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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.
Uncl assi fi ed Uncl ass i fi ed 123
Form DOT F 1700~7 (8-&9)
22. Prlco
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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
2
3
4
5
6
7
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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.
Page
1 1 4 6
9
9 9
15
15 15 16 23
25
25
26
29
34
37
39 41
43
45
50
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Table
2.1
4.1
4.2
4.3
4.4
4.5
4.6
vi
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.
Page
75
76
77
78
79
80
81
82
83
84
85
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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 .
Page
86
87
88
89
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 .
Page
110
111
112
113
114
115
120
121
122
123
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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
t. ~ r
r-1 L..
r If_ :.
r.··· 1
t.
3
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
I
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
J
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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
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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 .•
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.;!
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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
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1 L..
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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.
..--!
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1 L ._
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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 .
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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
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t.
1
1 i L._
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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-(
~
Positive Negative at Support
I I
o 0.1 0.2
Relative Diaphragm Stiffness
FIG. 4.9 MAXIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 5, b/a = 0.1
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- - ~ -
Positive Negative at Support
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
Relative Diaphragm Stiffness
FIG. 4.11 MAXIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 10, b/a = 0.05
-
)
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
6. Girder A 0 Girder B
~ 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-
Positive Negative at Support
o I I
o 0.1 0.2 0.3 0.4 0 0.1 0.2
Relative Diaphragm Stiffness
FIG. 4.12 MAXIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 10, b/a = 0.1
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
6. Girder A 0 Girder B
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
Relative Diaphragm Stiffness
FIG. 4.14 MAXIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 20, b/a = 0.05
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 '[ ..
Relative Diaphragm Stiffness
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
FIG. 4.16 MAXIMUM MOMENT COEFFICIENTS DUE TO 4W LOADING VERSUS DIAPHRAGM STIFFNESS; H = 20, b/a = 0.2
-
~
-
-, --
-
-
')
~
-
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
Relative Diaphragm Stiffness
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
FIG. 4.18 (CONTINUED)
~ 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
FIG. 4.18 (CONTINUED)
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
Distance From Simply Supported End/a
FIG. 4.1B (CONTINUED)
• ''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
FIG. 4.18 (CONTINUED)
~ ,., 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
Distance From Simply Supported End/a
FIG. 4.18 (CONTINUED)
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
Distance From Simply Supported End/a
FIG. 4.18 (CONTINUED)
~~~,~~~ 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
Distance From Simply Supported End/a
FIG. 4.1B (CONTINUED)
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
Distance From Simply Supported End/a
FIG. 4.18 (CONTINUED)
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
Distance From Simply Supported End/a
FIG. 4.18 (CONTINUED)
~. ~ ..... "~ ;, 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
Distance From Simply Supported End/a
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
Relative Girder Stiffness H
FIG. 4.21 RELATIONSHIPS BETWEEN MAXIMUM POSITIVE MOMENTS IN GIRDERS AND RELATIVE GIRDER STIFFNESS, H; NO DIAPHRAGM, 4W LOADING, b = 7 FT
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
FIG. 4.22 RELATIONSHIPS BETWEEN MAXIMUM POSITIVE MOMENTS IN GIRDERS AND RELATIVE GIRDER STIFFNESS, H; NO DIAPHRAGM, 4W LOADING, b = 9 FT
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
Relative Girder Stiffness H
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' -
Relative Girder Stiffness H
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
Relative Diaphragm Stiffness
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
Relative Diaphragm Stiffness
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
Influence Lines for Maximum Negative Moments in Girders at Interior Support - 4W Loading
FIG. 4.30 (CONTINUED)
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)
Influence Lines for Maximum Negative Moments in Girders at Interior Support - 4W Loading
FIG. 4.30 (CONTINUED)
<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
Positive Moment Envelopes of Girders for 3 Span Continuous Bridge (4W Loading)
Influence Lines for Maximum Negative Moments in Girders at Interior Support - 4W Loading
FIG. 4.30 (CONTINUED)
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
Positive Moment Envelopes of Girders for 3 Span Continuous Bridge (4W Loading)
O~ ~
Influence Lines for Maximum Negative Moments in Girders at Interior Support - 4W Loading
FIG. 4.30 (CONTINUED)
~
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
Beam Spacing in Feet
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
Beam Spacing in Feet
FIG. 5.4 MAXIMUM NEGATIVE MOMENTS DUE TO 4W LOADING COMPARED WITH AASHO DESIGN VALUES; INTERIOR BEAMS
Recommended