PRESENTATION AND DISCUSSION OF RESULTS

4.1 General Discussion

The main objective of this investigation is to study the effects

of diaphragms on the load distribution characteristics of continuous slab-

girder bridges. The effects of diaphragms on" load distribution of simply

supported bridges were investigated by Veletsos and Siess (14), Wei (13),

and Sithichaikasem (12), and others (19). It is difficult to make a

direct comparison between the results of an analysis of a continuous

bridge and a simply supported one because of the change of the "effective"

span length due to the negative moment at the interior support. The

studies of Veletsos and Siess, Wei, and Sithichaikasem paved the way and

brought out the significant parameters that influence the load distribu

tion characteristics of the bridgeso Most of the attention will be fo-

cused on the moments in the beams since the magnitudes of the beam

moments are the governing parameter which control the design of the ,

girders. In addition to the effects of the diaphragms, the influences of

the bridge parameters, H, b/a and b on moment distributions were also

studied. The results are presented as follows:

1. Effects of diaphragms stiffness on distribution of a point

load in continuous bridges,

2. Effects of location of diaphragms on load distribution,

3. Effects of diaphragms stiffness on load distribution of

a 4W (4-wheel) loading,

26

40 Effects of bridge parameters on moment distribution,

5. Comparison of the effects of diaphragms for a simply supported

bridge and a two span continuous bridge,

6. Effects of diaphragms on moments from truck loadings, and

7. Effects of diaphragms on three-span bridges.

Based on the results of the previous investigation of simply

supported bridges, Ref. 12, the effects of warping were found to be neg

ligible and the effects of torsional stiffness of the beam are important

only when the torsional stiffness of the beam approaches that of a box

section. In this analysis the warping coefficient is taken to be zero

and the torsional stiffness is calculated based on the transformed sec-

tions of the prestressed girders.

4.2 Effects of Diaphragm Stiffness and Location on Distribution of a Point Load in Continuous Bridges

Moment envelopes of beams A, B, C are shown in Fig. 4.1 to 4.3,

respectively for single loads moving along the beams. The influence lines

of moment at the interior supports of beams A, B, C are shown on Fig. 4.4

to Fig. 4.6. The figures show that the diaphragms reduce the moment at

the vicinity of the locations of diaphragm and that a diaphragm is more

effective in moment reduction for the interior beams than for the edge

beam. In all cases, the absolute maximum positive moment is reduced and

the location of the maximum moment is displaced. Considering the relative

diaphragm stiffness of k = 0.4, it seems that for a point loading, the

greatest moment reduction occurs when the diaphragm is at 4/10 point of

the span from the simply supported end. The diaphragms cause less

r .--1 I t _ ..

[

r

E~ .....

. .

[ ,:' t'

r •. J:.

r i ..

27

drastic changes in the negative moments at the interior supports. Fig.

4.4 to 406 show that the reduction in negative moment is approximately

the same for all beams, and it seems that there is no single optimum

location of diaphragm, when the maximum negative moment at the

support is concerned.

The effects of location of diaphragms on load distributions in

simply supported bridges were studies by Sithichaikasem (12). The results

of his study showed that the diaphragms have the greatest effect on the

load distribution at the immediate vicinity of the diaphragms and the

best location of the diaphragm is 'where the maximum moment would occur.

For a simply supported bridge, the optimum location is at the midspan

The effects of adding more than one diaphragm at other locations for a

simply supported bridge were also studied. The results of the analyses

showed that the effects of having one diaphragm at mid-span or two dia

phragms at 5/12 point or two diaphragms at 1/4 points and one at mid-

span are practically the same.

For a continuous bridge, the maximum positive moment will no

longer occur at the midspan but rather the maximum will be somewhere

away from the midspan towards the simply supported end. Three locations

of diaphragms were studied: a) one diaphragm at midspan of each span,

b) one diaphragm at the 4/10 point of each span, and c) one diaphragm

at each 1/3 point of each span.

The positive moment envelopes and the influence lines for nega

tive moments at the interior supports are shown in Fig. 401 to Fig. 4.6.

The moment coefficients, em' due to 4W loadings are shown in Fig. 4.7.

For the maximum positive moment in the beams, the effects of

28

one diaphragm at midspan of each span or two diaphragms at the 1/3 points

of each span are practically the same~

The bridge with one diaphragm at 4/10 point is slightly differ-

ent than the other two in three ways:

1. At k = 0.05, the moments in beams A and B are approximately

the same, whereas for the other two cases, the beam B

moment is greater than the beam A moment,

2. The maximum moment in the exterior girder increases much

more rapidly and becomes the controlling girder once the

relative diaphragm stiffness is greater than 0.05, and

3. At k = 0.4 the difference in moment among the girders are

the greatest and the magnitude of the positive moment in

the exterior girder is the greatest.

The variation of maximum negative moment at the interior support

bears no resemblance to that of the positive moments in the girders. The

best load distribution occurs when the diaphragm in at the 4/10 point

with the k value ranging from 0.1 to 0.4, although no combination of dia-

phragms results in a reduction in negative moment in excess of 7.5 percent.

For the bridge with diaphragms at midspans, the optimum k value is approx

imately 0.10 With 1/3 point diaphragms, the optim~m k is slightly smaller.

It can be concluded that the optimum diaphragm is one at the 4/10

point, with a k value between 0.05 to 0.075~ Such a flexible diaphragm is

rarely practical or realistic because it would have to be either way

shallow or very thin. Host of the diaphragms cast in prestressed concrete

girder highway bridges in Illinois are 8-in. thick and about 0.8 the depth

of the girders, and have k values of approximately 0.3 to 0.4. In view

r [

[ r~:

(;." Ii

[

[

1 L

T ,

29

of this fact, the best choice in lieu of a flexible diaphragm at 4/10

point is a midspan diaphragm, since stiff diaphragm at this point has

the least detrimental influence on load distribution characteristics of

the bridge.

For this particular bridge, no arrangement of diaphragms is

capable of causing a significant reduction in maximum moments, and from

a cost-effectiveness point of view, no diaphragm is undoubedly the opti

mum since a flexible diaphragm will cost about the same as a stiff one.

4.3 Effects of Diaphragm Stiffness on Load Distribution of 4W Loading

Figo 4.8 to Fig. 4.16 show the effects of diaphragm stiffness

for a 4W loading, where the four wheels are spaced as shown in Fig. 2.3.

As discussed in Ref. 14, the influence of diaphragm on load distribution

will be smaller for a multiple load loading than for a point load. Thus,

the change {n moment coefficients of a 4W loading with various diaphragm

stiffness will be small.

For bridges with H = 5, bla = 0.05 and 0.10, beam A is always

the controlling girder regardless of beam spacing, for both positive

and negative moments. For a small beam spacing of 5,ft, the C values m

(em = p~) for the maximum positive moments of beam Band C are fairly

clbse together, but this is not the case for the negative moment at the

support. When b = 5 ft and bla = 0.1, the Cm values at the supports

for beams A and B are the same (Fig4 4.9), in contrast to the distinct

differences in negative moments of beams A, Band C when b/a = 0.05

(Fig. 4.8). With the larger bla ratio of 0.20, the load distribution

30

characteristics are markedly different from that of the previous cases,

as shown in Fig. 4.10. When b = 5 ft, the center b~am, beam C, is th~"

controlling girder while the moment in beam A and B are fai~ly close io-

gethero With the larger beam spacings of 7 ft and 9 ft, the controlling

beam is beam B, but the moment in girder C is only slightly less than

girder B,

When k ~ 000, beam A has the smallest C , but as k incre~i~s, m

the moments in beam A quickly catch up with the moments in beam C and 'are

greater than in beam C at k = 0.40 In any case, for H = 5, bla = 0.2,

and b = 5, 7 or 9 ft, the controlling girder is never the edge girder

and the increase in diaphragm stiffness helps to bring the moments in

the beams closer to the same values~ For the bridges with H = 5, b/a.=

0.05 and 0.10, and b = 5, 7 or 9 ft the controlling girder is always

,the edge girder, beam A, and the increase in diaphragm stiffness only

does more harm than good.

Tables 401 and 4.2 show the moment coefficients as k

changes from zero to 0.4. It is obvious that C for beam C m

always decreases (Ref. 14) and has the largest percentage change.

It seems that for a same H value and bla ratio, the largest changes

in C ,for positive moment, with various k values, occur in bridges with m b = 7.ft. For the case of negative moment c~efficients, the larger the

beam spacing, the greater the change in C as k varies. m

As the relative stiffness of the glrders increases to H = 10

and 20, the behavior of the bridges with bla = 0,05 is approximately the

same as that for H = 5 (Fig. 4.11 and Fig. 4.14). ,For the bridges with

H =, 10.and 20 and b/a·~ 0.10, the controlling girders, when there are

r---f L.~._

f' {

roo

1: t II

31

no diaphragm, are still the same as in the case of H = 5 - beam B or

beam C. But as t~e diaphragm stiffness increases, the moments in beam

A quickly become the controlling moments. The higher the relative

girder stiffness, H, the greater is the value of k needed for beam A to

attain the largest C. The optimum diaphragm stiffness for H = 10 is m

from 0.03 to 0.05, and for H = 20 is from 0.05 to 0.08. The behavior

of the negative moment at the support is approximately the same as that

of the maximum positive moment. As k varies from 0.00 to 0.4, the change

of C is greater than that for H = 10 and 20 than for H = 5, but the m trend is the same with beam C having the greatest change in Cm"

When b/a = 0.2, the bridges with H = 5, 10, and 20 are similar

in that the interior beams are the controlling beams. As H increases,

the difference in moments between the interior and edge girders increases,

as can be seen in the moment values given in Tables 4.1 and 4.2. Even

when k reaches a value of 0.4, girder A still has the smallest value of

C and this is also true for the negative moments at the support. The m optimum diaphragm stiffness for these bridges is k = 0.4 or higher.

So far we are concerned with the maximum moment in the loaded

span or at the support. To complete the picture of the effects of dia

phragm stiffness on a 4W loading, the effects of the load distribution

characteristics in the unloaded span are shown in Fig. 4.17. Comparing

Fig. 4.17 and Fig. 4.8 for the bridge with H = 5, b/a = 0.05 and b = 7

ft, we see that the effects of diaphragms on this bridge, which already

has a good load distribution, are practically nonexistant. However, it

is noted that the maximum negative moment at the mid-span of the unloaded

span is only 42 to 45 percent of the maximum moment at the support instead

32

of the 50 percent value found for a beam.

Compa~ing Fig. 4.12 and Figo 4.16 to Fig. 4.18, there are big

differences between the behavior of the negative moment at the midspan

of the unloaded span and the maximum moment at the support. For the

bridge with H = 10, b/a = 001, and b = 7 ft, the moment distribution at

midspan of the unloaded span is unique in itself. It resembles neither the

behavior of the maximum positive moment nor the maximum negative moment.

The midspan moments in beams A, Band C of the unloaded span are 54.5,

42.0, adn 34.0 percent of the respective maximum negative moments at

the support when k = 0.0 and the moments stay nearly constant as k

increases.

For the bridge with H = 20, b/a = 0.2 and b = 7 ft, the moments

in the beams of the unloaded span do not show the large difference in

moments in the beams as exhibited in the beam moments at the support,

The maximum negative moment at midspan of the unloaded span of beams

A, Band Care 56.6, 45.0, and 46.0 percent of the respective maximum

negative moments at the support. As k increases, the percentage of moments

increases for beam A and decreases for beams Band C.

It can be concluded that the moments in the unloaded span of

the two span continuous bridge are more evenly distributed than either

the support negative moments or the positive moments in the loaded span.

The influence lines at various sections of a number of repre

sentative bridges due to a 4W loading are shown on Fig. 4.18.

The positive moment envelope and the influence lines at various section

of a two span continuous beam is shown on Fig. 4.19, for purposes of

comparison. For a beam, the influence lines are always concave upwards

I~

I l 0

[

f:0~ L

(;::

[

[

10

00

[

r ,

33

while the influence lines for beam A of the bridges exhibit a concave

convex-convex-concave sequence, indicating that at some points the slabs

are helping to support the beams. The influence lines for the interior

beams are always concave upwards and they can be interpreted as the beams

always support the slabs.

The addition of the diaphragms may not drastically change the

maximum moments in the beams but the location of the maximum moment is

changed. The higher the diaphragm stiffness, the greater is the shift

of the location of the maximum positive moment. As the diaphragmis

stiffness increases, the location of the maximum positive moment of beam

A shifts from 0.42 of the span from the simply supported end towards mid

span, but the location of the maximum positive moment of beam Band C

shift from 0.42 from the simply supported end towards the 0.3 point.

It seems that the amount of shift is not significantly affected by the

ratio of the girder's stiffness to slab stiffness, H. It seems that

the shift is more dependent of the beam spacing, b, and the b/a ratio.

In general, with the exception of beam C, the smaller the beam spacing,

the lesser is the shift and the larger the b/a ration, the larger is

the shift. There is no apparent relationship between the shifting of

the maximum moment location mbeam C and either b or bfa. The maximum

shifting of maximum moment location is in beam C and the higher the dia

phragm's stiffness, the greater is the shift. The following are typical

values of shifting when the value of k varies from 0.0 to 0.4.

The point of maximum location is expressed as the fraction of

a span length from the simply supported end; with the following values

being typical:

34

For beam A the change is approximately from 0.44 to 0.5

For beam B the change is approximately from 0.42 to 0.38

For beam C the change is approximately from 0.42 to 0.36

The most severely affected bridge is the one with H = 20, b/a =

0.05 and b = 7 ft. The changes are:

Beam A from 0.44 to 0.5

Beam B from 0.42 to 0038

Beam C from 0.42 to 0.3

The change in maximum moment location for a 4W loading will also

change the location of the maximum moment from the truck loading. In order

to accomodate this shifting, the profile of the prestressing steel in the

prestressed concrete girder may have to be modified.

4.4 Effects of Bridge Parameters on Moment Distribution

Fig. 4~20 to Fig. 4.25 shows the effects of bridge parameters on

the moment distributions in the beams. The moment coefficients shown are

the maximum moment coefficients due to a 4W (four wheels) loading on a

two span continuous bridge without intermed'iate diaphragms. In these

figures, the maximum positive moment coefficients are compared to those

of a simply supported bridges. The solid lines are for continuous bridges

and the dotted lines are for simply supported bridges. The data points

for the simply supported bridges are taken from Ref. 12, Fig. 5.27.

For a beam spacing of 5 ft (Fig. 4.20), it can be seen that the

moment coefficients of a continuous bridge for beam A and beam B are not

much affected by the relative girder stiffness, H. For beam A, they vary

from 0.19 to 0.20 and for beam B, they vary from 0.185 to 0,192. The same

r .--I

I L ..

[

r~'

[

35

trend occurs in the simple supported bridge for beam A. The moment

coefficients of the center beam, beam C, increase as H increases and the

magnitude of the moment coefficient is a function of both the relative

girder stiffness, H, and b/a ratioo It seems that the larger the b/a

ratio 1 the larger the increase in C values as H increases. The simply m supported bridges also exhibit this monotonic increase in C with Hand m the curves for the simply supported bridge are approximately parallel

to those forthe continuous ones~ If we make a comparison of the moment

coefficients between a simply supported bridge and a continuous bridge,

we will find that a much larger decrease in moment occurs in beam A than

in the interior beams. Table 4.3 shows the percentage decrease in moment

coefficients in beams for a continuous bridge relative to a simply

supported bridge. The effects of continuity decreases the moment in

beam A about 17 percent or more, while for the interior beams the

decrease in moment coefficient usually varies from 9 to 15 percent.

F~g. 4.21 shows the effects of H on C values for beams A, B, m

and C when the beam spacing is 7 ft. For a continuous bridge, except

when b/a = 0.05, the moment coefficients for beam A decreases as H in-

creases. The moment coefficients increase as H increases for b/a = 0.05.

For beams Band C, the moment coefficients always increase as H increases

and as b/a decreases. The change in Cm for beam A with H is much smaller

as compared to beams Band C. Only in the case when b/a = 0.05 is beam

A the controlling beam. When bja equals to 0.10 and 0.20, the controlling

girder is beam B. If we compare the Cm of beam A for continuous bridges

and simply supported bridges, we will find that for b/a = 0.05, the Cm of the simply supported bridge stays at a level of 0.3 and is unaffected

36

by H, whereas for a continuous bridge, C increases as H increases. When m

b/a = 0.10, Cm first decreases and then increases with the increase of H.

For beam B, the curves for the simply supported bridges are fairly similar

to those of the continuous bridge. Table 4.3 shows the percentage decrease

in Cm of beam A, varying from 16.6 to 25 percent and for beam B, varying

from 10 to 13.6 percent, comparing positive moments in continuous and

simply supported spans.

Fig. 4.22 shows the effects of H on the C for beams A, B, and m C when b equals to 9 ft. For a continuous bridge, when b/a = 0.05, C

m for beam A increases with H and for b/a = 0.1 and 0.2, Cm decreases with

H. For a simply supported bridge, C for beam A increases with H for m both b/a = 0.05 and 0.10. For the interior beams, regardless of the' type

of bridges, C always increases as H increase. It seems that the increase m

in Cm with H of a simpJy supported bridge is larger than for the continuous

bri dge for beam C, otherwfs·2:;. the trend is qu i te simi 1 a r to the case when

b equals to 7 ft.

It can be concluded that the edge girder A is the controlling

girder only for the cases when b/a = 0.05 for various beam spacing and

H value. For other b/a ratios, the interior beams will be the controlling

griders.

The decrease in' positive moment coefficients because of con-

tinuity can be separated into two categories; one for exterior girders

and one for the interior girders. The edge girders always experience a

larger decrease in positive moment coefficient ranging from 15 to 25

percent with an average of 18.6 percent. For the interior beams the

decrease in C is smaller, ranging from 9 to 16 percent with an average m

r-I L .•

[

r

( ... :.: '.

.;!

I: [

I;:·

[: .r ....

j

37

of 12.7 percent.

The variations of maximum negative moment coefficients at the

supports with respect to b, b/a and H are similar to those of the maximum

positive moment coefficients. In general, the Cm values for the interior

beams, beam Band C, increase with Hand b/a ratio (Fig. 4.23 to Fig. 4.2S)

but the rate of increase is smaller than for the maximum positive moment

coefficients. For the edge girder, girder A, the maximum negative moment

coefficients are fairly insensitive of the variation of H. The maximum

negative moment coefficient occurs when b/a = O.OS, and Cm decreases as

b/a increases. When b/a = 0.05, C increases as H increases and when b/a . m

= 0.10 and 0.20, Cm either stays constant or decreases slightly.

Recalling that for a two span continuous prismatic beam sub

jected to a moving point load, the ratio of the maximum positive moment

to the maximum negative moment is about 2.17; it can be seen from Table

4.4 that the relationships between the maximum positive and maximum

negative moments in a two span bridge are about the same as for the two

span prismatic beam in spite of the fact that the loading is different.

4.5 Comparison of the Effects of Diaphragms on Simply Supported and Continuous Bridges

Table 4.S shows the changes with increasing diaphragm stiffness,

in the moment coefficients of the girders due to a 4 wheel loading moving

on simply supported and two span continuous bridges with H = 20, b/a =

0.10, and b = 7 ft. The data of the simply supported bridge are taken

from Ref. 12, Fig. 4.3S. The changes in moment coefficients in the girders

are expressed as percentage change relative to the maximum moments in the

38

girders when k = 0.00 (i.e., no diaphragms).

Regardless of the type of bridge, the change in moment coefficients,

whether it is an increase or decrease of positive or negative moment at

k = 0.05 is about 40 to 60 percent of that when k = 0.4. That is to say

that the change in maximum moments in the beams are not very sensitive to

the diaphragm's stiffness once the diaphragms are installed. Depending on

the beam spacing of the bridge, the effects of diaphragms are more'~pro

nounced in a simply supported bridge than in a continuous bridge, As k

increases from zero to 0.4, the percentage change in e of the edge beam m

for both the simply supported and continuous bridges are approximately the

same. For the interior girders, as k increases the change in em of the

simply supported bridge is greater than in the continuous bridge. When k

reaches the value of 0.4, the percentage change in e of the simply supported m

bridge are approximately twice that of the continuous bridge, especially

for the bridges with large beam spacing. The largest change in moment

coefficients for both types of bridges occur in the interior beams. At

a k value of 0.4, the change in e ranges from 20 to 30 percent or more. m for girder C of a simply supported bridge, and it may be considerably less

for girder B apd in continuous bridges.

It was shown in Ref. 12 that the effects of the diaphragms

decreases at the sections far -away from the location of the diaphragm.

But it is not necessarily so for the negative moments at the supports.

Table 4.5 shows the percentage change in maximum negative moment at the

supports. Table 4.5 shows the percentage change in maximum negative moment

coefficients as a function of the diaphragm's stiffness. When k = 0.05,

the percentage changes in maximum negative moment coefficients are approx-

[

[

r

f?;:: L

1 11 L.

r-.

39

imately the same as for the maximum positive moment, and are slightly greater

than the changes in maximum positive moment when k = 0.4.

4.6 Effects of Diaphragms on Moments from Truck Loadings

Having the influence lines for 4W loadings, the effects of two

trucks with AASHO HS load distributions, running in the same direction

along the bridge, can be analysed by the method of superposition. The

spacings of the axles and the distributions of loads are shown in Fig. 2.3b.

When considering the moments due to truck loading, the direction in which

the trucks are going will affect the maximum moments in the beams. The

absolute maximum moments in the beams due to two trucks running in either

direction are designated as the maximum moments. As discussed in Ref. 14,

the influence of the diaphragms will decrease as the number of load in

creases; so only the bridges which are significantly affected by the dia

phragms will be discussed. Fig. 4.26 shows the maximum moment coefficients

in beams A, B, and C due to two HS truck loadings on the bridges with H =

20, bja = 0.2, and b = 5, 7, and 9 ft.

For the bridge with b = 5 ft, the length of one span is only

25 ft long and the full length of the truck cannot be parked in one span.

Therefore, when considering the maximum positive moment for this case

when b = 5 ft, only the two heavy axles of the trucks were considered.

The maximum positive moments due to two axiles are slightly greater than

those from one axle placed at the point of maximum moment in the moment

envelope. There a~e virtually no changes at all in em for girders A and

B and only a slight decrease in moment for beam C as k varies from 0.0

to 0.4. The diaphragms just cannot redistribute the loads in beam C to

40

beams A and B.

The maximum positive moment induced in the beams due to 2 trucks

running along the bridge are affected by the direction in which the trucks

are going. Two solutions, one for the trucks running towards the interior

support and one for the truck running away from the interior support were

obtained and the maximum positive moments that would be induced in the beams

are taken as the maximum moments. When the length of one span is small,

the maximum positive moment may be due to the loading of the two rear heavy

axl es of the trucks alone and the 1 i ghter axl es are some'where beyond the

pier.

For the bridge with b = 7 ft, the edge beam moment never has

the controlling moment when k varies from 0.0 to 0.1. At higher diaphragm

stiffnesses, loading due to two heavy axles induce a slightly larger moments

in the beams. They are shown as dotted lines in Fig. 4.26. For the bridge

with b = 9 ft, the span length is long enough that loading due to the two

heavy axles along can no longer develop the maximum positive moment. Two

trucks running away from the interior support will induce the absolute

maximum positive moments in the beams. In all cases, the diaphragms tend

to reduce the moments in the interior beams and increase the moments in the

exterior girders. A relative diaphragm stiffness of 0.4 for b = 7 ft and

0.3 for b = 9 ft will help to improve the load distribution of the positive

moment of the bridge, reducing the maximum positive live load moments by 10

to 15 percent.

The maximum negative moment at the supports are found by assuming

that the two heavy rear axles of the trucks are right on the point of max-

imum moment in the influence line for moment at the interior supports. By

,---i

r .. ' .' r·.··

1 L..

[

r. l-I

1. .

..

r I

41

doing so, we have the freedom of varying the spacing of the rear axles as

long as they are kept within th~'limit of 14 to 30 ft. If the spacing of

the axles thus found, is more than 30 ft, it is assumed that the two rear

axles will have a spacing of 30 ft and they are placed symmetrically with

respect to the interior support.

The 30 ft axle spacing limit controlled for the case of girder A

when b = 7 ft and for all girders when b = 9 ft.

The graph of moment coefficients of the negative moment at the

interior support against k are plotted on Fig. 4.26. For b = 5 ft, the

maximum negative moments occur at beam C and for b = 7 and 9 ft, the max

imum moments occur at the interior beams. Diaphragms help to redistribute

the loads, but there is a limit to their usefulness and their effects on

load distribution for negative moment are never as prominent as for positive

moment.

4.7 Effects of Diaphragms on Three Span Bridge

In order to get an insight on the effects of diaphragm on multi

span bridges, a three span continuous bridge was analysed. By comparing

the results of the analyses of the three span continuous bridge and the two

span continuous bridge, a more general conclusion on the effects of dia

phragms could be made. A three span continuous bridge with H = 20, b/a =

0.10 and b = 7 ft was analyzed and the maximum positive moment coefficients

in each span and the maximum negative moment coefficients at the interior

supports due to a 4W loading moving along the bridge were tabulated in

Table 4.6.

The maximum positive moment coefficients for all beams of the end

42

span of the three span bridge are slightly lower than in the two span

bridge, and the reverse is true for the negative moment at the support.

The results suggest that the three. span bridge is a little bit stiffer than

the two span bridge. The variation of the maximum positive C values of m

the end span of three span bridge with k is the same as for the two span

bridge, and there are slightly higher variations in C for the maximum m

negative moment coefficients at interior supports and the maximum positive

moment coefficient at interior span.

The trend of the maximum positive moment coefficients for the

interior span is a little different from the end span. Fig. 4.27 shows

the relation of Cm against k for the maximum positive moments and maximum

negative moments. It can be seen that the change in C of the center span m when k varies from zero to 0.40 are slightly greater than that in the end

span. Physically, the results indicate that the end span of the three span

continuous bridge is less sensitive to the effects of the diaphragms than

is the interior span. This may be due to the increase in the effective

b/a ratio and the concurrent increase in the H value because of the effects

of the negative moments at the interior supports.

The influence lines for the moment at the supports and at the

center of each span are shown in Fig. 4.28 and Fig. 4.29 and their trends

are comparable to the two span continuous bridge. Thus it can be concluded

that most probably, the effects of diaphragms on a multi span bridge are

not much different than that of the two span continuous bridge.

..--!

r ! i :

1 L ._

[

r r--) I L.. ....

[

43

Chapter 5

COMPARISON OF AASHO DESIGN MOMENT COEFFICIENTS WITH THEORETICAL DESIGN MOMENT COEFFICIENTS

Comparison of the beam design moment coefficients for slab'and

girder bridges from the 1969 AASHO specifications, Section 1.3.1, and the

coefficients as obtained from this investigation are shown in Figures 5.1

to 5.4. The loading applied on the bridge is the 4 wheel loading rather

than a two-truck loading. For the edge beams, the data points for b/a

= 0.05 give the upper bound for the maximum positive moment coefficients

for all beam spacings studied; and b/a = 0.2 gives the lower bound for

the maximum positive moment coefficients. The AASHO is conservative for

the beam spacing of 5 ft, but is unconser~ative for beam spacing of 7 and

9 ft. The calculated Cm values are approximately 9.6 percent and 5.1

percent higher than AASHO predicted C values for b = 7 and 9 ft respect-m ively.

values.

The percentage values are based on the AASHO recommended C m

The interior beams positive moment coefficients show a much

larger scattering than the edge beam moments, when plotted verses the

beam spacing, b. For b = 5 ft, the data points for beam C have a much

larger scattering than beam B while for b = 7 and 9 ft, the scattering

for beam C is only slightly greater than for beam B. The large scatter

ing of data points indi~ates that the Cm values cannot be solely pre-

dicted by one parameter, the beam spacing, aloneo The C values as m calculated from AASHO are quite unconservative for some bridges. As

beam spacing decreases, the differences between the maximum positive

moment coefficients and the AASHO Cm values increase. At b = 5 ft, the

44

maximum difference is about 31.9 percent greater than the AASHO value and

at b = 7 and 9 ft the maximum differences are approximately 13.6 percent

and 9.5 percent higher, respectively. It can be seen that most of the

points that fall above the AASHO line are those with b/a = 0.1 and 0.2.

The behavior of the maximum negative moment coefficients are

approximately the same as the maximum positive moment coefficients ex-

cept that less scattering is observed in the interior beams moment values.

The AASHO recommended C values for the edge beams are conservative for m

the beam spacing of 5 ft only and they are generally unconservative for

beam spacing of 7 and 9 ft. The upper bound for the Cm values are those

of b/a = 0.05 and the lower bound are those for b/a = 0.2. The maximum

differences of the calculated C values to AASHO C values are approxi-m m mately 14.2 and 11.7 percent higher for beam spacing of 7 and 9 ft. For

the interior beams the AASHO recommended Cm values are unconservative for

many bridges. The trend is similar to that of the positive moment co

efficients with the maximum deviation from the AASHO line of 36.7, 16.5

and 13 percent for beam spacing of 5, 7 and 9 ft.

, The comparison cited above is only confined to a 4 wheel load

ing and the results may not necessarily reflect the load distribution of

truck loadings.

While it appears that some changes should be made in the AASHO

load distribution values, this study has not been comprehensive enough

to develop design recommendations.

r I .--{ ! i .

[

r.~

i 1. "

45

Chapter 6

CONCLUSIONS AND RECOMMENDATIONS

From the analyses of continuous bridges with various diaphragm

stiffness and various bridge properties, the conclusions can be summarized

as follows:

The effects of continuity:

1. The variation in the maximum positive moments of simply

supported bridges due to the effects of the diaphragms is

larger than in continuous bridges. The load distribution

(of a 4-wheel loading) in continuous bridges without diaphragm

is similar to that in simply supported bridges without dia

phragms. The effects of continuity tend to cause a greater

reduction in maximum positive moment in the edge girder,

beam A; with an average reduction of 19 percent, than in the

interior girders, with an average reduction of 13 percent,

as compared to simply supported bridges of the same span,

beam spacing, and H values.

2. The effects of continuity greatly stiffen the bridges.

3. The average ratio of the maximum positive moment, due to 4-

wheel loads, to maximum negative moment at the support is

approximately 2.17 (range: 2.00 to 2.38). This ratio is

about the same as in a continuous beam.

The effects of diaphragms on moments due to 4W loadings:

1. Diaphragms are always helpful in reducing the maximum moments

in the loaded girders if the load is a single point load.

46

2. For bridges with b/a = 0.20, the maximum moment always occurs

in beam C, regardless of the beam spacing and loading condi-

tion. Stiff midspan diaphragms, with a relative diaphragm

stiffness of 0.40 or above, are always helpful in improving

the load distribution of the bridge especially with large,

beam spacings. The reductions of maximum moment range

from 5 to 24 percent, with an average of 12 percent.

3. For bridges with b/a = 0.10, a flexible midspan diaphragm

in each span may be helpful in improving the load distribu

tion. For H = 10 the optimum relative diaphragm stiffness

is approximately from 0.03 to 0.05, giving an average reduc-

tion of maximum moment of 2 percent, and for H = 20, the

optimum diaphragm stiffness is from 0.05 to 0.08, with an

average reduction of maximum moment of 6 percent.

As the diaphragm st'iffness increases beyond the optimum

stiffness, the maximum moment in the exterior beams will

increase to values greater than the absolute maximum moments

in the beams of the bridge without diaphfagms.

4. Bridges with b/a equal to or less than 0.05 do not need

any diaphragm. Diaphragms will do more harm than good in

these bridges.

5. By adding a diaphragm at midspan of each span of a two

span continuous bridge, the location of the maximum positive

moment will be displaced. For the edge girders, the locations

of maximum positive moments will tend to go towards midspan

and for the interior girders, the points of maximum moments

r

L .. r':

.. ( ',' ..

[ -, [.',"

I ..

. ' ...

1 ,~

i. .. ~

47

will tend to go away from the midspan towards the 0.3 point,

from the simply supported end.

6. Midspan moments in the unloaded span of two span continuous

bridges are much more evenly distributed than in the loaded

span. For b/a smaller than 0.1, the moments in the girders

are nearly independent of diaphragm stiffness. The midspan

moments in the interior beams are always less than 50 percent

of the maximum negative moments at the interior supports,

while the midspan moments in the edge girders may be less

than or equal to 50 percent of the interior support moments,

depending on the diaphragm stiffness. For the bridge with

fairly bad load distribution characteristics (i.e., H = 20,

b/a = 0.2), the moment distribution at the midspan of the

unloaded span is much more uniform than that at the support.

The increase in diaphragm stiffness tends to decrease the

moments in the interior girders and raise the moments in

the edge girders. The moments in the interior girders of

these bridges are always less than half of the moment at

the interior support and the moments in the edge girders are

always greater than half of the moment at the interior

support.

The results of the analysis of a three-span continuous bridge show

that the three span bridge is a stiffer structure than the two span continuous

bridge. The maximum positive moments in the end span of the three span

bridge are slightly less than that of the two span bridge. The maximum

negative moments at the interior supports are slightly higher for the three

48

span bridge than the two span bridge. The effects of diaphragms on the three

span bridge are primarily the same as those in the two span bridge, although

the change in the maximum positive moments in the center span as a function

of the diaphragm's stiffness is slightly higher than that of the end span.

The optimum relative diaphragm stiffness of the 'three span continuous

bridge with H = 20, b/a = 0.10, and b = 7 ft is approximately 0.07, which

is slightly higher than for a similar two span bridge. In general, the

behavior of the three span bridge is similar to the two span continuous

bridge.

The comparison of the AASHO recommended C values for 4 wheel m

loading with the calculated C values for the positive and negative moments m

are only conservative for the edge beams with 5 ft beam spacing.

C values are generally unconservative for other cases. m

The AASHO

The large scattering of data points for C values indicates that m the distribution coefficient cannot be based on the beam spacing alone.

Besides b, the beam spacing, both Hand bja are important parameters but

it seems that a distribution factor as a function of band b/a will give

a more realistic value.

According to AASHO recommendation, diaphragms should be installed

in bridges with spans more than 40 ft. Although flexible diaphragms, with

relative stiffness of 0.05, may slightly improve the load distribution

characteristics in some of the bridges with small b/a ratio, such diaphragms

are seldom pr'actical in terms of cost effectiveness, and it would be more

economical to build these bridges without diaphragm. The results of this

investigation show that only bridges with large b/a ratio could benefit

from the addition' of diaphragms. For these bridges, it generally would be

-I

:

r I L.

r:r L

[

(,

t.

1

1 i L._

f " j ... ~ . •

49

more economical to increase the number of prestressing strands in the pre

stressed concrete girders to resist the load than to use diaphragms to dis

tribute the loads to other beams. Therefore, unless necessary for temporary

erection purpose, it is recommended that diaphragms should not be installed

in straight highway bridges, whether they are simply supported or continuous.

r r-i I

L ..

rI l i.. -.

50

Chapter 7

SUMMARY

The effects of diaphragms and continuity on load distribution

of five beam, two and three span continuous bridges were studied. The

method of analysis was based on Fourier harmonic analysis. For the com

plete analysis, four computer programs running on IBM 360-75 were used.

Given the geometric and member properties of the slabs and girders, the

first two computer programs will calculate all the internal forces in the

girders of the bridge under point loads. The third program combines the

point loads by superposition to simulate the desired loading condition and

finds the maximum moments in the girders due to the specified loading.

The fourth program then plots out the influence lines.

The diaphragms are assumed to provide torsional and shearing

restraint at the junction of the girders and the diaphragms, and the

girder supports are assumed to be nonyielding. The major concern in this

investigation is the effects of diaphragms on load distribution in the

slab and girder highway bridges. Based on the investigation of the effects

of diaphragms on simply supported bridges, the selected parameters for

this study are H, the relative stiffness of the girders to the slabs; bfa,

the aspect ratio; b, the beam spacing; and k, the relative stiffness of

the diaphragms. The range of these parameters are: H from 5 to 20, b/a

from 0.05 to 0.2, b from 5 ft to 9 ft, and k from 0.0 to 1000.0. The

criterion of comparison is, in general, the maximum moments in the girders,

both positive and negative, as produced by the 4-wheel concentrated loading,

and the truck loading. The effects of the location and stiffness of dia-

51

phragms on the load distribution of a point load was studied. The

results of this preliminary investigation indicated that the best location

of the diaphragm would be a midspan diaphragm in each span and for the

sake of studying the effects of diaphragm, the variation of k from 0.0 to

0.4 was adequate.

With each bridge geometry, diaphragm stiffness, and girder

property, the effects of diaphragms under a loading consisting of four

point loads spaced according to the AASHO recommended wheel spacing were

studied. The variation of the maximum moments, both positive and negative

in the girders with various diaphragms stiffnesses were studied. A com-

parison of the simply supported bridges as reported in Ref. 12 with the

two span continuous bridges was also made. The analyses also gave the

influence lines for moments at various sections due to the 4-wheel load-

ing. With this information, the effects of diaphragms on the maximum

moments in the girders due to truck loadings were also investigated.

Reference was made to the AASHO recommended design C values for both m

positive and negative moment for girders due to a 4-wheel loading, and

the AASHO recommended Cm values was compared to the calculated Cm values

as obtained from this investigation.

A three span continuous bridge was analyzed to give an insight

to the load distribution in the multi-span bridge. In the discussion,

only the moments in the girders are considered, but the internal forces

in the girders are also available from the computer output. It is con-

eluded that except for temporary erection purposes, diaphragms are not

required in straight highway bridges.

j i L.-. , •. ;: ..•. '

L

r

[

I

L..:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

52

LIST OF REFERENCES

IIStandard Specifications for Highway Bridges,1I American Association of State Highway Officials (AASHO), 1969.

Guyon, Y., "Calcul des Ponts Larges a Poutres Multiples Solidarises par des Entretoises," Annales des Ponts et Chausees, Paris, 1946, September-October, pp. 553-612.

Massonnet, C., IIMethod de Calcul des Ponts a Poutres Multiples tenant Compte de Leur Resistance a la Torsion,1I Publications International Association for Bridge and Structural Engineering, Zurich, 1950, Vol. 10, pp. 147-182.

Rowe, R. E., "A Load Distribution Theory for Bridge Slabs Allowing for the Effect of Poisson's Ratio," Cone. Res., 7 No. 20, pp. 69-78 (1955)

Dean, D. L., "Analysis of Slab-Stringer Diaphragm Systems," Final Report Part III, Report on Project ERD-llO--7-4, Department of Civil Engineering, North Carolina State University at Raleigh, June 1970.

--....... Newmark, N. Mo, "A Distribution Procedure for the Analysis of Slabs " ,/ Continuous over Flexible Beams,1I University of Illinois Engineering f

Experiment Station, Bulletin 304, 1938. rV) '

Newmark, N. M. and Co P. Siess, IIMoments in I-Beam Bridges,1I University') ,''6-''', of Illinois, Engineering Experiment Station, Bulletin 336, 1942. \ ~j

Zienkiewicz, O. C. and Y. K. Cheung, liThe Finite Element Method in Structura 1 and Conti nuous Mechani cs," McGraw-Hi'll, London.

(-'1,'\' \-) 'u",

Gustafson, W. C., "Analysis of Eccentrically Stiffened Skewed Plate 1;\\"": Structures," Ph.D. Thesis, University of Illinois, Urbana, Illinois, \),._,/J 1966.

Lazarides, T. 0., "The Design and Analysis of Open-Work Prestressed ~ )~ Concrete Beam Grillages," Civil Eng. and=Pub. Works i ..... Rev., 47, No. 552, P. 471 (1952).

VanMorn, D. A. and D. Motargemi, "Theoretical Analysis of Load Distribution,in Prestressed Concrete Box-Beam Bridges, "Lehigh University Fritz Engineering Laboratory, Report No. 315.9, 1969.

12. Sithichaikasem, S. "and W. L. Gamble, "Effects of .Diaphragms in Bridges.with Prestressed Concrete I-Section Girders," Civil Engineering Studies, .Structural Research Series No. 383, Department of Civil Engineering, University of Illinois, Urbana, February 1972.

13.

14.

15~

16.

17.

18.

19.

20.

53 ,

?\V'\l

Wei, B. C. F., "Effect of Di aphragms in I-Beam Bri dges," Ph. D. 1 /\~ Thesis, University of Illinois, Urbana, Illinois, 1951. 7'~}

tNr~,

Siess, C. P. and A. S. Veletsos, "Distribution of Loads to Girders! "<) in Slab-and-Girder Bridges: Theoretical Analyses and Their Relation ( ... \~?J to Field Tests, IICivil Engineering Studies, Structural Research .\ Series No. S-13, Department of Civil Engineering, University of ) Illinois, Urbana, 1953.

IIDesign.of Highway Bridges in Prestressed Concrete,1I Portland Cement.Association, Structural and Railway Bureau, Concrete Information, 1959.

Gamble, W. L., D. M. Houdeshell and T. C. Anderson, IIField Investigation of a.Prestressed Concrete Highway Bridge Located in Douglas County, I11inois,1I .Civil Engineering Studies, Structural Research Series No. 375, Department of Civil Engineering, University of Illinois, Urbana, June 1970.

Goldberg, J .. E. andH. L. Leve, JlTheory of Prismatic Folded Plate Structures,JI IABSE, Zurich Switzerland, No. 87, 1957, pp. 59-86.

Fenves, S. J., "Computer Methods in Civil Engineering, PrenticeHall International Series, p. 41.

Lin, C. S. and D. A. VanHorn, liThe Effect of Midspan Diaphragms on Load Distribution in A Prestressed Concrete Box-Beam Bridge, Philade1phia .. Bridge,JI Lehigh University Fritz Engineering Laboratory Report No. 315.6.

Siess, C. P. and I. M. Viest, JlStudies of Slab and Beam Highway Bridges,.Part.V, Test. of Continuous Right I-Beam Bridges," University of Illinois Engineering Experiment Station, Bulletin 416.

{ I

~

I l .:.

r I t •

t~.··.:··~~.· 11

[:

[

\ I L:

H

5

10

20

54

Table 2.1

COMBINATIONS OF PARAMETERS FOR THE STUDIES OF EFFECTS OF DIAPHRAGMS IN TWO SPAN CONTINUOUS BRIDGES

Slab b/a b . a Thickness

(ft) (ft) (iri.r

0.05 5 100 7025 7 140 6000 9 180 7.75

0.10 5 50 7.00-7 70 7.25 9 90 8.00

0.20 5 25 9.50 7 35 9.50 9 ·45 7.50

0.05 5 100 6.00 7 140 6.25 9 180 6.50

0.10 5 50 7.25 7 70 6.00 9 90 7~75

0.20 5 25 7.00 7 35 7.25 9 45 8.00

0.05 5 100 6050 7 140 7075 9 180 6.25

0.10 5 50 6.00 7 70 6.25 9 90 6.50

0.20 5 25 7.25 7 35 6.00 9 45 7.75

1-; I

55 r··-· 1 :.

Table 4.1 r r--

~1AXH~ut~ POSITIVE MOMENT COEFFICIENTS I I

DUE TO 4W LOADING L.

[~ b Beam A Beam B Beam C r-: b/a

.. H (ft) k = 0.0 k = 004 k = 0.0 k = 0.4 k = 0.0 k = 0.4

5 0.05 5 0.198 0.201 0.183 o. 182 0.178 0.177 r-l.;._

7 0.241 0.248 0.223 0.221 O. 194 0.189 9 0.277 0.282 0.247 0.246 0.210 0.204

0.10 5 0.196 0.202 0.186 0.186 0. 192 0. 187 r~

U 7 0.243 0.257 0.243 0.236 0.225 0.207 .. -

9 0.285 00297 0.276 0.268 0.259 00234 0.20 5 0.190 0.193 0.185 0.190 0.214 0.204 I .. :. 7 00228 0.243 0.270 0.253 0.265 0.241

9 0.269 0.286 0.316 00292 0.315 0.281

10 0.05 5 0.200 0.206 O. 186 0.184 0.183 0. 181 [ 7 0.248 0.261 0~233 0.229 0.206 0.195 9 0.287 0.297 0.261 0.257 0.227 0.216

0.10 5 0.196 0.207 0.189 0.190 0.203 0.194 [ 7 0.241 0.264 0.257 0.243 0.248 0.215 9 0.285 0.305 0.293 0.278 0.288 00246

0.20 5 o ~ 191 0.194 0.189 0.193 0.233 0.215 I' 7 0.225 0.268 0.287 0.250 0.283 0.221 9 0.265 0.292 00347 0.306 0.346 0.298

20 0.05 5 0.200 0.210 0.188 o. 186 0. 189 O. 184 [ 7 0.252 0.272 0.242 0.235 0.219 0.201 9 0.289 0.306 00272 00265 0.248 0.226

0.10 5 0.193 00208 O. 191 0.192 0.214 0.200 r" '" 7 0.236 0.269 0.272 0.248 0.270 00221 9 0.281 0.311 0.315 0.287 00320 00257

0.20 5 0.192 0.195 0.193 00195 0.251 00222 . 7 0.220 0.246 0.302 0.270 0.298 0.259 1

t.._ 9 0.259 0.294 0.371 0.313 00368 0.307

I~ .' f'" I 1

i 1 l_ I .. :: z. i. ~

b H b/a (ft)

5 0005 5 7 9

0.10 5 7 9

0.20 5 7 9

10 0.05 5 7 9

0.10 5 7 9

0.20 5 7 9

20 0.05 5 . 7

9 0.10 5

7 9

0.20 5 7 9

56

MAXIMUM NEGATIVE r~ONENT COEFFICIENTS DUE TO 4W LOADING

Beam A . Beam B k.= 0.0 .k = 0.4 k = 0,0 k = 004

-00093 -0.095 -00087 -0.087 -00119 -0. 121 -0.102 -00101 -0. 138 -0.138 -00110 -0. 110 -00088 -0.092 -00089 -00088 -0. 113 -0. 120 -0.112 -00109 -0.135 ~0.139 -0.126 -0.122 -0.085 -0.085 -0.085 -00089 -0.102 -0.108 -0.127 -00119 -0. 121 -0. 129 -0.150 -0. 136

-0.093 -0.096 -00089 -00087 -00120 -0.124 -00107 -0.105 -0. 141 -0. 142 -00115 -0.114 -0.088 -00093 -0.090 -0.089 -00110 -00121 -0.119 -0.112 -0.133 -00 141 -00136 -0. 126 -0.086 -00086 -00087 -00090 -0.101 -00119 -00134 -00116 -0. 118 -00'30 -0.164 -00142

-00092 -0.097 -00090 -00088 -0.120 -0. 127 -00111 -00107 -0. 140 -0. 144 -0~121 -0.118 -00087 -0.094 -0.088 -00089 -0.107 -00122 -00126 -00114 -0.129 -00142 -00148 -0 ~ 130 -0.087 -0.086 -0.088 -00090 -00099 -00109 -00140 -00125 -0.116 -00130 -00173 -00145

Beam C k = 0.0 k = 004

-0.076 . -00075 -00087 -0.084 -00092 -0.089 -0.082 -00080 -0.107 -0.096 -0.123 -0.107 -00097 -00092 -00125 -00115 -0. 149 -0. 134

-0.077 -0~076 -00095 -00087 -0.104 . -00095 -0.088 -0.084 -0.118 -00099 -00137 -0.112 -0.107 -OoO~7 -0.132 -0.100 -O~ 161 -0. 140

-00080 -00077 -0.103 -00089 -0.116 -00100 -00095 -0.088 -0.127 -0.101 -00150 -00116 -0.117 -0.100 -0.137 -00122 -0.169 -0. 143

b

5

7

9

*

H

20 10 5

20 10 5

20 10 5

57

Table 403

EFFECTS OF CONTINUITY ON THE REDUCTION OF MAXIMUM POSITIVE MOMENT COEFFICIENTS IN PERCENT IN BEAMS RELATIVE TO

MOMENTS IN SIMPLY SUPPORTED BRIDGES

b/a = 0005 b/a=0.10 b/a = 0.20 Maximum Moment Occurs in Beam

* * 16.6, 14.0 17.0, 17.5 8.7 * * 16.6, 14.8 17.0, 13. 1 8.7 A and C

* 23.5 14.7 12.6

* * 16.6, 10.7 25.3, 13.3 13.8 * * 1606, 10.0 19.3, 11 .6 13.6 A and B

* 19.8 11 .2 13.4

* * 16.6, 12.0 21 .6, 12.5 13.7 * * 14.6, 1204 19.6, 12.2 10.9 A and B

16. 1 18.6 12.5

Reduction in controlling moment in Beam A. All other values are reductions in controlling interior beam moments.

\ L.. ...

r-i j ~ •• i-

u:········ . .... ;.

1-r·-

! ~

L

f :1 i . t:

H

5

10

20

58

Table 404

RATIO OF MAXIMUM POSITIVE MOMENT COEFFICIENTS TO MAXIMUM NEGATIVE MOMENT COEFFICIENTS, 4W LOADING

b/a b Beam A Beam B Beam C (ft)

0.05 5 2013 2.10 2.34 7 2003 2019 2022 9 2.00 2.24 2.28

0.10 5 2.22 2.09 2.34 7 2. 15 2.17 2.10 9 2.11 2.19 2. 11

0.20 5 2.23 2.17 2.20 7 2.24 2013 2.12 9 2.22 2, 11 2. 11

0.05 5 2.15 2.08 2038 7 2006 2018 2.17 9 2.04 2.27 2.18

0010 5 2.23 2010 2030 7 2019 2016 2.10 9 2.14 2. 15 2 0 10

0.20 5 2~22 2017 2.18 7 2.23 2.14 2014 9 2.25 2.12 2.15

0.05 5 2.17 2009 2.36 7 2.·10 2018 2013 9 2.06 2.24 2014

0010 5 2.22 2.17 2.25 7 2020 2.16 2.13 9 2. 18 2012 2.13

0.20 5 2020 2.19 2.15 7 2.20 2016 2011 9 2~23 2014 2.18

r--

59

r"~ Table 4.5 f-::

f COMPARISON OF THE EFFECTS OF DIAPHRAGMS ON SIMPLY SUPPORTED

BRIDGE AND CONTINUOUS BRIDGES SUBJECTED TO 4W LOADING r--

\

L. ...

Beam b Cmk=O ok = 0.05 ok = 004 [ (ft)

A 5 0.232 + 3.4 + 8.6 r B 5 0.257 - 9.7 - 21.0 C 5 0.257 - '9.7 - 21.0

-0 (/) r---t

OJ +-> \ +-> ... s:::: A 7 0.290 + 5.9 + 13.8 ~o OJ

t. _

ON E B 7 0.300 - 5.7 - 12.6 0.. 0 0..11 :;s C 7 0.319 - 12.2 - 29.5

U :::l r--V') :::c • OJ ...

0> >, OJ or- A 9 0.345 + 5.5 + 12.5

.-- 0') II +-> 0..-0 or- B 9 00359 - 9.2 - 16.2 Eo"" ro (/) r or- ~ -..... 0 C 9 0.361 - 14.4 - 36.2 V') c:c ..0 0..

'.

A 5 0.193 + 2. 1 + 7.7 B 5 O. 191 + 0.5 + 0.5 [ (/)

:::l C 5 0.214 - 3.8 - 6.5 0.--:::l • CI)

s:::: 0 +-> or- s:::: A 7 0.236 + 5.5 + 14. 1 [ +-> II OJ

s:::: E B 7 0.272 - 4.4 - 8.8 OnjO u-.....:::::E C 7 0.270 - 7.8 - 18. 1 ..0 s:::: OJ nj ... >

I:, 0..0°c- A 9 0.281 + 5.0 + 10.7 V') N +-> 0,... B 9 0.315 5.4 - 8.9 o I! (/)

3: 0 C 9 0.320 - 9. 1 - 19.7 I- :::c 0-

A 5 0.087 + 1 . 1 + 8.0 r i.:;:

(/) B 5 0.088 + 2.3 + 1 . 1 :::l +-> C 5 0.095 - 3.2 - 7.4 ,.- , 0.-- ro :::l . s:::: 0 +->

or- s:::: A 7 00 '107 + 4.7 + 14.0 +-> II OJ s:::: E B 7 o. 126 - 400 - 9.5 oroo

u -.....:;s C 7 O. 127 - 7 . 1 - 20.5 i ..0 s:::: OJ L . nj ... > +-> 0..0 or- ~ A 9 O. 129 + 3.9 + 10.0 V')N4->O

roo.. B 9 o. 148 - 6.8 - 12.2 r-: o II 0') 0.. 3: OJ:::l C 9 0.150 - 8.0 - 22.7 I-:::CZV')

Cmk=0.05 Cmk=O.O r'

°k=0.05 = x 100 Cmk=O.O

°k=0.40 Cmk=0.4 Cmk=o'.O x 100 ~ =

Cmk=O.O L_

f: .. ~ .':

60

THREE SPAN CONTINUOUS BRIDGE H = 20, b/a = OG1, b = 7 FT MAXIMUM MOMENTS COEFFICIENTS DUE TO 4W LOADING

Beam

c A 0,,230 + 502 + 1305 QJ0r- C E> ct:S ~ or- ~ 0. B 00265 - 4.5 - 7.9 E~C(/)

or- or- QJ XtnE-o C 0.263 - 6.5 - 1705 ct:Soo~ ~o..~1.J.J

C A OG193 + 6.7 + 16.6 QJor- s-E> 0 ~ or- ~ or- B 0.229 - 6.9 - 1104 E~cS-

0r-"r- QJ OJ XtnE~ C 0.228 - 8.8 - 21 .5 ct:Soo~ :Eo..~1o---j

~ A 00112 + 6G3 1502 QJct:S E> -I-' ~ or- ~ s- B 0.130 - 4.6 902 E~~O

or- ct:S QJ 0. xmEo. C 00 131 - 7.6 21 u 4 ct:SQJO~

:E z: :E (/)

C - C °k=0.05 = mk=0005 mk~OoO

>< 100 Cmk=OoO

C - C °k=0.40 = mk=0040 mk=OoO x 100 C mk=OoO

r

1.-: .,' L

rI

I I t

(':

r.

f L_

L_ ..

L ...

61

><

w <..!:l 0 ........ 0:::. Ci:l

w :r: ~

u.... 0

~ :::> 0 >-c::t: -J

N

+-> C

<..!:l -u.... 0

r-::>

62

Z

I' 'I I h

I

tw d d

I

I· ,I t

Z

FIG. 2.2 IDEALIZATION OF THE GIRDER CROSS SECTION

[--..

---~

r

[

r~

r-1 L. __ .

I [

[

1--. :.

[ I!"_ 0. j

\

l_.

[---

-,

63

P P P P X 6 1 4' 6 1

A B D E

I. b .1. b .1 ' b .1 ' b .1

a. Position of 4 Wheel Loads at Distance x from Edge Beam

P 4P 4P - - - -----,-

6' P 4P - - ----+--

P 4P 4P - - - ----f-------------- ..... ~---

P 4P 4P - - - -----'--

14' 14'*

* Axle spacing variable, 14 to 40 ft.

b. Wheel Spacing of Two HS Truck Loadings

FIG. 2.3 POSITION OF LOADS REPRESENTING TRUCKS

64

A B C D b b b I b

, r " u l - - -

.. " z

a. Vertical Forces Acting on Diaphragm

A B C D

b. Couples Acting on Diaphragm

FIG. 3.1 FORCES AND COUPLES ACTING ON DIAPHRAGM

E

r

y , (~

E

r-I

i." "

r· .,

I L "

[

r r--1 t ..

E [:

[

£: [:

[

f' ... ! , ~ .....

65

n .~ ~ ~ ~ A B C D E

a. Cross Section of Bridge with Diaphragm

b b b

A B C D E

b. Forces and Couples Replacing Diaphragm or Support

A B C

O~C ; i JO

~Ol

c. Displacement of Cross Section of Bridge

FIG. 3.2 FORCES AND DISPLACEMENTS AT CROSS SECTION CONTAINING DIAPHRAGM

.. :,~:"! .. ~

b

b

b

b

b

b

b

b

r·· ... ~···--l

I . f~

-

-

-

-

-H71 ~

1

,ta:

..... "'"

a ~~ a LI r r-l

? ':l Lt· ~ t:. 7 Q 77i ~, 0~2~ 0.3a ·0~4~ 0.5a 0.6a O.Ba 0.9a ;

13 14 15 16 17·

a. Sections Along .the Bridge Where Internal Forces Are Computed for One Diaphragm at 4/10 or 5/10 Point of Spans

nJ~ .,2. 3 4 5 6 7.8'" 9'" 1 0 11

0.1a 0.33a 0.4a 0.5a 0.67a 0.8a O.ga a 12 13 14 15

b. Sections Along the Bridge Where Internal Forces Are Computed for Two Diaphragms at One-Third Points of Spans

~ 16 17

FIG. 3.3 LOCATIONS OF SECTIONS CONSIDERED FOR INFLUENCE COEFFICIENTS

~ .... ~ fit"·..,·""" ~ .. " .... , 'Pi-ii, ~ M ,., ,-i -J .. \ ~ ~'J I \:: ~ r ;-'~'-'l ~"

(J) (J)

,.,..... .. ,' .. ! --'-1

:>

:>

:>

:>

" a , I,a t a -,

iiJR, " ") A ~ ~., n n , " , , ~ 13 14 15 16 ~~ 6 O.33a O.5a O.67a O.83a a 1 i2

c. Sections Along the Bridge Where Internal Forces Are Computed, Three Span Continuous Bridge

FIG. 3.3 (CONTINUED)

17

m '-.J

Beam A Beam B

1 .0

k = 0 k = 0.05

0.5~ ~~~k=O.l k = 0.2 0.5

k = 0.4 k = 1000

ol ~ A B C D "-~ E A 8 C D E

m co

1 .0 1 Beam C

0.5

o I I

A -8 C D E

FIG. 3.4 GRAPHS TO SHOW THE CURVE FITTING OF INFLUENCE COORDINATES AT A, 8, C, 0, E

...-...,. ... J1Il r·-~·····-

. --.. '~ ~ r~~' r .. • .. -··~

'"~ ~ M M ~ M ~-l ~ ~ ~--l ~ ~'''~ ,. --1 {; , •.. : ::." I~ ;.'1 \.,; -

'10

CO or- 0

I .. ~ r-

......, C QJ

E a

:E:

5 Static Moment

Total Moment in Five Beams

4

3

2 ~ -0 0 0 0 OA

~~V- 8

t I I I I • i I~ O ' I I I " I ~ I 5 10 15 ~~ x AX R A

Number of Harmonics Used 10 kip Concentrated Load at Midspan of Beam A, Five Girders, Simply Supported, H = 20, b/a = 0.05, b = 8 ft.

FIG. 3.5 EFFECTS OF NUMBER OF HARMONICS ON THE ACCURACY OF THE SOLUTION, BEAM A LOADED

oo~~

90~~

80~c

70%

60%

50% m ~

40%

30%

20%

1 O~,

0

r co 'r- 0

I ... ~ r--

+-> C W E 0

:E

III-'~ r ...... ~«.O ..... _-_ ... '"'4

5 Static Moment

Total Moment in Five Beams

4

3

2

B

A

C D

_~ ----- -w ------------ w _--

a 5 1 a 1 5 2 0 2 5 3'0 3 5

M

Number of Harmonics Used

10 kip Concentrated Load at Midspan of Beam B, Five Girders. Simply Supported, H = 20, b/a = 0.05, b = 8 ft.

L'·

FIG. 3.6 EFFECTS OF NUMBER OF HARMONICS ON THE ACCURACY OF THE SOLUTION, BEAM B LOADED

r ... -.,.~ M ~.~ ,., ,,~ ~-"l I

,., ~ .... - - .':':'1 ' I. .:: '.'

.., n

oo~,

90%

80~~

70~·

60~t

50;

~ 40%

30'\

20 11

10',':·

40

~ ~

-.......J 0

~--!'I u

- --l

5

4

r 3 co or- 0

I r-

:::..::: r-

+-' C 2 OJ E 0

:::E:

01

Static Moment .. II Total Moment in Five Beams

I I I I 5 10 15 20

Number of Harmonics Used 10 kip Concentrated Load at Midspan of Beam C, Five Girders, Simply Supported, H = 20, b/a = 0.05, b = 8 ft.

FIG. 3.7 ; EFFECTS OF NUMBER OF HARMONICS ON THE ACCURACY OF THE SOLUTION, BEAM C LOADED

11 oo~c

907c

80%

70%

60%

50% '-J

40%

30~~

0 l 20% E

1 O/~

I

r co ·,.....0

I .. ~ ..-

~ c OJ E 0

:£:

1 .0 I / "1111 ~

k = 1000. k = 0.40 k = 0.20

0.5 t- / IIL--k=0.10 k = 0.05 k = 0

OooV I ID I

Beam A Int. Support

k = 0 / k = 0.05 k = 0.10

l.0r- I I I ,--k = 0.20 k = 0.40 k = 1000

I / ~ 0.5

I

0.0 J I D I ~

Beam C Int. Support

ID

Beam B

~--k = 0 r----k = 0.05

k = 0.10 k = 0.20 k = 0.40

~k = 1000.

Int.

-- - - - - Di aphragm

FIG. 4.1 POSITIVE MOMENT ENVELOPES OF GIRDERS DUE TO A 10 KIP CONCENTRATED LOAD MOVING ALONG THE BRIDGE, ONE DIAPHRAGM AT MIDSPAN OF EACH SPAN; H = 20, bja = 0.1, b = 8 FT

~ .. ~ .... ~~ r~'-'-... -.-.~, ~ ! r--"''1I ~ ~ M ~ ~ ~ ~~-~J

j . ~ ;: .. ;:.,: ...., I; ~-l l

'-J N

~.~ ... -1

r CO 'r- 0

I "' ~ r--

+-l C OJ E 0 :a:

1 .0

0.5

0.0

1 .0

0.5

k = 1000 k = 0.40 k = 0.20 k = 0.10 k = 0.05

L---k=O

10 Beam A

j

Int. Support

D

Beam B

.----k = 0 " k = 0.05 .,---k = 0.10 ~--k = 0.20

k 0.40 k = 1000

tnt. Support

---- -- Diaphragm

O.O~ L-_~

Int. Support

FIG. 4.2 POSITIVE MOMENT ENVELOPES OF GIRDERS DUE TO A 10 KIP CONCENTRATED LOAD MOVING ALONG THE BRIDGE, ONE DIAPHRAGM AT 4/10 POINT OF EACH SPAN; H = 20, b/a = 0.1, b = 8 FT

---J W

r co 'r- 0

I ,.. ~ .--

+J C OJ E 0 ~

1. a

0.5

0.0

1 .5

0.5

k = loDe k = 0.40 k = 0.20 k = 0.10

,----k = 0.05 '"'----T- k = 0 o

Int. Support

0 = 0.05 = 0.10 = 0.20 = 0.40 = 1000

I ID

Beam B

:0

------ Diaphragm

o 0.05 0.1 P 0.20 0.40 1000

Int. Support

'-.J ~

I : 0 )I 10 I I 0.0 K Beam C

r·······-· i

Int. Support

FIG. 4.3 POSITIVE MOMENT ENVELOPES OF GIRDERS DUE TO A 10 KIP CONCENTRATED LOAD MOVING ALONG THE BRIDGE, TWO DIAPHRAGMS AT 1/3 POINT OF EACH SPAN OF THE BRIDGE~ H = 20, b/a = 0.1, b = 8 FT

M. ...... ~ r ..... ·'··· .. ." .. " ~::r.;-1 ~ M ~ ~ r~ r--'-l ~ ~ ,. ..• - .. -, ..,... .... ·1

I .. ,':: ~ t I ) 1fIlr"'-~

... \

i

0.0

-0.5J-

r co 'r- 0

I .. ~ r--

~

c OJ 0.0 E a

L

-0.5 L I

-1.0

I I I I ~ ~ ------------------ ~

Beam A

Int. Support ----.,

~

IIILk=~~ k = 0.1 k = 0.4 k = 1000

l- I I I Lk = 0 = 0.1 - 0.4

1000

. Beam B

--- Diaphragm

FIG. 4.4 INFLUENCE LINES FOR NEGATIVE MOMENT AT SUPPORT OF GIRDERS DUE TO A 10 KIP CONCENTRATED LOAD MOVING ALONG THE BRIDGE, ONE DIAPHRAGM AT MIDSPAN OF EACH SPAN; H = 20, b/a = 0.1, b = 8 FT

Int. Support ---,

----.)

<..n

'10

co 'r- 0

I .. :::..:::: r--

+J C OJ E a :E

0.0

-0.5

,-----k = 0 k = 0.1 k = 0.4 k = 1000

Int. Support

:D

o = 0.1

k = 0.4 '--k = 1000

-1 .ol. _________ ---::---------...... Beam A Beam B

k = 0 k = 0.1 k = 0.4

L..--_ k = 1 000

Int. Support

-l.OL· ________________________________ __ Beam C

Diaphragm

FIG. 4.5 INFLUENCE LINES FOR NEGATIVE MOMENT AT SUPPORT OF GIRDERS DUE TO A 10 KIP CONCENTRATED LOAD MOVING ALONG THE BRIDGE, ONE DIAPHRAGM AT 4/10 POINT OF EACH SPAN; H = 20, b/a = 0.1, b = 8 FT

~""~t-\l~ r····--- .~~ ,...'+'-'-".'" 1-···_ .. ,·,\

It"'~' ~ ,.., r-1 ~ r:i~ I i ,-

I ~. [~~'J ~ ~ ~l

Int. Support

~

'-J m

".".,.......~ .. . L.

---1

10 ;0 I I

k = 0

• 0 -1. 01 k = a 1 '71~ \ k : 0: 4 ~~ Beam A k - 1000

4J c::: ~ Int.

:E: Support o

-0.5 k = 0 k = o. 1 k = 0.4

'----k = 1000 -1. 0 L' _________________ ..J

Beam C

Beam B

- - - Di aphragm

o = 0.1 = 0.4 = 1000

FIG. 4.6 INFLUENCE LINES FOR NEGATIVE MOMENT AT SUPPORT OF GIRDERS DUE TO A 10 KIP CONCENTRATED LOAD MOVING ALONG THE BRIDGE, TWO DIAPHRAGMS AT 1/3 POINT OF EACH SPAN; H = 20, b/a = 0.1, b = 8 FT

'-.J '-.J

78 f"::; I

0.3 I I I t " : '

6. Girder A r 0 Girder B

0.2 0 Girder C -r--r--

L_ (~~ " (,> 0 c:r ~ ....!: [';' O. 1 ~

One Diaphragm at Midspan

0 I I I r r' ~ 0.3 1 l~.;

0.2 r "' r

to O. 1 0..

......... One Diaphragm of 4/10 Point [ ~ s::: OJ E 0

::E:

0 [ E u

0.3 I I I

[',; 0.2 - r

'l" '" :l' *",..;i

~ A 1\ ~

Two Diaphragms at 1/3 Point ~ I..;;J

w

O. 1 - ~

Positive -

Negative at Support 1

I I I ~

i o 0 0.4 0 0.1 0.2 0.3 0.4 Relative Diaphragm Stiffness r

i .. :

FIG. 4.7 MAXIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 20, b/a = O. 1 , b = 8 FT

i 1 i.

t 1 i

0.3

0.2

0.1

0.3

0.2

ro a.. .......... +-' C Q)

0.1 E 0

:::E

E u

0

0.4

0.3

0.2

0.1

o

79 I I I I I

6. Girder A 0 Girder B

J\ 1\

( ;: ~ ,.I ~~

~

l l u. u 0 I

0 Girder C r-

- - ~--t " >-l r.....r---v- ).l v l,r--U

b = 5 ft

~ I 1 I I

o

b 7 ft

b 9 ft

Positive Negative at Support

o 0.1 0.2 0.3 0.4 0 0.1 0.2

Relative Diaphragm Stiffness

FIG. 4.8 MAXIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 5, b/a = 0.05

I

-

""') ~

b = 5 ft I

b = 7 ft

b 9 ft

0.3 0.4

ro 0.... ........... +-' C (J)

E 0 :E

E u

0.3 80

I 1 T

" " " 0.2 ~~

r' 0 u CJ

0.1 -

b = 5 ft , I I

o

0.3

0.2

0.1

b = 7 ft

0

0.4

b 9 ft

~

d

-

~

o o

I

Girder A Girder B Girder C

..I U U

I

I

-

I

I

I I

A

b = 5 ft

I I

I I

1\

b = 7 ft

I I

I I

b = 9 ft

0.3~~~~ __ --~-----------y ~

0.2

O. 1

o

-

A " .A

l~ ~ ~

'-' i...! 'I-(

~


I I

o 0.1 0.2



I

0.3

-

) J

-

... )

-

-

)

..I J

0.4

r--I

I L _

[

[' .•... "

f

r . 1

! :L ,:

1 ~ L.:

81 0.3 r r I I r I

[j. Girder A 0 Girder B

0.2 h-..o ..... - 0 Girder C ~ .. ":: 'X - --v v ......

0.1 ~ - ~ p ...,

b == 5 ft Ip ~ £

b 5 ft I I I I I I

0

O.3~, t I

~ r I I

....... u .....

ow .. I , - '-l"

0.2 ~ - ~ -rt:l 0.. ........... 4-l c: ClJ E' (~ 0 ,

0 :r ~ A

O. 1 - - 'r .... ""I..l'"

E b == 7 ft b == 7 ft u

I I I I I I

0

0.4 J • I T I I

b == 9 ft b 9 ft 0.3 ~ ...... -I- ..... ~) ~ .....

i.. .. ~ ~ ;r .... .....

0.2 ~ - I- -

1>---0.-. '-J -A A A

4..- .... ~ -u

o. 1 I- - ~ -


0 I I 1 I I I

0 o. 1 0.2 0.3 0.4 0 O. 1 0.2 0.3 0.4 Relative Diaphragm Stiffness

FIG. 4.10 MAXIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 5, bja = 0.2

rc 0-......... -+-l c (])

E 0

:E:

E u

0.3

0.2

O. 1

o

0.3

0.2

0.1

J

1\ " (~

"'" Q

i-

I

I

A 1\

~ ( ..... -0---<> ~ ~

pt..;" -

-

1 0

0.4

I

.. .... Q

I

J

A

......

r-t

I

82 J I I r

A Girder A 0 Girder B 0 Girder C

ool t-

- "I- A .A .. :-~ >.I.

b = 5 ft b = 5 ft I . I I I

I

)

... ~

-

b = 7 ft b = 7 ft

L

b = 9 ft b = 9 ft 0.3 ~~~~ _______ ~~ ____________ ~

0.2

O. 1

Positive o

o 0.1

Negative at Support

0.2 0.3 0.4 0 0.1 0.2 0.3



-

)

0.4

r l. ...

[

l I

I ..

r L

t ...

ro Cl... '-+-' c Q.) ~

0 2:

E U

83 0.3 I I I I I


~ 1\ 1\ 0 Girder C ~ 0.2 \,r--J --v

0.1 r- - rJ:--....o,..... t:::! CJ w

b = 5 ft

0 I I I I I

0.3 I I

0.2

(~ 1\

l;r ~ L..I LJ 0.1 ~

b 7 ft

0 1 I

0.4 I I

b 9 ft

0.3

0.2

I\ 1\ 1\ 1\ H

1~ ...... ~ '-'

o. 1 r-


o I I

o 0.1 0.2 0.3 0.4 0 0.1 0.2



I

-

-

b = 5 ft

I

I

-

>.'

b = 7 ft

I

I

b 9 ft -

-

_J

1

0.3 0.4

84 IJ.- -;,

l. 0.3 6 Girder A 0 Girder B r 0 Girder C

0.2

r-

! 1.. -__

0.1 r ,.

b 5 ft b 5 ft

0 r 0.3 0- I I I

~ I I I

r-~.{"\.. t

'-' u j w -? L __ A

" 1\ ~

~

0.2 - - r- - [ rt'l 0-

~.f\ ........... I;;; +l - """ ~ C 1'1. A (!) O. 1 ~ - r--u- "-l'"' -E

0 :::

b = 7 ft b = 7 ft [ E

u 0 I I I I I J

0.4 I [

b = 9 ft b = 9 ft I' . ~ ;i ~ 0.3

r , . i' ... ~~

0.2 r' -

'--~

t i 0.1 L.:

Positive Negative at Support [ 0

0 O. 1 0.4 0 O. 1 0.2 0.3 014 Relative Diaphragm Stiffness

FIG. 4.13 ~,1AXIMU~1 MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 10, b/a = 0.2 1

j L _"

0.3

:J.2

0.1

0

0.3

0.2

f"O 0-......... 4-.l C Q)

0.1 E 0 z

= u 0

0.4

0.3

0.2

0.1

o

85 T I , I I I


A • A 1\ 0 Girder C - -r---:::'" ~

~ J

I-

o

- t- 1\ " " .. }-f

I~ \...I

b = 5 ft b 5 ft I I 1 I I I

b 7 ft b 7 ft

I I

b 9 ft

t-

t-

A A

,r .......

t--8 0 '-'

t- 1J -

I I

0.4 0 0.1 0.2



I

b = 9 ft

I

0.3

-

-

-

rl

)

0.4

.... ~ 86 1.

0.3 J I I I I I i .. ~

~ Girder A 0 Girder B 0

[ ~ A 0 Girder C I

0.2 ~ ~ - -'\.J' ....,

r-.

L. 0.1 ~ - ~ - [ ,. ,r--u I

b = 5 ft b = 5 ft

, I I I 1 I r' 0

0.3 I I I r··-

i L ..

[2 0.2 - .. ~:: "

ro 0... '-.... +-> s::::: [ <lJ

~ E 1\

0 ... :::: O. 1 ;r: 0 ...,. ,

E b 7 ft b = 7 ft [ u

I I I 0 r 0.4

b 9 ft b 9 ft [ 0.3

[ ... • p'

0.2

1! i

O. 1 L~

Positive Negative at Support [ 0

0 0.4 0 O. .4 '[ ..


FIG. 4.15 MAXIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 20, b/a = 0.1 i

1 L.._

I ! .,

0.3

0.2

O. 1

0

0.3 ()~).. A ~ ...,

" I""'"' ---u-~

0.2 I-rt:l

0.... ........... +-' C QJ

E 0

:E: O. 1 ~

E u

1 0

0.4

0.3

0.2

O. 1

Positive

o o 0.1

87 1 I I

t::. Girder A 0 Girder B 0 Girder C

t-

.}---n. ..... ,...., I- ...., ..... ;;:

(~ <;1 ~ tl· 0

b 5 ft b = 5 ft

I 1 I

I I J I I I

LJ ......., " '-'

- ~

V---0- ,...,

6----0-..... .....

'" '-'

A

- ;r ..... ..... <...l

b = 7 ft b = 7 ft

t I I I I

I I I

9 ft b = 9 ft

~

(~ -c:::r -u -1'l

.. '-' .... .. --

Negative at Support

I I I

0.2 0.3 0.4 0 0.1 0.2 0.3 Relative Diaphragm Stiffness


-

~

-

-, --

-

-

')

~

-

0.4

88

'l Girder A o Girder B o Girder C

0.2 I I I

H = 5, b/a = 0.05, b = 7 ft 0.1 - -

l,----u----v

o I I I

~ s:: <lJ E 0

0.2 ::E I I I

<lJ n::1 > c....

~ ro <lJ o. 1 z

H = 10, b/a = 0.1, b = 7 ft - -

II ~

...., \""'1.

~ E

u ~ ;r:

~

.0 J I I·

'0.2 I I I

H = 20, b/a = 0.2, b = 7 ft o. 1 I-- -

--". r .... ~

0 I I I 0 o. 1 0.2 0.3 0.4


FIG. 4.17 MAXIMUM NEGATIVE MOMENT COEFFICIENTS AT MIDSPAN OF UNLOADED SPAN DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS

r 1 .'

L ,.

[

r

F: ~'l"

I: .. , [

'

I~:·.

L ...

2. 51

Beam A k = 0

2.0l- P = 10k ips

1 .5

1.0

~I,~ 0.5

1 ~

~ ~I~ +J c QJ E 0

;:::: 0

-0.5

-1. 0

o 0.4 0.9

-1.5l' ____ -l ____ ~ ____ ~ ____ ~L_ ____ L_ ____ L_ ____ ~ ____ ~ ____ ~ __ ~~

0.1 0.2 0.3 0.5 0.6 0.7 0.8 1.0 Distance From Simply Supported End/a

FIG. 4.18 INFLUENCE LINES FOR MOMENTS AT VARIOUS SECTIONS ALONG TWO SPAN CONTINUOUS BRIDGE DUE TO 4W LOADING; H = 20, b/a = 0.1, b = 7 FT

~ ... ,.... .... " r--- -<-,,-

2.5&

2.0

1.5

1.0

Beam B k = 0 P = 10k ips

11,~ 0.5

1

$// ~

+-' c: OJ E 0

::2:

0

-0.5

-1.0

-1.51 I I , I

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Distance From Simply Supported Endja

1iI'~' ~.:ori""'" I .

r' ........ ~.~~l~ ~ : f


~ M M ~ M r"~l

0.8 0.9

~ 11 ~··--l

~ 1 ~

1.0

~1 fI'H'W'''' '11

'1

'10 co 'r- 0

I ., ::,.:::: r-

+J C QJ

E

2.5

Beam C k = 0

2.oL P = 10k ips

1 .5

1.0

~ 0

-0.5

-1 ,0

-1.5L, ____ -L ____ ~----~--~~--~~----~--~~--~~--~~--~ o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Distance from Simply Supported End/a


U)

jIlo ' ........ : ~ ..... 'tiI: r-''''''''

2.5i------~------~-------T-------r------~------~------~------T-------r-----~

Beam A ./ k = 0.05

2.0 P = 10 ki ps

1 .5

l.0

co ·,....0

I ., r ~r- 0.5

+J C Q)

E 0

::E

l/ff/ ~

0

-0.5

-1 .0

-1.5, I I I o 0.1 ~ ~ -- 0.5 0.6

Distance From Simply Supported End/a

FIG. 4.1B (CONTINUED)

~~' !","lnuOl r' ,_. '"' I

"{I~.· ~ H . I ,- - ~ I ::: .. ~ M ~ n ,....----., i

~~ I

0.7 O.B 0.9 1.0

~--'-l I J

~~ r -1

1O N

~, 1

2.51----------------~------~------~------~-------r------~------~-------r------~

Beam B k = 0.05

2.d-- P = 10 kips

1.

r co 'r- a

I "' ~ r--

-l-l 0.5 c QJ I //// ~ ~ ~~'''"~

lD E w 0 ~

0

-0.5

-1.0

-1.51~ ____ ~ ____ ~ ____ ~ ____ ~ ____ ~~ ____ ~ ____ ~ ____ ~ ____ ~ __ ~ o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 O.B 0.9 1.0



• ''Iot'-l .... ~

:1 r~ .... u ••

2.5 -. ---r----r------,---...----~-___r--__,_--,__-~r_-___,

Beam C k = 0.05

2.01- P = 10 kips

1 .5

1 .0

cl8 'r- 0 I ... ~ r-

~ U.brf//~ c OJ E 0

::E 0

-1.0

-1.5 o

~. fRIlOoi)!>tJII , ...

l ~~~ ",'.,

~~~"1'g

0.5 0.6 0.7 0.8 0.9 1.0 Distance From Simply Supported End/a


~ ,., r'1 ~ r.~~ ~1 "II'!I~:'-"~ , .•• f :!:,:: . .;. M ~"--l --"~l

J ~ '1

I

95

r-________ ~--------~--------T---------~------~~~------~-------------------O

c:::( c

E ro II Q.J co ..::.:::

~ ______ d_ ______ ~------~------~~----~~------~------~------~ L.(')

N o

OOO'L 'U~->l - +uawo~J

o o

I

01

o

co

o

r-

0

~

o

L.(')

0

<::::l'"

0

M

0

N

0

0

o

ro '-... ""0 C

L1.J

-t-l S-0 0.. 0.. ::::s

(/)

>, Cl W

0.. ~ E z:

........ (/) l-

E z: 0

0 U S-

LI-

Q) CO u c ro <::::l'"

-+-l Vl

Cl <..!J

LI-

co ·0 Or-

E ro II II (J)

CO ~ CL

96

0

O"l

0

CO

0

r--..

0

1..0

0

L.(')

0

c::::r-

0

(V)

0

N

0

o

L-______ ~~--------~~------~~--------~--------~ .~--------~--------~------~~O L.(')

N o

000' L _ "I uawol.1 'u~-)l + V'4

o I

. 1 l' t,

r I

L

i L_

r~

~

ro .......... '"'0

r -, -C I

W I i

'"'0 (J)

+-> :.... [ 0 .,

CL CL :::l

(/)

>, Cl (': r-- W CL :::J E z:

(/) r-z:

[ E 0 0 U s...

l.J....

(J) CO U

[ c ro c::::r-

+-> Vl

(.!J

Cl ......... l.J....

" ,~

i

r fr 0::.:: . ....,'";

r co 'r- 0

I " ~ r-

+-l C OJ E 0 ~

--~--~--~~--~--~~--~--~~ 2.51

Beam C k = 0.1

2.0 l- P = 10k ips

1.5

1 .0

////~

0

-0.5

-l.0

~ ~~~,~

-1.5, I I I I I I I I I I

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0



U)

-.......J

r co 'r- 0

I " :::.c::: ...-

+J C OJ E 0

:E:

'."".-r..,1f,,1IC\ ,..,-.,..-

2.5

Beam A k = 0.2 P = 10k ips

1 .

1.0

0.5

1/// ~ ~~ I 1..0 ex>

0

-0.5

-1. 0

-1.5~1 ______ ~ ______ ~ ______ ~ ______ ~ ______ ~ ______ ~ ______ ~ ______ ~ ______ ~ ______ ~

o O. 1

.~' ~ I

0.2 0.3 0.4 0.5 0.6 0.7



~~~,~~~ L, . / t, ~ J ~ n r--~

i -.,

0.8 0.9 1.0

~ r'"----'

~ ~~~

r co 'r- 0

I ... ~ r--

+-> c <lJ E 0 ~

2.51

Beam B k = 0.2

2.0 l-- p = 10k ips

1.5

0.5

////~ ~ ~~,~

0

-1.0

-1. 5 I I I I I I I I I I o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 O.B 0.9 1.0



lO lO

100

__ --------~--------~--------~--------~------~_'}_------~--------~--------~o

LO

N

(/)

CL

~

N U ·0

Or--

E ro II Cli

co ~ CL

o N

LO 0

000' l ·u~->I

LO LO 0 0

0 0 I

- +uawoW

L..()

r--I

co o

o

o

M

o

N

0

0

0

ro -..... -0 C

w

-0 CJ +-l S-0 CL CL :::s

V> 0

>, w ~

CL z E

or- l-V> Z

E 0 u

0 S-

I..L-

CJ co

u c ~ ro +-l (/) <..!J

or- t--<

0 I..L-

CIt ~' • .2

l~

r r --I

I l --

f l __

r: r" , i . L.~

tf.::;

[ '.'"

[,.

[

[

(.

r f. ,...-..i

,-i

i i

!

1 i

..

L. ,,'

'10 co ·,.....0

I ,.. ~ r-

+-l C OJ E o

:::E

2.5ri------.------.------,-----~r_-----r------~-----.------~------~-----

Beam A k = 0.4

2.0'- P = 10 kips

1 .5

1 .0

0.5

O~ IU L ~ I I I :7

-0.5

-1 .0

- 1 .5 I I o ~ ~ O. I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0



o

r co or- 0

I " ~ .--

+J c (lJ

E 0 :E

~, ,!.,~ .... ,.,. ~ .. -~-

2.5_i------~------~----~------~------_r------._------._----_,------_.------,

Beam B k = 0.4

2.0L P = 10 kips

1 .5

1.0

0.5

////~

0

-0.5

-1. 0

~ ~~"'~

-1.5~, ______________ ~ ____ ~ ______ ~ ______ ~ ______ ~ ______ ~ ____ ~ ______ ~ ______ ~

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7



~. ~ ..... "~ ;, j . ~ r-1 ~ ~ If:'~~

;u . . i ~ r---l .. . -..,

0.8

~ I. !;

0.9 1.0

r-'l :-J

0 N

.,...-..

r co 'r- a

I n

~ r--

+.J c <lJ E 0

:E

2.5~i------~--------~-------r------~--------T--------r------~--------T--------r------~

Beam C k = 0.4

2.0t- P = 10 kips

1 .5

1 .0

0.5

/// ~~

0

-0.5

-1 .0

./" ----------~ ~ " ""

-1.5 I ________________________ ~ ____ ~~ ____ ~~ ____ ~~ ____ ~~ ____ ~ ______ ~~ ____ ~

o O. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


FIG. 4.18 ~ONTINUED)

a w

+> s::: <lJlrO Eo.. 0 ~

\I

E u

......... ,~ ",. ........ ~.-

0.20

0.15

0.10

0.05l-- /// / / ./ ~~ ~ "" ,",~ -I

O~ ,/ >

-0.05

-0.10 I I I I I I I ,

-~- --......

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

~.

Distance From Simply Supported Endja

FIG. 4.19 POSITIVE MOMENT ENVELOPE AND INFLUENCE LINES FOR A TWO SPAN CONTINUOUS BEAM DUE TO A CONCENTRATED LOAD

~ r' .,_. '. "f.~~ " M ~ ~~ r---l ~ M P1 I ~' .... : ,,~~J L . r-.-l :--,

0 +=>

~ ~!. ~

0.3

0.2 -

0

......, 0.3 c OJ E 0

:E:

OJ ro > 0.. 0.2 !"'"""

......,

Vl 0

0..

O. 1 -E

u

0

0.3

0.2

O. 1

o o

105

I I I

Q...... 8 - --------, ...... - --------

/\ " L.J

-Beam A

I I I

I I I

'"'11 Q

.... =

-Beam B

I I J

0-----0------

Beam C

5 10 15 20

Relative Girder Stiffness H

6

0

0

b/a 0.05

b/a o. 1

b/a 0.2

Continuous Bri dge. Simp 1 y Supported Bridge

FIG. 4.20 RELATIONSHIPS BETWEEN MAXIMUM POSITIVE MOMENTS IN GIRDERS AND RELATIVE GIRDER STIFFNESS, H; NO DIAPHRAGM, 4W LOADING, b = 5 FT

+-> c OJ E o ~

OJ ro > 0...

.r-

+-> U'l o

D..

II

E u

106

0.3~ ____ ~~ ____ ~

0.2 .......

0.11-

o ________ ~ ______ ~ ____ ~~~--~

O. 31---r---:-::zE~=~--...-..y

0.2

O. 1

Beam B

o

0.3r---r----:c:::::==r==---Q

0.2

0.1

Beam C o

o 5 10 15 20



r ri

t .

f ::. L

r

[

0.2

O. 1

0

0.3

+-> c:: QJ

E 0.2 0 z::: QJ ro > 0..

or-

+-> U1 0

O. 1 0..

II

E u

0

0.3

0.2

0.1

o o

107

0-----o----------=-:2 6.-------

Beam A

----_-0--..--

Beam B

Beam C

5 10 15 20 Relative Girder Stiffness H

6 b/a 0.05

o b/a 0.1

o b/ a = 0.2

--- Continuous Bridge Simply Supported Bridge


108

0.3 I I I

6 b/a 0.05 0 b/a = 0.1

0.2 r- 0 b/a 0.2 -Beam A Continuous

Bridge --- Simply

0.1 r- 1\ - Supported ~ t~ Bridge

0 I I I

0.3 I I I

+-> c ClJ E 0 Beam B ~ 0.2 ~ -ClJ > C"O

.r- 0.. +-> ro

ClJ z:

O. 1 r- () -II 0 tr'

E u

o ~ ____ ~1 ______ ~1 _______ ~1 ____ ~

0.3 I I I

Beam C 0.2 - -

- 0- ~ n ...,r 0.1

~

o I I I

o 5 10 15 20 Relative Girder Stiffness H

FIG. 4.23 RELATIONSHIPS BETWEEN MAXIMUM NEGATIVE MOMENTS IN GIRDERS AND RELATIVE GIRDER STIFFNESS, H; NO DIAPHRAGM, 4W LOADING, b = 5 FT

" r,',

.. [':,""',

r

f~~" L

I" [

[

. I:'

f

109

O.3~----~,------~,------~------~ I

6 b/a = 0.05 Beam A 0 b/a = o. 1

0 b/a = 0.2 -0.2 f-Continuous Bridge

o. 1 r-1\

V " --- Simply

~) Supported u Bridge

0 I 1 1

0.3 I I I

+J s:: <lJ E 0 Beam B

::E

<lJ ro 0.2 - -> CL.

+J rc

<lJ z:

0.1 g-- -Q-0- _l

~ LS

II

E u

0 I I I

O.

Beam C o.

o. 1

o -o------~5----~1-0------~1~5------2~0


FIG. 4.24 RELATIONSHIPS BETWEEN MAXIMUM NEGATIVE MOMENTS IN GIRDERS AND RELATIVE GIRDER STIFFNESS, H; NO DIAPHRAGM, 4W LOADING, b = 7 FT

110 .,.. ~

1·' 0.3 I I r r' 6 b/a 0.05

Beam A 0 b/a = 0.1

0.2 - 0 b/a 0.2 L .. _ -Continuous

1\ 1\ Bridge

[ ~ .,

f} ~ --- Simply O. 1 "1

~ - Supported Bridge r

0 I I I

L_.~ 0.3

E ..... ~ -.

+oJ Beam B c 0.2 [" ClJ

E ..

0 :E

ClJ > ro [ or- CL

+oJ ro O. 1 ClJ z:

[ "', II "

U - 0

(: 0.3

r Beam C 1·::ii

0.2 1" . :~

: '---

0.1 l ;: [~

0 0 5 10 15 20 l' -


FIG. 4.25 RELATIONSHIPS BET~JEEN MAXIMU~1 NEGATIVE ~·10~1ENTS IN ~

GIRDERS AND RELATIVE GIRDER STIFFNESS, H' NO DIA- .~ j 1

PHRAGM, 4W LOADING, b = 9 FT i.~

tr.: -

f t ;

E u

111 0.75 _-----r-----r---..,.....----,

6 Gi rder A b = 5 ft o Girder B

0.50 o Girder C

0.25

o L-______ ~ ______ ~ ______ ~ ____ __

0.75 _-----r---....,..----r-------,

b 7 ft

0.50

0.25

o ~ ______ ~ ______ ~ ____________ ___

I T I

b = 9 ft

0.75 t- -

-

I\. " - ..,

0.25 t- -Positive

Negative at Support

o L-______ i~ ______ ~I ______ ~I ____ ~

o 0.1 0.2 0.3 0.4 0 0.1 0.2


FIG. 4.26 MAXIMUM MOMENTS DUE TO TWO-TRUCK LOADING VERSUS DIAPHRAGM STIFFNESS; H = 20, b/a = 0.2

0.3 0.4

+-l C Q) ro E CL 0

::E:

E u

11 2

0.3r-----~------~----~------~

0.2

0.1 Max. Positive Moment at End Span

O~ ____ ~ ______ ~ ______ ~ ____ ~

0.3

0.2

O. 1

0

0.3

0.2

O. 1

T I

-(~ 1\

Ir

>-

'-' u '>-!

Max. Nega t i ve ~1oment at

I I

Max. Positive Moment at Interior Span

I

-

.~ -

Support

I

O ________ ~ ____ ~~ ____ ~~ ____ ~

o 0.1 0.2 0.3 0.4


6 Girder A o Girder B o Girder C

FIG. 4.27 MAXIMUM MOMENTS IN GIRDERS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; THREE SPAN CONTINUOUS BRIDGE, H = 20, b/a = 0.1, b = 7 FT

"'--'~

i l "

r ~

! L._

r r-'-" j \ ~~ ....

[

r· .. .,

[

[

I'.

L.,

i 1

L.:.

r c: C) or- C)

I '" :::.:::: r--

+J c: OJ E 0

::E

0

fJ Girder A o Girder B o Girder C

FIG. 4.28 INFLUENCE LINES FOR NEGATIVE MOMENTS AT INTERIOR SUPPORT OF THREE SPAN CONTINUOUS BRIDGE DUE TO 4W LOADING; H = 20, b/a = 0.1, b = 7 FT

w

.- .. ~~~" rv~.-"-

'1° co 'r- 0

I " ~.r-

+-> c OJ E o

::a:

3.0

2.0

1 .0

o

~,

6 Gi rder A

o Gi rder B

o Gi rder C

.p:.

FIG. 4.29 INFLUENCE LINES FOR MIDSPAN POSITIVE MOMENTS IN GIRDERS OF THREE SPAN CONTINUOUS BRIDGE DUE TO 4W LOADING; H = 20, b/a = 0.1, b = 7 FT

,.'t";""";l~'" r .,' ''',,"' !~i~' ~.~ ", M ~~" r.r, r-' !J~ r. ;~'-1 -., ..... \of{C"!I'

I ' ,oj . :" '.'

L ............... ~l. .'~ i: . .ill~

L:! .. ;· .....

.10

co 'r- 0

I '" ~ r-

+J C OJ E o

::E

L~

3

2

o

~ .. L,'(~ jj~

H = 20, b/ a = o. 1 , b = 7 ft, k = 0

... ... IiIIIiI ~ "-'!I!"-.

6 Girder A o Gi rder B o Girder C

~ h

Int. Support

• I \i, .. ',. .J

Positive Moment Envelopes of Girders for 4 Span Continuous Bridge (4W Loading)

.,,,,,~ .• ,J

OK Y

Influence Lines for Maximum Negative Moments in Girders at Interior Support - 4W Loading

FIG. 4.30 POSITIVE MOMENT ENVELOPES AND NEGATIVE MOMENT INFLUENCE LINES FOR THREE SPAN CONTINUOUS BRIDGES WITH VARIOUS DIAPHRAGM STIFFNESS

~ ~, •.. "._.J .... ~~" .. J li4iJ'"".u

i I (J1

<l.

I

'10 CO 'r- 0

I '" ~ r-

+J c: OJ E: o

:::!::

3

2 r

o

H = 20, b/a = 0.1 b = 7 ft, k = 0.05 6 Girder A

~~~ 0 Girder B 0 Girder C

Int. Support

Postivie Moment Envelopes of Girders for 3 Span Continuous Bridge (4W Loading)

or y



ct I (J")

<l.

I

l"' ....... ,... tl·.,~ ~MMi L--.4 ~ ~ L . .J. w....... ___ ... IIWIIIII ~ ~~ l·· .. "" .• .J 1'<>,."..,-4 ~. ...' ... ,..... • ... .,...,,1 ~k-i.w

3 I

2t-

0

r CO 'r- 0

I " ~ .--

+J C OJ E 0 ~

-1.0

-0.5

o

H = 20, b/a = 0.1 b = 7 ft, k = 0.1 6 Girder A

0 Girder B ~~ 0 Girder C

Int. Support Positive Moment Envelopes of Girders for 3 Span Continuous Bridge (4W Loading)



<t. , --' --' '-.J

Cl.

I

'10 co ·,.....0

I " ~ r-

4-l C OJ E o

::::E

3 I

H = 20, b/a = 0,1 b = 7 ft, k = 0.2 ~ Girder A

2 J- ~~ o Girder B 0 Girder C

Ox A

-1.0

-0.5

o

Int. Support




Cf.

I co

~

I

\ :; 't',' L ':' 'I l.;· L' ";".1 : ." I;.':':":U L'i~._J ~ ........ j I , 'I J I: J" .• " ..... _ .:~ ~ __ .i ~.... __ W~ ~ __ ..... ....... ~ ~ .... _ . ..J ... ,,"'~ ""'~H' ... 4gj i""." ...• i ~ .... ,.".. "",,~,.

'1

0 CO

'r- 0 I ... ~ ..-

+-> C OJ E o

::E::

3

H = 20, b/a = 0.1, 6 Girder A b = 7 ft, k = 0.4 o Girder B

2 '- ~~ 0 Girder C

0, l

- 1 .

-0.

Int. Support


O~ ~



~

I \.D

<l

I

120 0.4~------~----~~----~------~-------r------~----~

0.3

ClJ > ttl 0.2 .~ CL 40..)

Vl o

CL

/I

E u

O. 1

o

b;t 20 10 5

0.05 • I> 0

O. 1 II I 0

0.2 • • t::.

No Diaphragm

5 6 7 8 9

Beam Spacing in Feet

FIG. 5.1 MAXIMUM POSITIVE MOMENTS DUE TO 4W LOADING COMPARED WITH AASHO DESIGN VALUES; BEAM A

., I

~

.~

j

1

]

l ]

I

'.: .. >1 1

J

... -~ I

, 1 -J

:-1 . ; ~:.::~

I J ~ (. -

Q) > 'r- ro +-l 0....

V1 o

0....,

E u

1 21

0.4--------~------~----~~----~~----~------~------~

0.3

0.2

0.1

o

~ 20

0.05 • 0.1 • 0.2 4

10

()

II

A

eel U

E E ro ro OJ Q)

eel eel

No Diaphragm

5

5

0

0

6.

6

eel U

E E ro ro Q) OJ

eel eel

7


8

EE roro Q)Q)

eel eel

9

FIG. 5.2 MAXIMUM POSITIVE MOMENTS DUE TO 4W LOADING COMPARED WITH AASHO DESIGN VALUES; INTERIOR BEAMS

0.4

O.

+-' C ill E 0

:::E

ill > rUo

.'-- CL O. +-'

ro

ill z::

" E

u

O. 1

o

1 22

~ 20 10 5

0.05 • () 0

0.1 .. IJ 0

0.2 A £ 6

t

No Diaphragm

5 6 7 8 9 Beam Spacing in Feet

FIG. 5.3 MAXIMUM NEGATIVE MOMENTS DUE TO 4W LOADING COMPARED WITH AASHO DESIGN VALUES; BEAM A

I i

j

1 l

.-:.1

]

]

]

]

"I I

-.J

:.J

]

-·· .. '1 'j b:::a

1 I

...-I

Q)ro >0....

or-.j-l

ro Q)

:z::

u E

123

0.4------~------~----~------~------~----~----~

0.3

0.2

o. 1

0

~ 20 10 5

0.05 • ~ 0

001 II II 0

0.2 A A t:.

AA ~~

~

A ~ ~ t)

()

0 0

co u co u co u E E E E E E ro ro ro ro ro ro Q) Q) Q) Q) Q) Q) co co co co co co

No Diaphragm

5 6 7 8 9


FIG. 5.4 MAXIMUM NEGATIVE MOMENTS DUE TO 4W LOADING COMPARED WITH AASHO DESIGN VALUES; INTERIOR BEAMS

effects of diaphragms in continuous slab and girder … · 2015-05-29 · point load in continuous...

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