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Fritz Engineering LaboratoryReport 223.10
LATERAL LOAD DISTRIBUTION IN MULTI-BEAM BRIDGES
by
Alfred Roesli
Progress Report 10 - Prestressed Concrete Bridge Members
Part of an Investigation Sponsored by:
PENNSYLVANIA DEPARTME~r OF HIGHWAYSBUREAU OF PUBLIC ROADSREINFORCED CONCRETE RESEARCH COUNCILJOHN A. ROEBLING'S SONS CORPORATIONAMERICAN STEEL AND WIRE DIVISION,
. U.S. STEEL CORPORATIONAMERICAN-MARIET.TA COMPANY,
CONCRETE PRODUCTS DIVISION
July 1955Lehigh University
Institute of ResearchBethlehem, Pennsylvania
~ ~. ~-...:'_*.- .... : • ..- - • -'_.. ,',,' ~ ..
(
TABLE OF CONTENTS
Page No.
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I.
II.
III.
IV.
V.
VI.
ABSTRACT
INTRODUCTION1. General2. Introduction to the Theory of Orthotropic
Plates
DIFFERENTIAL EQUATION FOR THE MULTI=BEAM BRIDGE1. Derivation of Differential Equations2. Experi.mental Determination of the Relation
Between a and f33. Assumption for the Relation Between a and f34. Limiting Cases of Di.fferential Equation5. Boundary Conditions
INTEGRATION OF THE DIFFERENTIAL EQUATION1. General Considerations2. Plate Strip of Infinite Width Under Line
Loading3. Solution for the Bridge of Finite Width4. Explicit Form of Formulas5. Influence Surfaces
LATERAL LOAD DISTRIBUTION1. General2. Definition3. Approximation for the Lateral Load Distribution4. Properties of the Coefficient of Lateral Load
Distribution
NUMERICAL CALCULATIONS1. General2. Accuracy of Results3. Descri.ption of the Tables4. Coefficients of Lateral Load Distribution
COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS1. General2. Deflecti.ons3. Coefficients of Lateral Load Distri.bution
1
33
7
1717
19212324
2525
26334647
51515154
57
6060636566
69696971
VII. USE OF THE COEFFICIENTS OF LATERAL LOAD DISTRIBUTION 74
NOTATIONS
LIST OF REFERENCES
110
113
LIST OF TABLES
Page No.\
Table I-Formulas for the plate Strip of InfiniteWidth
Table 11- Formulas for General Case
Table III - Formulas for Articulated Plate
Tables IV through IX
Tables X through XV
Tables XVI through XXI
Tables XXII through XXVII
81
82
83
84
85
86
87
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Table XXVIII - Coefficients of Lateral Load Distribution(a/h= 1.7) 88
Table XXIX - Coefficients of Lateral Load Distribution(a/h = 1.0) 89
Figure
LIST OF FIGlJRES
Page No.
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Multi~Be&~ Bridge With Lateral Prestress 90Schematic Cross-Sectiol)l of a I\fu.1ti=Beam Bridge 90Corrugated Steel Sheet as Orthotropic Plate 90Gridwork System 91Forces and Moments on Dif,ferential Element 91AS3umed Re,lations Betw'ee,n a a.nd J3 92Deformation Produced by Constant Twisting Mbments 93Equivalent Corner Loading 93Plate Strip of Infi.nite Width Under Line Loading 94Bridge of Finite Width 94Assumption for the Longitudinal Distribution of
a Wheel Load 95Distribution of Load Applied at Center for Q = 0 96Distribution of Load Applied at Center for Q = 0.1 97Distribution of Load Applie·d, at Center for Q = 0.5 98Distribution of Load Applied at Center for
Isotropic Plate Q = 1.0 99Distribution of Load Applied at Edge for Q = 0 100Distribution of Load Applied at Edge for Q = 0.1 101Distribution of Load Applied, at Edge for Q = 0.5 102Distribution of Load Appli.ed at Edge for
Isotropic plate Q = 1.0 103Comparison Between Experimental and Theoretical
Deflections for Load Applied in. Bridge Axis 104Comparison Between. Experimental and Theoretical
Deflections for Load Applied at Edge 105Compari.son Between Experimental and Theoretical
Load Distribution Coefficients for Load atBridge Axis 106
23. Comparison Between Experimental and TheoreticalLoad Distribution Coefficients for Load at Edge 107
24. Determinati.on of Load Carried by Edge Beam 10825. Percent of Wheel Loads Carried by Center - and
Edge Beams in a 27 ft. Wide Bridge of VariableSpan 109
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ACKNOWLEDGEMENTS
This report concerns a part of a research program on
prestressed concrete, carried out at the Fritz Engineering
Laboratory, Lehigh university, Bethlehem, Pennsylvania of
which Professor W.J. Eney is Director.
This research program is guided and sponsored by the
Lehigh Prestressed Concrete Committee, Hr. A.E. Cummings,*
chairman, and is represented by the following organi.zations:
Pennsylvania Department of HighwaysU.S. Bureau of Public RoadsAmerican Steel & Wire Div., U.S. SteelJohn A. Roebling's Sons CorporationReinforced Concrete Research CouncilConcrete Products Company of America
The author is greatly indebted to Dr. C.E. Ekberg, Director
of this research program and Professor in charge of the pre-
sent report. His h~lp and advice, as well as that given by
the entire special committee for this doctoral work with
Dr. F.W. Schutz, Jr. as chairman, is sincerely appreciated.
Deep appreciation is expressed for the help given by
Dr. E. Bareiss who prepared the program for computation of the
numerical values. Tabular values were compiled on a contract
basis by Remington Rand, Inc., New York on their Univac Com-
puter.
The many suggestions and the untiring help received from
the entire staff of Fritz Engineering Labcratory and especially
* Deceased
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from the author's friends, Dr. B. Thllrlimann, who furnished
much technical advice, and Messrs. A.N. Sherbourne and
A. Smislova for help on the editing of this paper, are
sincerely acknowledged. The cooperation of Mr. I. Scott,
who prepared the figures, and Mrs. V. Olanovich, who typed
the original manuscript, is gratefully appreciated •
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ABSTRACT
A method is derived to analyze multi-beam bridges and"
especially to determine for what portion of the live load
each beam must be designed.
Present design procedure being rather conservative in
its scope, this analysis will represent a more realistic
solution for the problem and may lead to a more economical
design of such bridges'.
The method is based on the theory of orthotropic plates
and ',on the main assumption that the interaction of the beams'
provided by shear keys and lateral prest~ess exclude any
slip between the beams.
Formulas for deflections, moments and forces are
derived for bridges with various degrees of lateral prestress.
A limiting case was found in the "articulated plate", a
bridge with no lateral bending stiffness, but with a beam
connection which transmits the full shear force.
Numerical values are presented for the most important
loading conditions and for bridges of various sizes • For
design purposes the coefficients of lateral load distri-
bution may be used •
Among other results, it was found, that for two
standard trucks placed side by side on a 27 -f,t. wide bridge,
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the maximum load carried by a 3-ft. wide beam is 55% of
a wheel load, as compared to 80% recommended by the speci
fications, and is almost independent of the amount of
lateral prestress •
..
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I. INTRODUCTION
1. General:
The design of modex'n structures tends to utilize
.construction materials to an opti.m·...'m while an appropriate
factor of safety is sti.ll me.intained. This requi.res how
ever an accurate investigation of the stresses in the
structures. In many cases) the ~)i.mplified analysis based
upon the beam theory has to be replaced by a more exact
one, considering the structure as a two or three dimensional
one, such as a plate or a shell. It i.s h:>ped that the
following study is a contribution to this development.
The investigation described herein de,sd.s 'with the
lateral load distribution in multi··'bearn bridges. This
type of bridge is conetructed from precast beams made of
reinforced or prestressed concrete. They are placed side
by side on the abutments and joined together laterally
by steel rods -y;hich mayor may not be prestressed.. In order
to incr.ease the interaction hetvJeem the beams, continuous·
longitudinal shear keys are formed at the joints, e.g., by
dry packed mortar in a recess formed at the sides of each
preca~t unit. Fig. 1 shows an isometric view and Fig. 2
a cross-section of such a multi-be&'1l bri.dge.
.,
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The problem to be i.nvesti.gated i.(..; the interaction of
the' beams and the determination. of the portion of the load
each beam must carry should the load be applied to one of
the beams. Various degrees of lateral prestress will have
to be considered.
In the present design proce~~re of these bridges a
limited interaction of the beam~, is considered. It is
assumed that a loaded beam c81.rr.'ies 80% of the applied load
and that the remaining portion is distributed among the
adjacent beams. This assumption is based on an inter
pretation of design section 3.3,lb of the 1949 AASHO
Specification. (1)*
Past experience has shown, th.e,t bridges d.esigned
accordingly, a.re stiffer than expected and that: the in
dividual beams are unnecessarily heavy. (2) The application
of this bridge system is thus limited to short span lengths.
In order to increa.se the span length and to desi.gn such
bridges more economically, a more exact analysis is
necessaty.
Extensive investigations have been made on similar
bridge systems. Of most interest for the present study
* Numbers refer to List of References .
'..,...
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are the ones made on bridge~ formed by tv.10 systems of
parallel beams spaced at equ.al inte1:vals i.n the longi
tudinal and lateral direction. The beams are rigidly
connected at their points of intersection and support the
bridge deck. This type of constru,ction will hereafter be
referred to as a gridwork and is ~'5ih0wn in Fi.g. 4.
Several rn'ethods, some of 't\yhich a.:re only approximate,
have been developed to analyze gridwork systems. Massonnet
presents in Reference (3) a very helpful survey of these
methods. It appears that thB analY8i.s of a gridwork as
an orthotropic plate is 'very F.":fficient. The gridwork with
its discontinuous elastic properties is replaced by an
equivalent plate, having the same average bending stiff
nesses in the two directi'...~ns as existi.ng in the gridwork.
Y. Guyon applied this method to investigate the lateral
load distribution of beam and girder prestressed concrete
bridges. (4) To simplify the an81ysis he neglected the
torsional resistance of the beams. Based on this work
Massonnet extended the i.nvestigation and included the
torsional resistance of the beams as well as that of the
bridge deck. (5 )
Both of the above mentioned authors derived general methods
to analyze these structures and p:r:ep.:t.:r.'ed design ta.bles. The
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latter were set up for a live load of sinusoidal nature.
They considered this type of loa.ding as a sufficiently
close approximation of any loads encountered in the design
of bridges.
P.B. Morice and G. Little described some laboratory
tests on models of gridwork systems. (6) They concluded from
their experiments that the analysis of gridworks can well
be based on the theory of the orthotropic plate.
A bridge formed by beams with the sides in continuous
contact to each other could be considered as a limiting
case of a gridwork and the methods mentioned above could
be used for its analysis. However, it is shown in this
investigation that the assumptions on whi.ch these methods
are based do not generally hold for multi-beam bridges.,
A method is therefore derived in this work which will
generally hold for the- latter systems, and which is also
based on the theory of orthotropic plates. The theory is
first described and. then modifi.ed for multi-·beam bridges.
The resulting differential equation is solved, considering
the structure as a plate with two opposite edges simply
supported and the other two edges free. Ntlmerical cal
culations are presented for the most i.mportant loading
conditions, for bridges with different spans and widths, and
•
also for va.rious degrees of l,eteral prestress an.d dif-
ferent sizes of beams. Using the coefficients of the
lateral loa.d distribution the rE.:i;';jult~S% are presented in a
form suitable for the practical design of such bridges.
2. Introduc tion to the-Th.~ory of Orthotrop}.c _Plates:
An orthotropic plate is defined a~ a plete with
different bending stiffnesses D = EI in two orthogonal
directions x and y in the plai.!'.e. of the plate. These may
result either from different moduli of elasticity Ex and
Ey of the material i.n the t,IiTO direetions, or from different
moments of inertia Ixand I y per unit: width of the plate.
An example for the first kind. of Grthotropic plates
is a timber plate. Assuming the X~a:Ki3 par:allel and the
y-axis perpendicular to the grai.n ll the modulus of elasticity
in the y-direction Ey is ~ a.ccording to E. Seydel, (7) only
about 1/10 of that in the x·-direction.
A corrugated steel sheet is; a typical. example of an
orthotropic material of the secon.d kin.d. The modulus of
elasticity is the same in every direction; the material
itself is isotropic. Here the dtfferent bending stiffnesses
are functions of the shape of the shE.~et. The a.ve,rage
moment of inertia with respect to the neutral plane in the
•
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direction perpendicular to the generator is much smaller
than in the direction parallel to the generator. Fig. 3
shows this case.
The basic assumptions in the theory of the orthotropic
plate are identical with those i.n the theory of the homo-
, genous plate, namely:
(1) The thickness of the plate is small compared
with its other dimensions.
(2) The deflections ware small compared with the
thickness of the plate.
(3) The transverse stresses oz:: are small and their
influence on the deformation c.an be neglected.
For a right hand coordinate system (x,y,z) where x
and yare in the plane of thE: plate and parallel to' the
two distinct directions of the orthotropic plate, the" ...:.
differential equation for the deflection w parallel to the
z-direction is given by~ !
(la)
• L
(EI)xand (EI)y represent the two bending "stii:fnesses per
unit width of the plate. 2H is a coefficient containing
two parts. The first part is the twisting resistance of the
plate and a,second, smaller part is a function of the two
bending stiffnesses modified by the Poisson's ratio. p(xy)
,
denotes the intensity of the load at the point (xy)
parallel to z.
Dividing equation (1a) by (EI)x, the following
differential equation results which i.s mainly used here-
after:
P (:K1Y)= ·--(F~'fr~ (lb)
2{3 =(2) (3)
•
2{3 shall be called the c.oeffi.ci.(~nt of torsional
rigidity and (l' the ratio of the bencirJI.g stiffnesses.
The derivation of equation (1) i.s attributed to
J 0 Boussinesq (1874) and is given, fen: example) in
Reference 8, page 188. M.T. Huber (9~10) applied it for
various plate structures of techrdcal i.mportance: Con-
crete slabs reinforced in two directions for cracked
...'-
cross-sections, beam and slab construction.~ gri.dwork systems;
corrugated steel sheets and plywood plates.
For the discussion of a multi-bearn bridge a coordinate
system (x,y,z) is assumed as shown. in Fig. 1 wi.th x and y-
axis respectively parallel and perpend.icular to the beam
in the middle plane of the bridge 0 The z'~axis is positively
directed downward and the origin is. located at midspan of
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one support. This type of bridge can be treated as an
·orthotropic plate of the second kind Ex = Ey if, in addition
to the basic assumptions of the general theory, the following
assumptions are made:
1. The connections between the beams are such, that
the points of contact in two adjacent beams de-
form equally.
2. The number of beams is large enough, such that
the real structure can be rep1aarl by an idealized
one with continuous properties, in order that the
differential calculus can be applied.
Th~ first assumption implies that the connections.
prevent the beams from slipping against one another, and
that they transmit the full shear force. This can be
accomplished either by lateral prestress and/or by shear
keys as shown in Fig. 2. With lateral prestress only the
shear forces are transmitted by, the ·friction produced by
the prestress. If;4 is the coefficient of static friction
between two concrete surfaces and erp.dA is the lateral
'prestress applied on the"di~ferentia1 area dA a maximum
shear force of !/f,o-:podA can be transmitted by the frictionA
forces without a slip.
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The second assumption requires that the width of the
beams is small compared with the width of the bridge such
that the actual lateral bending stiffness Ely and the co
efficient 2H, which are both discontinuous, can be replaced
by average values. This will be discussed in the following
for Ely or a and in a later paragraph for 2H or 2~.
If, for a sufficiently high lateral prestress and
for any given load, the entire beam sides remain in com-
pression; the bridge will behave as a homogenous slab. The
benqing stiffness Ely in the lateral direction will be
constant and will b~ the same in the direction of the
beams; or a = 1.
Assuming a smaller prestress, the bending moments My
in the loaded slab may produce tensile stresses in the
Joints between the bearns. Since the joints do not have any
or only minor tensile strength (from the shear keys) these
tensile stresses will have to be carried by.the prestressing
~lements. The joints tend the~efore to open up, thus.
reducing the bending stiffness i.n this cross-section; or a ~l.
A limiting case with no lateral bending stiffness in
the joints may be reached if no lateral prestress is applied., .
In this case 'tpe joints are unable to transmit bending
mo~ents, but according to assumpti.on ~, the shear keys trans-
mit the full shear forces. Such a structure can be thought
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of being formed of beams with the adjacent sides connected
by continuous hinges along the beams 0 Herea.fter this type
of structure will be called an articulated plate.
Fig. 2 shows a schematic cross-section of the bridge
with the joints having a smaller lateral bending stiffness
than the beams.
The simplest way of obtaining average values for the
lateral bending stiffness would be by experiment on a
model bridge. This could be ac~ompli~hed by supporting
the bridge along the two edge beams instead of the supports
at the ends of 'the beams. Subjecting the plate to a load
which does not vary along the length of the plate, pro
duces a deformed surfa.ce of cylindrical shape which can
easily be measured o Comparing these deflections with
the theoretica~ ones of an isotropic plate with the same
boundary and loading conditions, an average values for Ely
can be obtained. This test can be done for various magni
tudes and locations of the lateral prestressing elements.
Dividing Ely by the flexural rigidity Elx per unit width
of the beams gives the coefficient a. This may vary as
shown previously between the two limiting values 0 and 1.
For design purposes it should be accurate enough to assume
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first an appropriate value of a and later to check whether
the lateral bending moment can safely be transmitted by
the reduced section of the joints o
In the application of the theory of the orthotropic
plate the main difficulty is not so much the determination
of the coefficient a but the determination of the coefficient
of the torsional rididity 2~o
Various simplified assumptions can be found in the
literature for different structures. Most of these can be
traced back to Huber (9, 10,'11).
For two-way rein~orced concrete slabs, Huber recom
mended the following approximation:
~ =~ (4)
Massonnet (3) used a similar expression in his investigation
of gridwork systems viz:
~ =;v;- (5 )
t••
·..
where f is a parameter of torsion varying between 0 and 1.
This approximation allowed him to consider any proportion
of torsional resistance of the beams and the bridge deck.
The most commonly applied expression for a gridwork
system as shown in Fig o 4 is:
'.-14-
(6)
where Cxand Cy are the torsional rigidities of the individual
stringers and beams, respectively, spaced at distances Cx
and cy apart. Dx is the average bendi.ng stiffness per unit
width of the gridwork in the x-direction.
The following relation is in general used to determine
the torsional rigidity of beams of rectangular cross-section
a.h:
."
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?C := KGh..Ja (7a)
where G = E is the shearing modulus, K a cross-section2 (1+lJ)
factor depending only on the ratiQ a/h. This factor is
tabulated in several engi.neering handbooks and textbooks,
for example, Reference 12, page 277. In general for con-
crete structures the effect: of Poisson v s ratio lJ is neglected
since its influence is small. (13)
'Using I, the moment of Inertia for rectangular sections,
the above formula can then be expressed as:
C = 6KEI (7b)
• It
'..Substituting this expression in equation (6) and using the
average bending stiffnesses per unit width of the gridwork
D -_ EI~Y Cy
~15-
one obtains:
f3 = 3 (Kx + a Ky ) (8)
'.•
A multi-beam bridge may be considered as a limiting
case of a gridwork system where the beams would correspond
to stringers and the transverse members, normally represente4
by beams in a gridwork system, would be replaced by a co~-
tinuous plate with average bending stiffness Dy • In equation
(8) Kx would then assume a value corresponding to a
particular a/h ratio of the beams and Ky the value cor
respondin~ to a plate. With a later value, Ky = 0.333,
the above equation would then take the form:
f3 = 3 (Kx + 0.333 a) (~)
••
The two functions (4) and (9) of f3 are plotted in
Fig. 6 for a varying between 0 and 1. Equation (4) re-
presents a parabola through the origin and the point a = 1,
f3 = 1, the axis coinciding with the a~axis. Equation (9)
represents a series of parallel lines, intersecting a =0,
such that the intercepts on the f3-axis are three times the
Kx values.
Considering now the limiting cases of the multi-beam
. bridge, it is apparent that for a = 1 the differential
.,
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.. .
~16~
equation must be identical with that of an isotropic
pla,te, which is:
d 4w + 2 a4w + 44,w;:;: p(xlY)() x~ dx2Jy2 dy4 EI (10)
This implies that ~ as well as a must be 1. For
the case of the articulated plate, or ~ = 0, the structure
maintains the torsional rigidity of the beams, which means
that ~ must be equal to' 3Kx .
Each of the two approximations for ~ given by (4) .and
(9) satisfies one of the above conditions. Equation (4)
yields the correct value for a = 1 but gives zero for a = o.
From equation (9), for a = lone obtains a value larger
than 1, and the correct value for a = O. The true function
for a lies therefore somewhere between these two approximat-
ions.
In the following chapter a,n attempt. is made to establish
a relation between a and ~.
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II. DIFFERENTIAL EQUATION FOR THE MULTI ~BEAM BRIDGE
1. Derivation of the Differentia.l Equation
The second assumption made previously for the analysis
of a multi-beam bridge as an orthotropic plate implies that
the number of beams making up the bridge is of such magni-
tude, that the real structure can be replaced py one of a
continuous nature which exhibits equivalent average elastic
properties. Hence a differential element, cut out of the
latter structure by two pairs of planes parallel to the xz
and yx planes, will have bending stiffnesses Elx and Ely
per unit width in the directions x and y respectively.
Fig. 5 shows the middle plane of such an element with
the forces and moments acting per unit width of the element.
The moments are designated by arrows with double heads and
their directions are defined by the right hand rule. p(xy)
represents the intensity of the applied load.
Three independent equations can be written, expressing
. the equilibrium of the forces and moments. These are:
_ 'dMy; "dMyxQx - dx + d Y (lla)
••
...
_~ ()MxyQy - dY + a x (llb)
(llc)
-18-""""'.
By substituting in equation (lIe) the expressions for
Qx and Qy as given in equations (lla) and (lIb) the
equilibrium equation for the entire element results:
(12)
..
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To represent this equation in terms of the deflection
of the plate w, the relation between the moments and the
deformations are used. In the following the effect of
Poisson's ratio is neglected. This agrees with general
practice for the investigation of reinforced concrete slabs .
This simplification allows the use of the relation
between moments and deformations from the elementary
beam theory, giving:
Q2Z a22'My = -Ely dY = -OEIx dY
(13a)
.. (13b)
Similar expressions can be written for the twisting
moments:
••'Mxy (14a) .
...Myx = - (14b)
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where ~x and ~y are two constants to be determined. By
substituting these expressions in equation (12) the
differential equation for the deflection of the multi-
beam bridge is obtained:
with ~x + a ~y = 2~ equation (lb) results.
2. Ex erimental Determination of the Relation Betweena and
In order to establish an experimental relation between
a and ~ the homogenous differential equation is considered:
A particular integral satisfying this equation is (Reference
15, page 164):
......
w = C·x·y
with C an arbitrary constant.
This solution yie~ds from equations (13)·and (14):
~2w= - EIx ax2 ·= 0
My = - aEI h.. = 0x ay2
(17)
(lBa)
(lBb)
-20-and
'd 2wMxy = - ~x EIx dXdY = - ~X Elx·C
Myx = -af3y , EI d 2w =x dX'Jy
Adding equations (19a) and (19b) the constant C can be
determined as:
(19a)
(19b)
1C = -EIx
Mxy + Mvxf3x + a ~y , (20)
and the deflection w results as
• 1 Mxy + M~xw = - EIx xyf3x + a y
.. or
.. 1 Mxy + Myxw = - EIx 2f3 xy
fFrom equations (18) and (19) it is evident that only
(2la)
(2lb)
....
.,.
constant twisting moments are acting on the bridge, deforming
the middle plane to the anticlastic surface given by
equation (21). This loading and deformation condition is
shown in Fig. 7.
In each element dy of the edges parallel to the y-
axis, the twisting moment Mxydy acting on it, can be
considered as being formed by two vertical and opposite\ .
shearing forces of magnitude Mxy at a distance dy apart,
(see Reference 8, page 47). It is apparent that these
"1'
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-21-
shearing forces balance the ones in the adjacent elements
and only the forces in the corners of the plate remain.
These are, ± Mxy from the edges parallel to the y-axis
and, with the same reason + Myx from the edges parallel
to the y-axts, giving equal resulting corner forces of .
shown in Fig. 8.
If such corner loads are applied to a small bridge,I
in the laboratory, an experimental relation between a and
P can be found from the observed deflections and equadbn
(2lb). Also the effect of lateral prestressing on the
value of ~ may be established.
3, Assumption for the Relation Between a and~: .
Since these test data are· not yet available an
assumption for P as a function of a must be made. This
assumed relation mUst be chosen so as to satisfy at least
the'cond!tions for the limiting cases.
(a) '; For a ~. 0 the twisting moments are carried only. by .
the beams along their longitudinal axes. With a torsional
rigidity of the beams as given in equation (7b), the
twisting moments which results from the deformation for
this case are:
Mxy = -~'2w
6KElx ~x~y
-22-
Myx = 0
(b) For'Q ~ 1 the bridge can be considered as an isotropic
plate with Mxy =Myx and 2f3 = 2 or for a = 1: f3x = f3y = 1.'j/~ t:J.Assuming that f3x ~ 6 K(l - a ) + a and tJy = 1 and
substituting these expressions in 2f3 = f3x + a f3y the
following approximate relation between a and~ is obtained:
.. .}'.t..f3 => 3K (1 - a ) -+ a (22)
•
This expression yields exact values for a = 0 and a =1 and
reasonably correct values in the intermediate range. It
represents a family of curves with K, the constant of
torsional rigidity for rectangular beams, as parameter.
In Fig. 6 some of the curves for variousa/h ratios are
plotted.
With this assumption and with Elx => EI the moments·and
forces result from equations (11), (13), (14), as follows:
Bending Moments:
... Mx => - q2w EIdX2
••
My ~ - 22wEI aay2
(23a)
(23b)
•-23-
Twisting Moments:
Shear Forces:
a2wMxy = - ;)x~Y EI(2J3
J2wMyx = - ~~y EI 0'
Qy =
0')
(2fl - <>J EI
(24a)
(24b)
(25a)
(25b)
•
•
Boundary Shear Forces: (Ref. 15, page 154)
Rx - Qx + ~;L -[~ + 2fl ~;;~ EI
Ry = Qy + ~yx = _[~ 0' + 213 '/3wl EI~x dy3 dx2d~
4. Limiting Cases of the Differential Equation:
(26b)
(26b)
As mentioned before for 0' = 1 the differential equation
of the orthotropic plate becomes the differential equation
of the isotropic plate:
'J4w ~4w p4w. 4 + 2 ax20 y2 + " 4 =JX .. dY
P (xlY)EI (10)
.--.
For the case when 0' is zero the differential equation reduces
to the one for the articulated plate:
P(X1Y)EI (27)
•
-24-
where 130 = 3 K.
5. Boundary Conditions
In order to solve the differential equation for a
given bridge deck, the boundary conditions (hereafter
called B.C.) for simple supports and free edges h~ve to
be formulated.
(a) Simply Supported Edges
For a simply supported edge parallel to the y-axis the
deflection w and. the bending moment Mx is zero. The fol~
lowing conditions are resulting:
w = 0
and'/lw = 0~x2
(b) Free Edge
(28a)
(28b)
...
Along a free edge parallel to the x-axis the bending
moment My and the boundary shear
must be zero. These conditions are fulfilled if
..,
~2w = 0~y2
(29a)
= 0 (29b)
-25-
III. INTEGRATION OF THE DIFFERENTIAL EQUATION
1. General Considerations
In the general theory of linear differential equations,
the solution of the given non-homogenous equation (lb)
can be obtained by adding to the general solution
of the homogenous equation a particular
solution of the non-homogenous equation, or (14):
(30)
•
•
I ~
!
....
••
WI is a particular solution of the complete differential
equation taking into consideration the effect of the loading .
In general this solution does not satisfy all the B.C. Hence
Wo has to be superimposed on wI to give the exact solution.
Wo is the solution of the homogenous equation:
d4w + 2f3 "d 4w + Q! d4w = 0dX4 d x 2J y 2 dy4 (31)
which represents the differential equation of the unloaded
plate with boundary forces and boundary moments acting on
it.
For wI the solution of the infinitely long plate strip
is used which satisfies the B.C. along the supports. This
solution will be derived in the following for the general
"
-26-
case 0 c::Q< 1. The limiting cases occur with Q = 1, the
isotropic plate, and with Q = 0, the articulate plat~.
2. plate Strip of Infinite Width Under Line Loading:
(a) General Case:
Fig. 9 shows a plate strip of infinite width simply
supported along the edges parallel to the y-direction. A
line load, applied along the x-axis, and representing the
live load, is given by the following sine series:
•
p (x) = 2P 2:.1Tc fTl::l
1 . muus~n --
m 1• m'Tr c • mrrxs~n-- s~n--
1 I (32a)
1 sin rnlJ sin m'6 si'n m~m )
or with 1!..u....=lJ1
p(x) = 2P !1'( m=l
1!C1 =(5 1LL- F
1 - \
(32b)
••
where P is the total load uniformly distributed over the
length 2c.
Since the deflection is symmetrical with respect
to the x-axis, only the portion with positive y will be
considered. This is done by cutting the plate along the
x-axis. The resulting parts can now be analyzed as unloaded
plates on which boundary forces and moments are acting of
•-27-
such magnitude as to restore the continuity. The deflection
w has therefore to satisfy the homogenous differential
e'quation (31) and the B.C.
The Levy solution (Ref. 8, page 125) may be assumed
for this equation and is written as follows:
w =~ Ym • sin rn ~m;~ )
=~ Wmm::l
(33)
•
where Y is a function of y only. As each term of the seriesm
mus~ satisfy the differential equation and all the boundary
conditions, it is sufficient to consider only one term.
The B.C. along x = 0, and x = 1 , expressed in equation
(28) are satisfied by sin m f. In order to determine Ym
the deflection Wm has to be inserted in the differential
equation (31), yielding the following fourth order linear
differential equation for Ym:
Ym = 0 (34)
Taking Ym = ery the characteristic equation results as:,
• • with the roots:
= 0 (35)
rl 2 3 4 = +~-. / {3 [1 + -V1 - "'J i (36), '. , - 1 V C¥ - {32
..-28-
On the basis of the assumed relation between a and {3 these
roots will always be real.
Taking:
•
and
k1 = -V ~ [1
k2 = -V ~ ~ --,; 1a' J'- {32
(37a)
(37b)
(38)
•
...
the general solution of the plate equation becomes:
where Am, Bm, Cm, Om, are arbitrary constants to be
determined from the B.C ..
Observing that the deflection and its derivatives
approach ze+,o at a large distance from the x-axis, it may
be concluded that the constants Am and Cm are zero.
From the conditions of symmetry:
•-29-
yielding:
(40)
•
A second equation results from the fact that the load at
y.= 0 is equally divided between the two halves of the
plate and that these loads have to be in equilibrium with
the appropriate shearing forces, or:
[QymJ y=o = - P2m
= ~ 1't ; sin musin lll"$ sin mt (41) .
with the expression (25) for Qy this equation reduces to:
Em [-kl a + kl (213-a~ + Dm [-k~ a + k2 (213-a~
_ P12 1 . 11'- EI'6 1/3 ITi4 s~n m s~nm~ (42)
Solving the two equations (40), (42) simultaneously one
finds:
= -1. . v.Iii4 s~D.m1f s~n mo (43a)
Dm = EI 't 11 3 (43b)
••
and the deflection results:
(44)
..
•
..
..
..
-30-
Using equations (23), (26) expressions can be found
for the following: Mx ' My, Mxy ' Myx , Qy. These are listed in
Table. (I). These expressions hold for values:
and
For the limiting values of a and ~, however, one must
proceed to the limit and this will be shown in the following
for the deflection w.
(b) Isotropic Plate:
For the case of the isotropic plate with a and ~ equal
to l? ~f i,~::easy tb seetha,t kl = k2 = 1. $ubstituting
these value,s in equation (44) results in indeterminate
expressions ~ for each term of the series. These are
examined as follows. Using the identity:
kle-k2m~-k2e-k1m~
a klk2(kf-k~ =
e-k2m~ (kl
-k2e{q -k2)m~)
a klk2 (kl+k2) (kl - k2)
a.
the right hand side can be resolved into the following
partial fractions:
e-k2m~ (l+e- (kl-k2)rn~ l-e- (kl -kt)m?)a klk2 2 (kl+k2) + 2 (kl-k2)
•
•
•
-31-
~he transition to the limit yields:
and the deflection of the isotropic plate is obtained
(Reference 8, page 169):
the formulas for the moments and forces are given in
Reference 16, page 26 •
(c) Articulated Plate:
For the case of 0:-.0 and (3-.{3o it is noted that:
'..
lim kl = lim ~ [1+ ~ l-fil CD 0()
0:... 0 ~{3 -{3o
For:
lim k2 = lim \I~ rl-~l - ~l(~..., ~ V l ~ J~~;Jdo
the Binom;nal Expansion is used:
i~ 1·0: 1 a2V1 - ~ = 1 -2 {32 + 8 ~ - ...
(464)
•-32-
and bearing in mind that, since ;2 is very small, higher
order terms can be neglected, the limiting value of k2
results as:
Furthermore, one ob~ains:
(46b)
..
•
..
(47)
Now, upon investigating equation (44) the general terms
can be partially rewritten as follows:
kle-k2m~ - k2e-klm~ e-k2m~ e-klml?a klk2 (k12. - k2'a..) - == a k2 (kt-kt>- a kl (k{..kt) .
Using the limiting:values as derived above, the second
expression becomes zero, the first becomes:
and the deflection w of the articulated plate results:
p~2w =.-EI-~-1i- J---~-r:2:"'?f3'"""o,...1 sin l1J\Isin ~}{ sin m~
(48)
... The expressions for Mx , Qy, and Mxy are derived from
equ~tions (23) to (26) and are given in Table (I). My
•
o •
••
-33-
for this case is zero. Equations 44, 45, and 48 hold for
the complete plate if instead of y the absolute values,'Y',
are used.
3. Solution ~or the Bridge of Finite Width:
(a) Orthotropic Plate:
The bridge of span 1 and width 2b is loaded with a
line loading as shown in Fig. 10.
Two different coordinate systems are used to simplify
the numerical calculations.
(x,y,z) is the coordinate system used for the solution
of the homogenous equation with the origin at the middle
of one support. All the values, such as deflections, forces
and moments related to this solution will be marked by an
index O. For the solution of the infinitely wide bridge
the coordinate system (xl, Yl' zl) is used with the x,axis
coinciding with the line of the applied load. An index 1
will designate the values in this second system. The
distances of the free edges from the two x-axes are denoted
as y = +b and Yl = e'b, and Yl = -e"b, where e'+e" = 2.
The midpoint of the applied load is given by the coordinates
(u, v), where v = (l-e 10
) • b is the distance from the x-
axis. The equations of transformation between the two
•-34-
coordinate systems are:
xl = x·
Yl = y-v
zl = z (49)
In later equations, similar abbreviations as in the
derivation of the particular solution, are used. These
abbreviations are given by:
£ _1-- b
•
A ='lI.£• 1
~ = k£
~I = ,.(~E+ e I~ I I (50)
As mentioned before, the solution of the non-homogenous
differential equation (lb) can be formed by adding the
general solution of th~ homogenous equation Wo to the
solution wl (equation (44) ) of the infinitely wide plate
strip in such a way that the B.C. at the free edges are
..... fulfilled.
Considering only one term of the series, the solution
of the homogenous equations as given in equation (39) will
•
•
•
..
-..
-35-
be used in the following transformation, using hyper~
bolic functions:
Where Am, Bm~ Cm, and Dm are a new set of constants. The
applications of this solution permit simplifications which
far outweigh the disadvantage of the more elaborate trans-
itions to the limit.
One term of the complete solution now takes the form
of
Wm = (Am coshklm~ + Ib:J coshk2m? + Gns:inhktm~ +D:nsinhk2~) sinm~+ Wim
(52)The requirement that each term of w satisfies the
B.C. permits the determination of the constants. The B.C.
as given by equation (29a and 29b) are:
for y = +b:
My = [My~ + [My~ Yl=e'b= 0 (a)y=b
Ry = [Ry~ y=b + [Ry~ yl=e'b = 0 (b)
My = [My~ y=-b +[ My~ yl=e"b = 0 (c)
Ry = ~y~ y=-b + [Ry~ yl=e".b = 0 (d)
(
(53)
1-"
-36-
substituting equation (52) in equation (53) the following
simultaneous equations can be formulated for each value of
m:
from equation (53a):
=
from equation (53c):
~y:l'yl=e 'b____~-!J'-.':'""-"---- = S1m'a EI sin m~ (54a)
•
from equation (53b):
[M.Yi! yl=-e"b. aEI sin m~ (54b)
Am k12 sinhlqm,(+Bmk22sinhk2mA+ Cnl<\cosh klmA + Dmk22 c.osh k2 m,(
= _ ~ [axIl yl=e'b _m3 ttr3 a klk2 E,I SinmT - S3m "(54c)
from equation (53d):
.. (54d)
•
••
-37-
Determinants will be used to solve this system of
equations. Observing the sYIIlmetrical and skew-symmetrical
properties, the determinant of the denominator reduces to
the following: (15)
al bl cl dl
al bl -cl -dl= 4 (albZ - aZbl) (cldZ - CZdl)
aZ bZ Cz dZ
dt(55)
-az -bZ cz where:
al = klZ cosh kl mA , etc •
For the same reason, simple expressions are obtained for
the third order determinants Upg , where p stands for the
eliminated column and g for the eliminated row.
Ull ~ U1Z = +ZbZ(cldZ -- cZdl) .
(56)
-38-
With these values the constant Am is a function
of the boundary values .Slm' S2m' S3m' S4m' is-found thus:
Ull Slm + U12 S2m +U13 + 83m +.U14 S4mAm = 4 (alb2 ~ a2b]) (Cld2 ... c2dl)
or:
and similarly:
Slm + S2m2 .
S3m - S4m2 (57a)
'.•
Bm =-a2 Slm + S2m a~ S3m - S4m
alb2-a2bl Z + alb2-a2bl 2
Cmd2 Slm S2m dl S3m + S4m
= cld2-c 2dl 2 cldZ- c2dl 2
-c2 Slm-S2m cl S3m + S4mOm = cld2-c2dl 2 + Cld2- c2dl 2,
As is shown below, Am and Bm contain the symmetrical part
(57b)
(57c)
(57d)
••
of wl' whereas Cm and Dm contain the skew-symmetrical.part
of wl- If wl is symmetrical with respect to the x-axis
the following identities hold:
-39-
and
or
and
which gives
= [RylJyl==b
Sm3 = - Sm4
em = Dm = O.
For wi being skew-symmetrical with respect to the x-axis and
• rMYl]L yl=b- ~M J- L YJ yl=-b rRylJ = fRylJ .L yl=b L· yl=-b
it follows that Am = Bm = O.
Replacing the coefficients of the determinants by the
terms given in equation (57) and using the following
abbreviations:
-40-1. .
jlmbl k2coshk2rn~
= alb2-a2bl = ktcosb~lmAsinhk2mA-k!sinhklmAcoshk2m~ (58e)... _.
.. al ktcoshklm,,(j2m =
alb2~a2bl - klcoshk:lmAsinhk2mA- kisinhklm,(coshk2mA (58f)
"'It-
dl k2s~_n.hk2mA
j3m = cld2- c 2dl - k~sinhklm~ coshk2mA- k.~coshklrnA sinhk2mA (58g)
Cl k~sinhk1m,(j4~ = cld2-c 2dl =- ktsin.hkJY,il~coshk2mA- k2coshk1mA sinhk2ql,{ (58h)
the constants can be written as;
Slm + SZm • 83 .. S4mJ mAm = ilm - 1Ji'i -• 2 2 (a)
Slm + S2m j S" - S4m,jID
Bm =-iZm 2 +'?"fifll -_._--2 (b)
..Slm ~ S2m . S3m + S4m
em = i3m 2 J3m 2 (c)
=-i4mSlm - S2m e S3m -} 84m
Dm 2 -+ J4~ 2 (cD, (59)
and the deflection of the bridge results:
.'.e.
w = Zl(t\ncoshkim? + Bmcoshkzm? + Cmsinhkim~ + IlmSinhkZm?)Sinmr+ wi
(60)
(b) Isotropic Plate:
As proved before, the deflections and therefore the
moments and shear forces of the infinitely long isotropic
".
•
e
•
-4l~
plate result from the equations of the orthotropic plate
by a transition to the limit a~l. In the following the
same will be done for the solution of the homogenous
equation (32). Since the analysis is the same, the
transition will be shown in more detail for the first two
terms of equation (51) and only the resulting expressions
will be given for the last terms.
The first two terms:
rewritten with equation (59) as:
Slm + S2m. S3m ~. S4m)(ilm . 2 -Jlm ----2 . coshklm?
(Slm + S2m S3m ; S4m]
+ -i2m 2 + j2m / coshk2m?can be transformed into:
and with the values given 1n equation (58) into:
....
e.
_ k~coshk2mAcoshklm? klcoshk1mAcoshk2m~
klcoshklmAsinhklm~ k~sinhklmxcoshk2mAS3m-84m
2
•
~ ..
..
...
".
-42-
Using the boundary values Myl and Ryl of the iso
.tropic plate it is easy to see that Slm + S2m as well2
as 83m - S4m are of finite magnitude. One needs there2
fore to investigate only the two remaining fractions.
The following steps are necessary:
1. Rewrite the fractions as functions of (kl + k2) and
2. Divide the numerator and denominator by (kl - k2).
3. Proceed to the limit (kl - k2)... O.
In order to clarify the operation the numerators Nl
and N2 and the denominator D will be investigated separately:
The first numerator becomes:
and with the following identities:
sinh (a+b) = sinhacoshb + coshasinhb·
cosh (a+b) = coshacoshb + sinhasinhb
it can be arranged into:
[kl +k2 kl -k2 kl +k2 kl -k2 ]
(kl-k2) sinhm~ 2 coshmA 2coshm~ .2 coshm~ 2·
•
•
'""
~43
+(kl+k2) [ sinhm,< k.l:+k2cosh.mt< kl -k2sinhmn kl +k2sinhmrykl-k;l2 2 or 2 2 J
[kl+1<2 kl -k2 k +k k -k ~- (kl+k2) coshrnA sinhmA coshmn 1 2coshmr> 1 2
2 2 .( 2 I 2
Dividing by (kl - k2) and proceeding to the limit (kl -k2)... O
yields:
Applying exactly the same operations to the second
numerator N2 and to the denominator D one obtains:
N2 : -(2coshmA + mtsinhm~)coshm? + m?cosh m~sinhm?
D: 3coshmA sinhmA -m~
Substituting these values in equation (32) and rearranging
the expresgion yields:
with the values A*m and B*m as given below. Similar
expressions are obtained for the later terms.
,.
•
. ... . .
-44-
The deflection of the homogenous bridge thu~ is
obta1n~d as:
The coefficients A*m to D*m are given by equation (59) if
i m and jm ~~e replaced by i*m and j*m as follows:
(pI)
.*~lm
m~coshmA- sinhmA=~--=--~---:----:--:-~
m~ - 3coshm~ sinhm~
•
•
".".
.* sinhmA~2m = mA - 3coshmA sinhmA
.* . mA s inhm~ - co shmA~3m =-mA+3coshmAsinhmA
.* coshmA~4m =-mA+3cosl:l.InA sinhm~
.* mAsinhm~+2coshm~. JIm = mA - 3coshmA sinluriA
.* coshmAJ2m = mA-3coshmAsinhmA
.* mAcoshmA+2sinhmAJ3m mA+3coshmAsinhmA
.'J< sinhm~J4m ~~mA+3coshm~sinhm~ (62)
-45-
and~
•
1,Q~2'l- '= 1.:__'-JI ''''1 2 IEJ[
IlliI m 11 .
r;A~
33m
(c) Articulat,e.:! .?lat~_~
rM,p·:tl vJI -~e"b.l':..L~:'="__
Sir..,m~
._L~11E=e~ bsinm~
_hI] YL=-~siri\1Jll
(a)
(b)
(c)
(d)(63)
'0
•
o.
/
For the li.u:d.ting c:~t:3:e a~O 'which includes kr'~oO,
k2-ko = ,,\~i and ~l;'7 ."= 0 >, it is easy to see from equation"Vz.(3~ .f •
(58) that i.n equation (59) Am and em v,anish and that amreduces to:
and re,arranging the expre,~sion, we obtaiR,
1Em = a kTilk2Sill\hk~t<rJA ~:ak2' tanhklm,{coshk2ID .
~. tq
[~!L.x.l.=~..Ub.:.....~yJ !l=a'e"b2EI sin ror
-46-
For a-O the second term in the denominator approaches
zero and:
It follows that:
yl=-e"b
(64a)
and similarly:
yl=e 'b+ [Ryl] yl=-e"b(64b)
."The deflection of the articulated plate can now be
written as:
QC
W = L (Bg coshkom~ + ng sinhkoml?) sin m~ + wIm:l \
4. Explicit Form of Formulas
(65)
...
a.
Equations (60), (61) and (65) can be written in explicit
form by replacing the boundary values Myl and Ryl by their
values obtained from equation (24) and (26) with equation
(44). The resulting expression is listed in Table II for
the general case of the orthotropic plate together with the
important formulas for moments and forces used in the design
~' ,
..
•
•
•
a.
-47-
of bridges. Table III contains these expressions for
the articulate plate., For the,case of the isotropic
plate, reference is made to the extensive study by
H. Olsen and F. Reinitzhuber of the rectangular plate with
two opposite edges simply supported and the two other
edges free. (16). This study contains all the formulas,
as well as influence surfaces for deflection and moments
for plates of various sizes.
5. Influence Surfaces:
In beam statics influence lines are commonly used in
studying the effect of concentrated loads or load systems •
The influence line for the moment, for example, describes
the variation of the moment at a given point of a beam
due to the passage of a single load across the beam.
In a similar way influence surfaces can be defined for
plate structures. The influence surface, for example ,for'
the moment at a fixed point (u,v) for a unit load is given
by a surface with the ordinate z in the point (x,y) equal
to the moment'MX produced at the point (u,v) by the unit
load placed at (x,y).
The following discussion of influence surfaces holds
as well for the isotropic as for the orthotropic plate, and
covers only those items which will be used in the next
-48-
chapter to derive the properties of the lateral load
distribution coefficientis. For i:l [!]ore complete investi-
gation includi.ng the derivation of the basic rules, one
is referred to K. Gi.rkmcu.!~~ pCil.ge 225. (11)
d . I. fThe two rna,i.n rules for the . er~v,at~on of in luence
surfaces are~
1 0 The deflection su.rface W = 'VI (x,y;u,v) due to
p = 1 a.t the point (1.1, v) i.i:,~ equal to the influence surface
..for the deflection at point
•
•
....
-".
2. The i.nfluence Si.l.rf:aces for arn.y derivative of the
deflection at the point (u.~v) if; obtained by differentiating
the deflection surface w = w (x,y,u,v) due. to P = 1 at
(u,v) with re.spect to the coordinates u and v .
It C~1n be proved that the above rules hold also for
the case in 'which the unit load is replaced by a line load,
applied over the di.stance 2;;-; parallel to the xg·axis and
with the mid-poi.nt coordin.'ites u and v. The proof is based
on the fact that, for the gi:\,7en li.ne loa.d at (u, v), the
deflect:i..on at (x,y) is equ$.l to the deflection at (u,v)
due to the line load at (x,y). Thts can b~ seen from
equation (1), Table II, by il'.~teT.'ch.anging the coordinates
(x,y) and (u~v)o
"
"
As the expression for the line load in the equation
for the deflection·is not affected by the differentiation
of the latter with respect to x and y or to u and v, it
follows that the second rule holds also ,for the line load.
Obviously one obtains in this case the influence surface
for this type of loading.
Fr.om the first rule it follows that equation (1) in
Table II also represents the function of the influence sur-
face for deflection at the point (u,v) due to the given
line load. This conclusion holds generally since it
follows directly from Maxwell's law of reciprical deflection§.
Applying the second rule to derive the influence line
for the moment Mxat the point (u,v) which is denoted by
[Mx ] u,v one obtains:
[Mot ] uiit =- EI ~~~ =_ EI 02w
dX2 (66)
It is easy to see that the resulting expression is identic~l.
with equation (2) in Table 1;1. It follows that this
(
equation also represents the function
the influence surface for the bending moment Mx at the
of
"'. point (u,v) for th~ given loading. This result will be
used in the following chapter to establish an important
property of the coefficients of lateral load distribution.
•
!
-50-
It must be noted that the above identity holds only
for the bending moment Mx and only because of the simple
boundary conditions of the plate for x = 0, and x =1 ,
and the fact that the effect of Poisson's ratio is
neglec ted. If needed, the func t ions "of the influence
surfaces for other moments and forces may be found with
out difficulty by differentiating equation (1) in Table ~~
according to the second rule. The same results can be
obtained from equation (3), (4), and (5) of Tab Le· II by
considering the coordinate (x,y) as fixed and the co
ordinates.of the midpoint of the loading (u,v) as variable.
In this case the equations represent the influence sur
faces for the point (x,y) •
...
I>
•
••
-51-
IV. LATERAl. LOAD DISTRIBUTION
1. General:
The formulas for any location of a live loading derived
in the foregoing chapter a.nd summarized in Tables I and II
permit an accurate design of multi-beam bridges. To
eliminate the extensive amount of calculation, numerical
values ar.e given in the later part of this study for de
flections, moments and shear forces at points along, and
for loading positions ~n the midspan cross-section of the
bridge. They were computed for bridges of different span
and width and also for various degrees of lateral prestress.
These loading positions at midspan of the bridge are,
in general, the most unfavorable position$ encountered in
the design of such bridges. However, in many cases, other
loading positions have to be considered for which only the
principal bending moment Mx would be required. The deter
mination of Mx ' however, may involve much numerical work.
In this case, an approximate determination of the bending
moment Mx can be made using coefficients of lateral load
distribution, introduced ~y Y. Guyon. (4)'
2. Definition
Considering again the bridge with the live load applied
as shown in Fig. 10, the average bending ~oment in a cross-
of
-52-
section x = Xs of the bridge is given by:
If+bMxav :: 2b-b
Mx dy (67a)
..
For the case where the effect of Poisson's ratio is neg-
lected this average moment is equal to the bending moment
per unit width obtained by replacing the applied live load,
which acts over distance 2c, by an equivalent uniformly
distributed load, acting over an ar~a 2c.2b, where 2bis
the width of the bridge.
By integrating equation (2) 1.n Table II the average
bending moment is obtained as :
•
+b
ib J Mx dy =-b
1 sinm1Jsin m~sinmfm3 )
(67b)
This expression developed in a sine-series is identical
with the average bending moment obtained from conventional
beam statics. This is true because the above mentioned
loading, which does not vary over the width of the bridge,
caus~s a deformation of cylindri.cal shape.
The coefficient o~ the lateral load distribution Sxy
for the point (x,y) subjected to the actual bending moment
Mx , may now be defined as:
,S~y - -::l~-i""'~~:----
2b Mx dy
-b
-53-
(68)
'.•
This is a non-dimensional coefficent and indicates the
portion of the average bending moments of the sections x = xs
which exist at the point (xs,y).
F'or the design of a multi-beam hridge one is especially
interested in the bending moment associated with each
beam incorporated in the bridge. For the beam i, for ex-
ample, this moment is s4bstantially the bending moment per
unit width at the mid-point of the beam, multiplied by its
width a; from equation (67a):
a Mxi = sxi tb JTb
-b f+b
sxiMxdY=n
-b
IVlx dy
(69)
as a percentage of the total cross-sectional moment
where n = 2b is the number of beams.a
one obtains:
Expressing this moment+b. (Mx dy
)-b
s*·x~sxin
100(70)
...Assuming now that the actual l~ve load is not directly
applied upon beam i, the bending moment at beam i may be
obtained from a consideration of the interaction of the beam
-...
-54-
as expressed by equation (60). This moment can be thought
of as being produced by a proportion of the actual load
ing, directly applied on an i.ndependent beam. It is
apparent that the facto~ of proportionality is identical
with the factor given by equation (70). S*xi also indicates
the distribution of the actual load applied on. one beam
among the other bearps. Hence, in the determi.nation of the
magnitude of the beam bending moments, each beam is con
sidered as being independent and subjected to a load
depending upon its value s*xi.
Similarly the coefficient sxi may be considered as
a measure of the lateral load distribution, and since it
is independent of the number of beams u, it is more suit
able for non-dimensional representation.
It is to be understood that the coefficient Sxy for the
point (xy) varies with the type and the location of the
applied live loading. Its introduction, therefore, does
not yield any advantage in the accurate determination of
the bending moments. However, it is very useful in approxi
mating the bending moments.
3. Approximation for the Lateral Load Distribution:
Y. Guyon (4) and Massonnet(5) computed the coefficients
of the lateral load distribution for a live load of~nusoidal
.\
-55-
type given by:
p(x) _. Po(71)
where Po is the load intensity at midspan of the bridge.
This expressi.on is i.dentical ~Jith the first term of the
sine-series of equation (32b) where P is replaced by:
Po_ D__2sinvsinl(
•
..
For this loading the moments and forces are given by the
formulas derived in Chapter III with rn = 1 .
With equation (2) in Table II and equation (67b)
the coefficient of the lateral load distribution for the
above loading results as:
SxyMx= _ .._.:
MA av (72-)
.~
where the expression in the bracket above is identical
with the expression in the bracket of equation (2) in
Table II.
It is easy to see that this coefficient of the lateral
load distribution depends only on the eccentricity of the
line live load wi.th respect to the bridge axis and on the
•
•
,-
~56-
ordinate y of the refer.pp~p. noint (xy). It is inde-
dependent of the abscissa x of the crosB-section under
consideration. It follows that for a particular loading
position, the moment curves of all beams are similar in
shape, with the moment ordi.nates dl~pending upon Sy. The
sarne relation can be derived for the deflection curves
and it follo\'iT8 that the cOE';;fficient Sy also represents the
ratio of the deflection at a pairl."!:; y of any crosse-section
to the average deflection of the cross-section.
Because of these consi.derable advantages and the fact
that their values can be more readily computed) Guyon and
Massonnet considered the coefficients for the sinusoidal
loading as a practical approximation to be used for any
type and location of loading encountered in the design of
bridges.
The application of thi.s approximation may be justified
for loading conditionG pr.escribed in some European speci-
fication. According to the .American specification for
l'l)highway bridges'" ,however, one must consider the effect
of concentrated wheel loads or truck axle loads. For such
types of loading whi.ch differ considerably from the sinusoi-
dal load, the above approximation coefficients lead to a
very favorable distribtltion of the load but induce a rather
small moment i.n. the directly loaded beam.
\
,.
e.
••
For a safer design~ especially for concentrated loads
and wheel loads, the following approximatiO!l is proposed.
This is based on the exact distribution coefficients com
puted for the midspan cross=section of the bridge correspond
ing to a wheel load placed at midspan. It i.s considered
that this posi.tion is generally the most unfavorable en
countered in bridge design. It may therefore be assumed
that this lateral load distribution is representative for
all cross-sections of the bri.dge subject to a similar load
application in each crosS-6ect~on. It is apparent that
variations might exist in the curve of lateral loa.d distri
bution from section to section along the bridge. However,
it may also be ass~~ed that, having derived the curve for
the worst condition of loading, that i.s, at midspan, it may
be applied to any section to yield a moment which would be
higher than the actu.al moment existing at that section.
Coefficients for lateral load distribution corresponding
to the most important loading conditions are computed and
given in the following chapter.
4. Properties of the Coeffi.cient of Lateral Load Distribution:
It was stated in Chapter III that equati.on (2) in Table II
representing the moment Mx at point (x,y) due to a load at
(u,v) also described the function of the influence surface
'\
~58-
for the moment Mx at the point (u ~ v) . F'rom this and using
equation (68) an important property of the lateral load
distribution coefficients follo,tJ'S: "The curve of the co-
efficients of the lateral load distribution in the midspan
cross-section for a load applied at the point v of the same
section, is also the influence line for the coefficient at
midspan and for the point v. n This means that the reciprocal
relation holds:
Syv = SV'lj (73)
or that the coefficient of the Lateral load di.stribution at
y due to a lo,a.d at v is equal to the coefficient at v due
to the load at y.
The se.cond property of these coefficients, though
obvious, is still worth mentioning, since it is useful for
checking purposes.
Keeping the definition of the coefficient in mind
(equation (68) ) and integrating Syv over the cross-section
one obtains:
..,
'..:; 2b
(74)
-59-
In performing this operation one may replace the integral
by the value of the area. under the syv-curve determined
using Simpsonvs rule:. In the next chapter the coefficients
are computed for the following points~
(-b ~ 3b = b = b, -4-~ 2 ~ 4'
• b..,- --2' +b)
Applying Sirnpsonos; rule to each point a.nd designating the
coeffici.ents for the va.rious points as 511' 52 .•.• 59
yields:
or:
This condition may be used to check the accuracy of
the distribution coefficients.
,'"
.,
'I,
0,
.,
'.
~60-
v. NUMERICAL CALCULATIONS
1. General:
The extensi.ve amoun.t of numerical work to compute exact
values for all formulas listed in Tables II and III was
reduced by making use of an electronic high speed computer.
Numerical values were computed for 9 points at equal inter
vals along the midspan cross·asection. The values for Mxy
were determined for points along the support, x = 0. Two
loading positions in the same cross-section were investi
gated, namely the load placed in the bridge axis with v = 0,
or e r = 1 (Fig. 10) and the load placed at one edge with
v =-b or e r = 2.
The following parameters were considered:
(a) The ratio of the bending stiffnesses a.
The computati.ons were made for a:::: 0, the case of
the arti.culated plate, and for Ol::: 0.1 and a = 0.5.
With these several a - values and the values given by
Olsen and Reinitzhuber(16) for the isotropic plate
a = 1, an interpolation for any intermediate a is
possible.
(b) The size of the beams •
The size of the beams, or more precisely the width
to depth ratio a/h of the beam making up the bridge
.,
..
-.
••
affects the coefficient of the torsional rigidity of
the orthotropic pi,ate, (equation 22). The following
two ratios which are used in present bridge design
were' considered in the cornputati.on~
a/h = 1.00 and a/h ~ 1.7
(c) The size of the bri.dge.
With span 1 and b the half width of the bridge,
computations were made for:
b~ = 0.5 0.375 0.25 0.125
For any intermediate bit ratio, moments and forces may be
obtained with sufficient practical accuracy by interpolating
between the given values.
(d) Line Loading
For the represe.ntation of a wheel load in the
form of a line load, the distribution of the load
in the longitudinal direction ~sho'Wn in Fig. 11.
It is assumed that the length of contact between
the wheel and the wearin~ surface is 4 inches in
the longitudinal direction and that the load is
then distributed at an angle of 45° through a
2-inch wearing surface plus the half depth h/2
of the beams. Thus a total longitudinal dist-
tribution of 2c = 8 in. + h. results.
..
•
•
-62-
This distribution, depending on the depth of the beams, is
in turn a function of the span of the bridge.
The present computations were made for one c~ ratio.
This was chosen as cll = 0.0318 or -rrc/l = 0.1, which
corresponds approximately to the value of c obtained from
the above distributio~ and for beam depths and spans
currently used for multi-beam bridges.
The writer is aware of the fact that the above assumption
for the distribution of a wheel load is not included in the
American Specifications for Highway Bridges(l). These
specifications do not consider any longitudinal distribution
of wheel loads. In bridges analyzed according to the
derived method however, a concentrated load would cause
infinitely large moments directly under the load. To avoid
these infinitely large moments, which are of a theoretical
nature only, the above longitudinal distribution over the
diStance 2c is assumed.
Any distribution of the load in the .lateral direction,
caused by the width of the tires and the wearing surface,
as well as the beam depth may be taken into account by
evaluating the prepared influence lines.
..
..
•
..
••
••
-63-
2. Accuracy of the Results:
The calculation of the values for the series expression
includes the terms up to and including ill = 19. This
limitation was set by the capaeity of the computing machine
in handling certain large parts of the series.
The accuracy of the results is therefore a function of
the convergence of the series in Table II, ~vhich in turn
depends mainly on the power of the factor directly after
the summation sign. Furthermore, it also depends on the
location of the points, for which series were computed, wi.th
respect to the applied load .
Check calculations have revealed that for points not
coinciding with the load location, the computed values, with
the exception of those for Qy' are exact to within three
significant figures. For points. directly under the load,
E= 0 (location of point) for a' = 1 (location of load) and
£= -1.00 for e' = 2 (see Tables IV - XXVII) the accuracy
is as follows:
(a) Deflections:
The convergence of the series with 1/m4 is excellent,
the calculated values check at least to within three
significant figures .
..
(b)
~64-
Moments. (l1x ' My, Mxy ): . ..The ser~es converge sat~sfactor~ly w~th 11m2 and the
•
•
..
•
given values are accurate to two significant figures.
(c) Shear Force Qy :
Since the power of m in the series expres~ion for Qy
is 1, the series converges very poorly. For points not
directly under the lo~d, ttle values ~ay be accurate to
within two significant figures. For the peak values, occurring
in the points directly under the loaq., only the first figure
is correct. A better valuernay be obtained by considering
equilibrium between ~he applied load and the shear force
(neglecting the influ,ence of twisting moments) yielding:
for e' = 1:
p- - 2.2y
e' = 2:
= -
~. 7085
p2C
'.
= ~. 15.7
•
..
-65-
In general, the accuracy of the results obtained may be
considered sufficient for design purposes, since only
slide rule precision is required.
3. DescriE,Sion of the Tables :.
The results of the numerical calculati.ons, are given
in Tables IV-XXVII (as obtained fro~ the computer). Each
table includes values determined for one combination of
parameters a, alh, bll, as listed in the heading of the
table. The left h~nd column shows the Q4antities w, Mx '
My, Qy and Mxy ' where each one is multiplied by a coefficient
which renders it dimensionless; the negative sign for My
and Qy should also be noted. The secop,d column indicates
the position of the load: e' = 1 (E' = 1)* represents
the load placed in the bridge axis and e' = 2 (E' = 2)
the load placed at the edge.
The top line shows the coeffici~nt £ = t determining
the distance of the points from the bridge axis for which
values were computed. The sign of £ is indicated in the
third column. For e' = 1 the values are given only for £
positive, since under tnis loading the values w, Mx , My are
symmetrical with respect tq the bridge axis; similarly Qy
* Note: Only capital letters are printed by the computer .••e' = E'
-66-
and MXy are skew-symmetrical to this axis.
For e' = 2 the values are given in the second and
third line for each qu~ntity. The second line includes,
as indicated by the sign in the third column, the values
for £ positive, whereas in the third line the values for
£ negative are listed. The values for £ = 0 are given in
both lines.
The computed values are represented in the following
form:
where
+ X • 10n
l""X~O.l
..
and the exponent is a positive or negative integer.
The tables include the numbers for x and n only and
have to be read as shown by the following example.
The value for wEI for e' gland £ = 0 in Table IVPfZ
is given by:
.33822 -1
which stands for:
4. Coefficients of Lateral Load Distribution:
The coefficients of lateral load distribution, as defined
by equation (68), were calculated and are given in Tables XXVIII
•
..
•
••
-67-
and XXIX for bridges composed of beams with a ratio
alh = 1.7 and a/h = 1.00 respectively. Included are the
coefficients for the isotropic plate or a = 1.0, which were
calculated from the bending moments given by Olsen and
Reintzhuber (16) .
The same results are plotted in Figs. 12-19. Figs. 12-15
show the coefficients for the load applied in ,the bridge axis,
or e' = 1. The upper graph includes the values for alh = 1.00,
whereas the lower graph shows those for alh = 1.70. For the
case of the isotropic plate (a = 1.0) the ratio alh is
meaningless .
Figs. 16-19 give the coefficients for alh = 1.7 with
the load applied at the edge £ = -1.00, and those for alh = 1.00
with the 10b1d at E= +1.00.
Each graph includes coefficient values for one a -value
and shows the effect of the various bridge sizes, described
by the parameter b~. This representation had to be chosen,
because of the unexpectedly small differences in the coeffi
cients for a fixed b1 -ratio and variable a-values. With
the exception of the peak values directly under the load the
coefficients are only slightly affected by a variation of a.
The effect of this variation is somewhat larger in wide
bridges (bh = 0.500) than in narrow and long bridges
(bit. = 0.125).
•
•
-68-
Similarly one may conclude that the coefficients
calculated for the two a/h ratios differ only by a
small amount. Significant differen~es are obtained for
small a -values and then only for the points directly
under the load.
It follows therefore that the size of the beams and
the amount of lateral prestress in a multi-beam bridge
affects the coefficien~s of lateral load ~istributton only
in the points near the applied load.
For the design of multi-beam bridges, it is recalled,
that Figs. 12-15 represent also the influence lines for
the coefficients of lateral load distribution at the
midpoint£ = 0 and Figs. 16-19 those at the edge points
of the bridge. The application will be shown in Chapter VII.
•
•
-69-
VI. COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS-~ - -- - ......;.;;-.-;.....,.,;.---
1. Gener:=l.l
A field test on a multi~beam bridge is described in
Reference 2. The following data is taken from this
report.
The tested highway bridge with a span of 34 ft. and a
width of 27 ft. is composed of 9 pr~fabricated) pretensioned
concrete beams 36 in. wide and 21. in. deep. placed side
by side the peams are connected together by a steel bolt
at midspan (without significant prestress) and dry-packed
shear keys. The geometrical parmoeters for this bridge are:
b 27f == 2(34)
1.7
= 0.4
o.
ft~ong other tests a single axle load of 47,000 lbs.
was con-:;entrated over the 'i.l7idth of one beam, first in the
bridge axis and secondly, 4 ft. 9 1/2 in. from the edge.
The results for these loading positions will be compared
with the theoretical values.
2. Deflections:
Fig. 20 shovJS the measured and theoretic8.1. deflection
curves for the load placed in the bridge axis. The,cross-
t.
..
•
n70~
section of the bridge and the loading position are
schematically indicated in the top of the graph.
The theoretical curves for a = 0, 0.1, and 0.5 were
obtained for b1 = 0.4 by interpolating between the values
for bit = 0.5 and bit = 0.375. The deflecti.ons were com
puted with a modulus of Elasticity of 6.68.106 psi, the
approximate value determined in the fi.eld test. Similarly
Fig. 21 shows the measured deflections for the second
actual loading and the theoretical deflections for the edge
loading.
Comparing the experimental and theoretical deflection
curves, one cannot completely exclude the possibility that
slip may have occurred between the beams in the tested
bridge. Thus the main assumption for the theoretical analysis
(no slip· between the beams)wQuld have been violated. From
observation of the curves this fact is more apparent for
the load applied in the bridge axis. On the other hand, the
maximum measured deflection, for the same loading position
is approached by the theoretical one, for a = 0, to within
12.5%.
With these facts in mind, one may conclude that the
results of the theoretical study are in satsifactory agree-
ment with the experimental ones ..
•
3. Coefficients of Lateral Load Distribution:
In Reference 2, the difficulties arising for determining,
the exact coefficients of lateral load distribution from the
results of the field tests are discussed. For the reasons in
dicated, only the approximate coefficients based on the dis
tribution of the deflections could be obtained.
These approximate coefficients are defined by equations_
(68) and (70) when Mx is replaced by the deflection w of the
same point and Mxav by the average deflection wav of the cross
section under consideration. The S~ coefficients so deter
mined are given again in Fig. 22 for the load applied in the
centerline and in Fig. 23 for the load applied on the edge.
The same:figures include the approximate coefficients com
puted from the theoretical deflections as well as the exact
S~i coefficients obtained from the theoretical moment dis
tribution. The given theoretical results correspond to a
value of a = o.
A comparison of the curves plotted in Fig. 22 reveals
a satisfactory agreement between the approximate coefficients
resulting from the experimental and tbetheoretical investi
gations. Both of these curves show a similar trend. The
curve representing the exact coefficients deviates from the
trend mentioned above to a larger extent. The deviation is
especially significant for the sections of the curves corres~
ponding to the directly loaded beam.
•
•
,
-72-
Similarly in the case of the edge loading (Fig. 23)
one observes a close agreement between the two approximate
values for the loaded edge-beams. Again, the corresponding
e:c.act curve shews a substantial difference.
To permit a better comparison the following table
includes the three different S*xi coefficients for the
loaded beam in both loading cases.
Approximate Load Exact LoadDistr:Lbution Distribution
Experimental Theoretical Theoretical
Center Loading 16.6% 15.2% 19.6%
Edge Loading 20.8% 22.6% 28.3%
First in the case of center loading, according to the
approximate coefficient obtained from the field tests, tl:\e
center-beam would have to be designed for 16.6% of the
applied load. However the exact coefficient indicates that
this beam carries 19. 6% of the load. The beam 'Ovould there-
fore be under-designed, whereby approximately one-fifth of
the load would have been neglected.
Secondly in the case of edge loading, the experimentally. .
determined portion.of the load to be carried by the edge
beam was fOurld to be 20.8%. The corresponding exact value
."
•
•
~73-
.~
•
•
.'
-74- '
VII. USE OF THE COEFFICIENTS OF LATERAL LOAD DISTRIBUTION
The following is an example showing how the coefficients
of lateral load distribution may ~e used in the design of a
multi-beam bridge.
It is assumed that the bridge has a span of 36 ft. and
otherwise is identical with the bri4ge described in Chapter
VI and in Reference 2. The overall width of the bridge is
27 ft. and includes two 8 in. curbs cast to the edge beams.
The nominal roadway is therefore 25 ft. 8 in. The geometrical
parameters for this bridge are:
b = 27 = 0.375r 2(36)
a = 36 1.7- =h 21
The live load to be considered in the design of the
bridge is given by the AASHO Specifications (1) and con-
sists of H20-s16-44 truck loadings.
For the determination of the design load of the edge
beam it is assumed that a = 0 or that no lateral prestress
is applied, The coefficients of lateral load distribution
for this case can be taken from Fig. 16 or Table XXVIII .
They are plotted to the correct scale in Fig. 24 below the
schematic cross-section of the bridge •.
-75-
On the latter, two truck axles are placed side by side
with the minimum spacing between, as given by the specifi-
cations and such that the sum or the areas Al' A2' A3 , A4
in Fig. 24 is a maximum.
Utilizing now the properties of the eurve plotted in
Fig. 24 as an influence line for the coefficient of lateral
load distribution of the edge beam, the portion of the load
P carried by this beam is given by equation (70) with n = 9,
the number of the beams of width a, as:
= 50.3%
•
.'
..,r.
This means that the edge beam carries 1.50.3% of a wheel load .
From the assu,rnption that the same ratio may be used for
other cross-sections of the bridge it follows, that the
edge beam has to be designed for 50.3% of the wheel loads
positioned for maximum effect.
Similarly it was found that the maxi.mum portion of
the wheel loads carried by the center beam is 50.2% or
practically the same as for the edge beams.
The same evaluation of the coefficient of lateral load
distribution was made for other a-values, as well as for
different bit ratios and also for alh = 1.,00. In all cases
it was assumed that the width of the bridge would be 27 ft.
" -;:"." " ...
-76-
•
The various bit ratios ~7ould then correspond to the
following bridge span$~
bh = 0.5 1= 27 ft.bit = 0.375 1 = 36 ft.bit = 0.25 1= 54 ft.b!l = 0.125 1 =108 ft.
For the case of a/h ='1.00 it was assumed that the
bridge would be formed 'by 15 beams, 21· inches wide.. The
re'sults of this evaluatio.n are summarized in Fig. 25.
The graph on: the left-hand side shows the portion of
the wheel load carried by the center beam as a function
of a and with b1 as parameter. The upper group of curves
includes the values obtained for a bridge formed :by 9
beams, whereas the. lower graph ~hows those for a bridge
composed of.15 beams.
In the right-hand graph the portion of the wheel
load carried by the edge beam is plotted in a similar
manner.
Both figures confirm the interesting result, that for .
the loading positions under consideration, the portions ,of
the load carried by the center beam and the edge beam are
practically the same. They vary only slight~y with the,
degree of lateral prestre.ss, described by the coefficient a.
In the case of a bridge with a span of 27 ft., composed of
i
•
•
-77-
9 beams, the largest percentage of a wheel load carried
by one beam amounts to 53.2%. For the same bridge com
posed of 15 beams it amounts to 32.9%. The percentages
are smaller for longer bridges and approach in the
limits the values corresponding to a uniform load dis
tribution, which are 44.5% and 26.7%.
A comparison of these results with the 80% prescribed
by the present specification reveals the obvious advantage
of an exact investigation .
1.
2.'.
•
•
-78-
VIII. CONCLUSIONS AND RECOMMENDATIONS
The main purpose of this theoretical study has been
~o provide information for the more economical design
of multi-beam bridges. The main results are summarized
below:
There was found to be a satisfactory agreement between
the theoretical and the available experimental results.
This may be considered as a confirmation of the estab
lished assumptions and the applied methods of analyzing
this type of structure.
The approximate distribution coefficients based on the
deflections are especially insufficient for the design
of the directly loaded beam. A safe design for wheel
and axle-loads must be based on the given exact load
distribution coefficients.
3. The size of ,the beams and the amount of lateral pre
stress affect the coefficient of lateral load distri-
bution only in the points near the applied load.
4. The curves representing the coefficients of lateral
load distribution for the midpoint - and edge - loading
can be considered as influence lines for the load
portion to be carried by the center - and the edge beam,.
Their evaluation, similar to those used in the beam -
7 •
6 .
5.
•
•
-79-
statics yield the design load for one beam expressed
as the percentage of the left or right wheel loads
of the standard truck.
For a 27 ft. wide bridge with two trucks placed side
by side, 'it is shown that the maximum load carried by
a 3 ft. wide beam is 55% of the right or left wheel
loads of one truck. This value is almost independent
of the amount of lateral prestress and varies only
slightly with the span of the bridge. It has to be
compared with the 80% recommended by the present
specification.
This significant reduction of the design load can
fully be utilized only if the connections completely
prevent the beams from slipping against one another.
The condition mentioned is mainly a technological one
to be examined by tests, which are strongly recommended.
In the meantime, the results of the theoretical study
may be used for the design of bridges if the interaction
of the beam is guaranteed by sufficieQtly high lateral
prestress.
8. The relatively small influence on the load distribution
which is caused by a variation of a shows clearly that
..
..
•
. 9.
-80-
the main advantage of laterally prestressing a mu~ti-
beam bridge is an increase in the shear resistance of
the connections between the beams .
Further experimental investigations on multi-beam
bridges by means of field and laboratory test are
recommended. The scope of these tests should incor-
porate the following points:
(a) Investigati.on of the influence of the magnitudeand the locations of the lateral prestressingelements on the coefficient a.
(b) Verification of the assumptions, especially theo~e con~erning the relation between a and ~.
(c) Investigation of the behavior of multi-beambridges under higher loads, determination of thecollapse load and the factor of safety •
....a) General Case
-81-
..
..
'. Mxy = 2{3- c( . 00. -L (-~ITV? kl m7J) .p 2"yP VI p....
2:., m2 e - e Sin mv sin my cos m{
(I)
(2)
(3)
(5)
b) Articulated Plate My :: 0
f,.I"r
.9.pt! =_ .L ~ -L eko m'11r m=1 m sin mv sin my sin m!
~= i. ~ I ;komT] . .P "'y m;' m2'" . 510 mv 510 my COS m€
(6)
(7)
(8)
(9)
• •
. ,i,.;.. '. ..
..:,;
•¥.- I 'f *sinmUSlnmySOnmC[:t(rellz""h - 1011,""',,_ t (-U,~IS sonh"'"",+ Uz"2rsiMbz!'kl-V'~ISCOsll~m"+ ~llztcoshhzmll~2.,1J V'-;t ..., ,J . n
~, 12S-c()! ·~slf1mU!lll'Imycoamc[±(·ekz"""-e~''''''')-.L(-U,ll,sonh~,..... + Uz~zsinhllz"'" -~k,CO$hll,"'" + Vz~hhzli'i·P 21l'yfj l/i _.a "'" m 2 •¥ Sz .
o Note the upper S'lin IS valid far .~lI-e'). Ih3 Io<oct s'gn " valod for 0<(1-0'),
(1)
(2)
(3)
(4)
(5 )
3
N. ' cosh to, mH'M "'.lm~ - (-r.) s,nh k,m~ cosh IczmA
(~3Nz' "nh k,m~ cosh I<zm~ - \ll,Jcash II,m~ 'IGlh kzm~ Ila· J~~-FJ'] ,
k,(-Ilz"Wl' -lo.zmAe") I-ktm"e' k,mU")o ' -.. e + e - \e .. •, k,
k'I:IL..~· -IL..~") (-.,m~e' ,.m~e·)DZ 'k,'\e" - e--' - f' - e"
.. _ I"'z"",e' -kz~"l (~2\eI<,m~' -kt"",e")"t- e +e - e +e
. " r • [p (I.+V'- p~) - 011
s' [$ (, -Hz) - ~I
'I' ..!.l.b
Notation:
• ' .l.b
see Table III & Fig. 10
Table II Formulas for General Case
Ico~I
-83-
M)ty • ..L I ~ sin mv sin my cos me [t tom~_ t (u,slnh kom'l) +~ cosh ko m'))]P "''1 "'_, m \.. (4)
•
M~. 0 • Note: the upper sign Is valid for .t (I - e')
the lower sign is valid for fe: (I - et)
Formulas for Articulated Plate
. £~
;•...i~t....~:'1 ;, '
li.
i';:~~,'Table m
• III j.OJ b
~ I; 1- el
b
AU'H4 .0TAIILE IV
BETA IUJ. .f»O
~/h a 1,7
'l'AIlUl 9
liTA lU..l. .6)lI
tlo/'h. 1."IJISII.Ql '," .00 0.25 0.50 0.75
-~I;~~~i--- ---_ ..__._._. .. ~ ..... ~._-_. ---_._.~ ----1----------;. .,. i' 1 • .JJlW " 1 .~" 1 .1905,} " 1 .161W1 • I
i' ~ • • 151~ " 1 .1\2s;l • 1 .lIIlQll " 2 •74IM • 2 •'10\27 • a!a' :I. .IS1~ " 1 .;al61 . 1 .0I9JQ4 " 1 .41713 • II .606)0 • I j"". . ---_. . -..._- ...,-"-- ._ ...._.- ---
"-.'~'67"--O1M.. I' 1 .J9lIIl4 0 .;l8S74 il .:41J27 0 •162'l"1 0,..I' 2 •. IS167 0 .1119S 0 .&7196 • 1 • '7)617 - 1 .6'l288 • 1•I' :I. ,1511>7 0 .21J~ 0 .lU33 0 .4m5 0 .1128) • 1
'.- .'-EPSILOIl .,.- .. 0.25 o.~ Go'1'5 ,..
-- .'''' -, .'_. ,Hi _, I + ,2'lIIll - I ~-1 .19h6. ,
.. 2 + ....11 - , .1,. -, ..,,,,.. ._.. ....". a_, 1 - .'fI61' •, ,.ata)6 • , ..... , ...... .....
_ . ..,.,. .;a -' , + ..... 0 .-m 0 .... • ..- •_, a· + .1569'l1 0 .11'JS) 0 oWm-1 .-., ......._, a .. .1569'l1 0 ,21IlIO 0 .J'499 0' .- 0 .......
-;r _, I • -.1"" 0 .''7»1 •) .1OMB - , ...--a ..... '1
I' 2 + .11. - I ..,,, • a .6II'l6 - a """.1 .~.'l\
.' I - .11527 - , .''''75 • , .~2lIl • , ...... ,•..". tr--'- ........ a-:n I' I + .1IUCl'7 + I .1»41 • , .B'J89 0 .-
f' a • .4865J 0 ,)11'" 0 ."'" 0 ..'"' • '1·.456ft ••
.' a - o4Il654 0 •.".50 0 .1ll6a9 • I .-ra +' .,"" t a
~I' I • .00ll00 ."''16 0 .1*, 0 o6JOOt - 1 ..... 1
f. a I • .148'l\l 0 .Wip· I ,6tan - I .J)01) • , ...- -,I' 2 i - .~ \) .210"11 0 .211661 0 .»615 0 ".",., iI
TAilLE VIa£TA~ .6JO
~/ll • 1.7
'UILI VIIBETA IIU ,4.U
~/h • 1.0
-.
~=._-.....,...--,-----.,...'-- ---.,...---~~--~_._-
i.t - 1.00 o.as 0.50 0.7SJ--~~
"!-i i " 1 • .W7'1· 1 .23047.1 .iIJJ629. 1 .18565" '1 .169?5- I
PI I";j. • .1(,97{.1 .1'281·1 .10734 - I .90\43· 2\.7'179?" a.' 2. • •W/?S "' .:al9S· I .289S7· I ,)'JlU· 1 0488'19" I1---1---+--1----+----- 1--.----- e. --- .. -.
II" I" I • I .40921 0 .m67 () .2al4J 0 .18993 0 .17126 0
r "'.:1. • 1."'n2/> 0 .1)25"" 0!.,Cl64? () .89192.1, .76911 - 1~l~r-:- .~'~~2~ __~ .~~ OJ ..31~ .. 0 .4Hfl2 0 .•~31 _ oj-;r ," 1 .. "~IOS~. 0 .16960· '1.1)397. 1 .16llO6 - 1 -.19929 - h
" 2 I • .Cil!)l:S - 1 .J62.i\. 1 I .26798 • 1 .15'.l44" 1 -.)llj'l6 - I>
I' 2 i - .Cil!36· lj.S5m • 1 1.6500'1 . 1 .?!T/M - I .4)~ - S
• " I I. .601IJ'7' 1 • 1)Q}6 1 il.-ij"r#.-O 1-;1917:1-'0 :;65Q06~i-'l-I I ~, 'i " 2 1," .41170 0 .2.0/183 ~ i .14)55 () .471J33· I -.~5 - Ii I! " 3.! - i .41168 0' .64m u i .10627 .. 1 '2263' ~_,,~~=a :
~ :" 1 I... ! ,oooao ,4J445". I .58j5Q - 1 .~ - '1 .io/:DXJ. I
II ' .' 2 I .!.,3816 0 .<f12fJfl. 1 .b8U15 - 1 ~11. '1.J7SllQ - I ,
~16 il! .181~5 () .24BO 0 .25"'1 01 .2SJ5J "
- - __ ·_e"···EPSIUlti ...- .00 0.2S 0.'" 0.'15 '.00._._--- ----W o' 1 I • .,~" - 1 .251\44 - 1 •Ul3Q4 • I .W". I .1Jtat. - ,
~ al • .131:14 • I .89195 _ 2 .6J767 _ I .!IOt" • a .4J927 • a,. a - .131:14 - , .19790 - I .)0236 • I ~-1 •'D'III9 - I
1\
---,1,1. 0' .. .71o:lll 0 .;rjJ92 0 .18'MI 0 .14411 0 .1llP7S 0r
I' 2 • .1307' 0 .1IlU68 - 1 .6JQ:lI9 • I .49562 - 1 04UU. I
0' 2 - .1)075 0 .'9986 0 .31592 0 .SJ8;l'I 0 .1-"'1). I
-';J 1" .. ooסס0. .00ll00 ooסס0• ooסס0. . .lIIlIllioI' 2 • .00ll00 .oooao ooסס0. .oooao ooסס0.
I' 01 - .0000Il .oooao ooסס0. .00ll00 .oooao
-~!I' I • .603iJ'1 .. 1 .1Ql!4;Z • 1 .451~ 0 .18561 0 .~9)" 6
I' 2 • .J61'19 0 .2Olll95 Q .1268'1 0 .S7868 - 1I-.~ _ 6
.0' 2 · ,;)61'19 0 .Sif.!IS 0 .1Cl289 t·' .:.lZlI61 • 1 ,l6Ofi .• a...... _.~ ..............~-'"' ....... '.''-''~'" --~"':.- --
.!iQlll4 ---,-,.--=M..,
.' 1 • .24641 0 .17439 0 .'0'722 0 1-.421'71 - 6I
.68946 - 1 .4lXIU • I l.lm2 - I I-.J'-'9 ~ 6
I " 2 .. .11011 0
I' 2 - ."011 0 ,169?~ 0 ~2544J 0 i .J6710 0 .,49a1 0
AU'HA .1TABLIl VI II
II£TA IlLl .4Zl
~l\ • l.0
!AaU: UlETA IIJ.L .4AlI
&/tI a 1.0
IPSllQIl .,- ,OIl 0.2' 0.50 0,75 . "OO~--~----
'Sl " I .. .2'N.bl - 1 .2:;2" - 1 .1989'1 • 1 ,16221 - 1 .14O'1ll - IPI'
,4~7" 2'.' . • .140?8 - 1 .97912 - 2 .714'2- 2 .)6271 - 2
0' 2 - .14D17 - I .a:no~ - 1 .30657 " 1 .4S1S7 - 1 .i>5i!64 - I
"ll I' 1 • .S22Jl () .29-"90 0 .20711 0 • 1!>lOll 0 .I4Q(,.< 0r- .1' 2 .I4Q42 0 .9?<>'IJ - I .706Q/, • I .S5S76 - 1 .47361 • 1•
f' 2 - .14Q<l2 0 .2D162 0 .J.!213 0 .~ 0 .12169 • 1
.~ " 1 • -.1:;212 o -.1S464 - ~ .115JOl - 1 .10541· I -."J9'7'Z7 ... 7I
0' 2 t .14&27. 1 .10359 • 1 .723;lS - 2 .~_2 ··~-710' 2 - .1~Z7 - 1 .21H2 - 1 .29117 • 1 .36761 - I -.2Il1074 - S-1----_..__.- .- _. --,--_._ ... 1----._--- --._--
.~ " 1 • .0030'7 .. 1 .11111 .. 1 .4671/7 0 .18~ 0-.14581 • 1
[I ~ t .)56/02 0 .21an 0 .~a~ 0 .!i4741 • 1 -.54.)4;l • 2
" 2 - .35641 0 .~'/l!71 0 .997'J8 0 .~4Jll .. 1 .14857 • 2._----..., " I • .0Q:XIll ,938'iJ - 1 .9J'I9O - I .605J2 • 1 .4QJ69 - 1
" 2 .. I .12062 0 .7'118) - 1 .461~ _ 1 .2S06J. 1 .1Sl148 - 1
•, a • i .1206Z o . .17'190 0 .1"" 0 .~ 01 .JJa44_~
R"1.Ql +,. .00 0.25 0.'" 0.75 '.lIO----'-- .----~ rl I • .l!lOt2 • , .23a'75 - I .aD652 - I .1&:l~ - I .16Sa4 - I
I' 2 • .16$14 • 1 .12462 - I .966DO • 01 .7745' - .6lll' . aI' 2 - .16~ - 1 .2211lt • I .3911'6 - , ,)9'05 - I .5Zl9J - ,
;. I' , • .41_ 0 .2'l'il8I 0 .21954 \) •Ul6'r5 c .16m 0
II 2 • .16m 0 ,1~ 0 .~-1 .'765016 • I .62556 • I
I' :I. · ,16575 0 .2Zrl7 0 .Jan4 0 ..48434 c .968~ 0
.~ 0' 1 • _.:141)' o -.197"J6 _ , •12M - I ,16671 - , .WIII. 6
.' 2 • .51368 • 1 .3911IO " I .28682 " 1 .'6420 - I .,"', • 6
0' a - .,U'IO - I .~"'.7J605 _ 1 .80148 _ I ,aB65a ~ 4._--_. ----- 1-------
-~ " 1 t .603iJ'1 • I .1Ill68 .. 1 ,476J7 0 .162'72 o -.'1415a • I
I' :I • .);l'lll' Q .1'16'Jb 0 .101~ 0 •.nOIla _ 1 _.Jill•• 1
I' I - .J2'Al6 0 .1l1llO 0 .9Cl9'71 0 .1'D~ • , .1:asD • "-~ "' I • .ooaoo .JfH14 • , .51409 - 1 .4'mO. , .... ·'1:
~, 2 • .1~ II • ~'7oUl> - I .601Q3 • , "''107 • , .,.,.. .' ...
_, :I. · • 'a<i41 0 .1624D 0 .~ 0 .~ c .326Jt 01'
EPSIUlII +,- .00 0.2S" O.SO .I o. 7~ 1.00..
~, " I ..- .JeI211 - 1 .J09'4 • 1 .1:hJtIJ - 1 T;ul!lY' - 1 .22'lllJ -
.229lU - 1 .\IH17 - , .1<009 - 1.1478J - 1 .14286 - 1" Z. T
1.4704J3 - 1" 2- - .ZI9ll4 • 1 .2861,0 - 1 .3641) - 1 .(,19~7 - 1
..__..- ---- _.. -~ " i 1" .6415J U •.>6712- 0 .2IIJ1' 0 .24560 0 .~ 0
.' 2- t .;u..SS u .19071 0 .16Z~ 0 .I4/>72 0 .14161 0
.' 2-. .Z34~ 0 .)00t:6 0 .JJJJ7'1 0 .5t!d'l~ 0 .11416 + 1
----
-;: " 1 1- ooסס0. ooסס0. ooסס0. ooסס0. ooסס0.
" 2- ~ ooסס0. .OOOCO ooסס0. .OOOCO ooסס0.
" 2- - ooסס0. .WOOl) ooסס0. ooסס0.
:~-~'s-1.~..._..- --......~-,_.-., ..- .............,- ..
-%1 " 1 + .-.:JJ07' 1 .17187' 1 .7371' 0 .311112. 0
" 2. T .7111if- 0 .41'J89 0 .2'J044 U .13&0 0 .6le71 • 5 i" 1- - .71'~ 0 .10l'i?2. >I .,..,. Y'''''' 1 • 16016~ , 2.. ,\r---"-" ..._.,c
..... ... i'
1Ij1 I' 1 + .~I· 0 .Ie~ o .122U1 0 .60441- I .12JlI4 - ,~
(i Z. ~I·I~ 01 .0389 "I .S~I)ll - 1 .41J24- 1 .1916U - 5'!I
.:15<176 'J! .)~t;, oi .. lZU. oJ! .492112. 0:,,2.. -j .I~ 01 ,
IlUI BEMllI\l~
f~ll
BETA ILL.L .6JQ
a/ll 1"•I ~IL.OII +,- .00 0.15 0.50 0.75 "lID
-Et " I + .)2660 - I .)0)69 - 1 .Z7JQ4 • I .25025 • , ..u"'. If'YW
" 2 .-U~' - 1 .19693'· I .non· I .15113 • , .\4JD • I+
" 2 - .-U~~ • 1 .~., .~7- I ..um - I .t'IMI - ,
~ " 1 + .5)0)6 0 .36'S5 0 ._fJ 0 .2fJ05 0 .:141" 0
0' 2 + .24121 0 .191l19 0 .1'l'OO6 0 .Isago 0 .142'16 0
" ;l - ."4121 0 .JO"'I 0 ~19 0 .~J 0 .106QJ. 1
-(t " 1 + -.1))01 o .,41)72 • 2 .'I'Ill9' .2 .IQJ72 - I -.74149 • .,
(' 2 + .17249 - , .14\)4- I .11361 _ 1 .~.2 _.2lI012 • .,
" 2 - .11'44'1 - ., .2'061! - 1 .25347 - I .aMIl7. 1 -.IS01? • 4
I-~0' 1 + ,i$D)07 .. I .17A1Xl .. I .14J2'I 0 .)OJ-U o -.t4680 • I
c' 2 .. .6<;0aa6 01 04S990 0\ .2llO>S 0 .1296' o -.91'180 • 2
I " 2 - .6982' 0: .10570 • 1 .171lJ .. I .,)4S9tl ., I ."137" a
l~ " I + ooסס0. .16.u'l - 1 .Cl4920 • 1 .6ll6IlI • I ,"~. I
, I " 2.. .1'lO96 0 .I)6IlJ 0 .'JO'01 - 1 .S4J71 - I .,J'7llXJ • I
! ; " 1 - .19096 0 .2'lJ8 I) .)IJO'Ij I) .36126 0 """" 0-----
Bt\- .175'I'ABU! X
BETA N.L1. •<{So
.;n .. 1.?
ALPHA .0
8t\- .)7~' .... lIU
BETA ILU .4Z<
a/l> • 1. 0
M.PHA .0
[J-;IL;A>j '.- .00 O.2~ 0.50 0.7S 1.00
WII " 1 -t" .~., .3214il • 1 .2~7~ _ I .222402 • 1 .21122 _ I
PI'" 2 .ll1l'" • 1 .'6289 • I .IJ'67. 1 .1\.'6 - 1 .101l~ - I....
Ift 2 - ,2111't - 1 .211190. 1 ,JllJ2? - I .~I _I .7ioJY5 - I
II .. ~ 1 • .744~~ 0 .J71065 0 .27'24J 0 .22h~ 0 .212\/8 0r
f' 2 • .21298 0 .1~ 0 .1)060 0 .11295 0 .107)0 0
.' 2 - .2129ll 0 ."J!f0h8 0 0414'!6 C .i.I,ojS 0 .U"8 .. I
II " 1 01- .1))000 ooסס0. ooסס0. ooסס0. -oo,,:rסס0," 2 + ooסס0. ooסס0. .OOUDO .OOCIJO ooסס0.
" 2 · ooסס0. ooסס0. ooסס0. ooסס0. ooסס0.
-~!(' 1 .~ .&)JIY1 ... I .1429) , 1 ,~JO 0 ,2~14Ol 0 .4377'1 - 6
" d- + .~~N 0 .")61~ 0 .218It 0 .10282 0-.17130.6
" J. · .~W70 0 .~1ll 0 .14280 • 1 ,2%U t I .16061 , 2--_.'-
I liT " 1 .. .lo4641 I) .182" 0 .1111lS 0 .rtnJO - I -.:«1361 • 6
i " 2~, .1W43 0 .101l44 0 .6t6I4iiJ • I .)IIlT.! - I -.24171 - 6
! [' ;l · .I~ 0 .;z;:)'/o 0 .)OJI" 0 .NTaJ 0 .492111 0"
o
o
1.00
.25715
v .149'11'
Bt\- .)7''I'ABU! XII
BETA IUL .1.30alb, 1.1
.oouoo .2&432..·1 .;,;;67f!J- 1 .4611.;1 - 1 ...WI - 1
.1~5113 C .12'/,.. 0 .~21- 1 .76ez;' - I .6BOI~ - 1
.~1)7 • 1 .I"lOO5"' I .109'/4 U .2«>55 0 -.80.ti'4 • 1
.511:w> 0 .36"1) .~3Jt u .%<0 - 1 -.~)<\Q( - I
.sgJ;.4 V .~ Q .141>12.·' 1 .~+ 1 .1~'"
[I 1
" ~
-.PHA .,
EPSIUlII '.- .00 O•.2.S" I U.Si' I G.'6
J.-W-,...-~"'l';-;-~-'-2.'+--:-+....,,:::--~"'7O---.-::+-:-~c=,~~::=~T=:~ :::: :c' ~ <-5016 - 1 .<''1632.- 1 .35421 - 1 .1I47s - 1 .o;prf2-- 1
II.. (, I {o .I,(J.!~ U -.;-;);.4-'0 -:jif;;3····0 -:-;;:.:J!-"cr ;[, ~+ ~2S7I" C .1..ll.i.iJ 0 .18'/.1 0 .1~
I-_-+~(':'":.~:+..::"":,+-~.I~.$71~'~~O~.~3~1q$~r-_::o~.4UOB~~1~U1'r·~~'100::::"-T·~~
,'I 't -.urI4 0 -.~56 I ,$1/{$ - 2. .14;0) - 1 -.~120 - ,
c':l ... • ~9416 • 1 .5OCI4S .'3&3? - I .2.."'9:14 - I -.,..,131 - 6
"... .~14 - 1 .iih88. 1 .7:"U.t.12-- 1 .,31',7 - 1 -.~- 4
•
TA8U1:llYIITA IILl. ,4U
a/tl .. 1.0IEPSIUlII +.- .00. O• .l~ o.m 0.71 lao 1:~
( .. ,JQUQ • .7.!/J3'j - 1 .27m - 1 .261rr _ 1 .14m • , 1I o' ~ • .14731 - I .al69J • 1 .176\J - I .1S2~ - , .IJJOS. I I
" ;l - .447JI - I .m21 - I ,.J641t - I ,44,Jjl • , .5:164' • ,
IIx " 1 + ,41)71 0 .'~7 0 .JO;l16 0 .;n,60 0 .'~ 0p-o' 2 .. .2~ 0 .207lIO 0 • \7"" 0 .1~IJ\ 0 .1)1'1'6 0
" 2 - .2"'J<I 0 .;1"04 0 .4V1'dI> 0 .~789 0 .9SJ~ 0- .
-~ " I • ••-U~J4 o -.2?lS/j _ I .'/07'J~ • 2 ,IAIlO'I • I • •.lOJ75 • S
.' 1 • .6~'n~ • 1 .~'111 _ I .41;:79 _ I .4021. I •• 10~)~.J
" ~ - .65714 • 1 .74601 - , .lll*l) - 1 .~., -.1119:1 - ..t------~._- --
"~ .' I+ ,lKIJ07 • I ,15'7, + 1 .CU840 0 .Zl4OlJ o -.~O - ,
c' ~ • ....17IfJ 0 .~5 0 .1J,II44 0 .2.01161 _ 1 -.W164 - I
c' a · ....7781 0 .75181 0 .14Jj", .2S62l> ., I .12liO"/ ., :I_._--- ...
~ " I • .0lXI00 .lJ'IA - , .J7n6. , ....,001, - I ~·'I
I .' 2 • .14'H1 0 ,,,.,. 0 •.",711 • , ..,. -, "'14'1·' Ic' a - .14'~ 0 .'";<1, 0 .2D6llO 0 •.:tIOfJ 0 ,zaul 0....
tABU ~lV
IIrTA IUL .4<Ualb • ~.O
..-' -1IPSlL.OII ; +,•. .00 0.2' O.S:O 0.7) 1.0e
~I' 1 + ,';3'112-. 1 .Jl0" . ,
.27)1' - I .~70 _ 1 .2Z))JI • I
e' +. .~~-I .1?5"roJ. , .14~b6 - 1 .121,'1 _ 1 .1\011 • 1 i
o' II - .~.I .29:J& - 1 .1i'319 - 1 .ljl)5i,I - 1 .f#7i - I
;. ", + .l6470 0 .~71~ C .l.~I<l5" 0 .~UI 0 .22St/P "
" Ol. t" ,aa~ 0 .\7!l45" 0 ,142bO C .1~ 0 .IOB'JO 0
" 2I
.Z2~~ 0 .301011 C .417'2i 0 .62.7.:&1 0 .IVJ.i'" 1-> " 1 • -.I4ka \I -.68671 - <- .103'19 - 1 .11"90 - 1-,I?9~-b
" ;Z ... .aa~9. , .17'6~ - 1 .1j):' .. 1 .~.1- .; _.1~~. ,
.' , · .U"'9 -, .:IS:lJl - 1 .")L,e~-- 1 .38591 - 1 -·'JIJ.J:'(l- r-
-~! " 1 .. .•!kJ)07." 1 • 14111~, I .624bS u .24~t<J u -.2l.$)- I
" 2- 1" ,~o 0 .3I.9~ 0 .'2D'153 0 .9189S" - 1 -.'2451 - I
.1' 2 - .~ 0 .~p 0 .1318J + 1 '~'I • lJ,8~~ .. l......, i' 1 +- ooסס0. .1069' - I .14524 - I .66ID _ I SJJ96 - I
i'
2 1" .167/1 0 .117/1 0 .77494 - 1 .4&.)) - , .:l6U)) • ,
" 2- · .11>711 0 ."2Z~? o . :'~'h?. 0 .J296ll 0 .J3579 0
---'-" _L-.
taaul nIllETA IIU .6JO
alb • loT~------,.----,-----
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1 PI" ,'2 +,"- - 1 ",.. - 1 .J8l2 -, .Jt:o' -, .." - ,
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i :" 2 I -I ,409S2 0 047)91 0.J7* 0 .7J'I\O 0 .'"'' + 1
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I 1'~_2..~+-.~ -__1~~~ ~~.~)ll4.~.~+~)IU'7 -, 046522 - ,r;;--rll'.' 1 + Il1JJO'7 + 11 24515 + 1 ' .1lJ9,U + 1 045000 0 -.2192'7 - ,i -=J-! i . I' i • , I
; ! I' 2 1 + I .10715 + II' .7129'/ 0 'I' 04J65) 0 i.3Illm 0\-.1?'a4'7 - 1i J I I
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: PI"" + : .J8101l _ 1 ; .J45S'1 - 1 i.J211S - I! .J06l13 _ 1 : :30211 - ~: .' 2 _: .J81O'f - 1 : .J;H06 - 1.•·.4~~~:.~ •.~1~.-__1~.6~.~ 1t·> h_"~-:';';;;2- ~ :-:-»OI~ 0 .448~ 0 i .4115'7 0 i.~ 0 i
.' 2 +:..4lIOS4 0 .)551" 0 .J'/555 0 i Jl1!79 0 I .)OJ,s o!i ." 2 ..IoOll" 0 .46IlO, 0 SnS2 0 ! .75162 0 ' .11m + 1 i
::-2":~ :=.-:::. := ::= := '-1.' 2 ' .oooao .•00000 .00000 I .0lIl00 .0lIl00 I
+ ~ ;a;.m-~·-I-~:~~ ~'~'0ll62 ~"r 04SWJ 0-.~~::..~.1100 + 1 .•~19~ 0 04S019J 0' .2D3 0 .1:U"O - 4 I
: i.1109J + 1 ••16~ + 1 ,.263;/4 + I, .~ +_ 1 •• V.o&.4_+~
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o . ,46111
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.0
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T4ILI UKlETA IIU .422
alb • 1.0
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I I . I !__•.'_2~,_~..-.J6/:IJ__5_-_1 , .43069 - .~ ; .51~.~6mS - 1 .~_:_~
II x.' +.lWoD96 0'.SJ9')2 0 ~ 0: .39251 0.J1896 0
r ,[. 2 + •.l~ 0 1.Ja)92 0 :... o;.~ o! ....151 0I I. I
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TAlII.! xv IIIIl£TA I8ll ,1:>.)0
alb. " :'7
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, .' ~ .401)0 - 1 j .4J676 - 1 ' ,4'If>:JJ - 1-' ,~1~ - 1 i .5'1'1S2 - 1 '1-- ,.__ .. ----+- - 'lilt" +' ."h)~J U.•51'1S6 0 .";'7'1< 0' .44145 0' .42li'll! 0
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(, " ' .7.llJ'/ - 1 .78'n4 - 1 .dOl.l~ - 1 .75010 - 1, .17785 - 4
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: .CIlIODO
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O .•J75~
o .17186
; .oooao1 .ODOOO
o •.l71~
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o .6OOIl2
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" 1
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.~1
.21444
o .18673 o .1:u9'f
o .10011
o .1)1716 - 1 .135"0 - 5
o 049S~ - 1 ' .262f>l! - 5
.' .ll!O7.l 0 .20160 o ,.<17~ 0 i .~J6 0: .22~'/_.1 ....1..._.....
o._.- .' 2
-_.j..2\444 o .2'11*11 o .•4:u03 o : 049282 o'
"2j .7)~ 0 .I\41b+l .1II7Jl+1 ,J684O+1 .125)4+4
" 2· - .167J.i 0 .1847', 0 .19'161 O'.~ 0!.aQ)ZJ 0" __ 1. _1.. _..1 1.... • __
O! .104l14 + 1....
.~167 - 1 .601., - 1t-
O' .4217) 0
o .72513o S13S4
.'''47 - 1 .~ - 1 .83571 - Ii .1807) - 4
TABLR IXIlIETA IIU .422
a/h • 1.0
0: .48.WI
.11105 - 1 .ll1tll - 1 • .uMJ> - 1 .....136 - 1
o .'14114.> 0 .IJO'19 0 .11?86 0;.". 0
.7'fll0 -
.421"14
.~:J07 + l' .a2J7' 1 .~51)5 O,.J4SllO 0 -.11'JQ.) "
.V594 0 ~ 0' .21819 0 .1)~ - 1 -.18/21 0
+ . -.21ot'l 0·-.:lE'176 - .•1),144 - 2 .I(l1J111 - 1 .6471~ - ~
+
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~-;s;u;- '+'-1 .00 I 0.25 I o.~ I O.75 ~ 1.00
~ ~ (' 11 •. .42'f1) - l' .424JJ - 11.41606 - 1 .40771 - 1: ~ - 1
r l 2' + .40010 - f .J6~79 - 1 .3'" - 1 .". - 1i .2JI9D - 1i
,4IYJ:H - 11 .440'6 - 1 ! .Jk1#fl - 1+-._ .. _._.. ..
.60051 0 .51%) 0;.46698 0 i .438"1tiI ,
,4.l174 0 •.l7I,J'J 0 i .34165 0; .3138' 0, .l.8974 0
fAIlLE IIUMA .1 Il£TA IlW. .J.U &A. .250
a/h • 1.0
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'-1Ay " 1: + i-·14D72 J,-.I.J8'f5 - 1; .6S236 - 2, .'1'17.11 - 2:-.£)89. - 61
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i ~I i .' 1, + 'I .IJJJJtY/ + l' .21)41 + 1 '.92'J69 0' .Yt'2a6 01-.2981~-1-p'! I 1 '
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~.;; I -~..:.~~~1.,~:1~~.~_~.:.2.-l-.411~~_~~~ + 21
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. • 1 I .' ~l -, .21212 "I .~~ 0 ,.Hl'IO 0: .J~15J 01 .JZlI" 0~. .L__, 1.-.- ...1 _
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fl ... •;a SI 1 .. .1Cl6W1 .. , .102$" 1 .9\I:I'l"P 0 •~1llI
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I' 2 - .al718 • 1 .JIO'llI4 • I .H:IJ49 • I .~+1 .1Sl1•• '..- oJI''' • ,
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.1 a · .:0413 0 .26OIIll 1/ .lIil1.u, 0 o2lM1 C ..... 0
~ .•AIJ!lfA .1. TABLE 1111
lIiITA IIU.l. .6JOalb .. 1.'7
.,IUII +,- .. 0.2~ 0.'" Ii, o.?S '.00
W.. , • .e1lW ~ , .&Il146O. , .Il26?S - " .816i11 '" , .B1:m - 1
I' 2 • 08'm - I .7ilI!8 • , .'n66a - I .~., .'l'6aIS • 1
I' 2 - 08,~ - I .~~1 o8'l8iO • 1 .9;oon - I .9'llI55 - I
';J .. I • .11S'O. 1 .tenlS. 1 ._'''4 0 .llO'UlI 0 .&lmV 0
II 2 , .~ 0 .a:r;u7 0 .a;o12 0 .~ 1/ olI0107 1/
., 2 - .ll9?lU 0 .96OlI6O 0 .~.,"~" .1~' I
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s' a • .GlllllIO .GlllllIO .GlllllIO .ClOl!l!lD •aml!O
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II , • ".1llIJO'I , I ..Ji"lQ + I .1972'r , I .~ o • .J69)l1 - S
cl 2 + .211:Q' I .14646 , , .'i1a>,) 0 04,J891 0 • 'ClIU'l - 'sc' a - .217U' I .Jt6S7· , o4llsn, I ,111469 t , .161l61 + 2
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.~1 1/ .11161l9 1/ .12490 1/ .6.168J - 1 oJ~· 7
II a + .<l4'1l102 1/ .111691 0 .12460 0 •~-I •'2~20 -6
I' 2. - o<l4llU 0 .7119"" 0 .J7~ 0 o4J60 0 04\lOIII2 0
.'
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alb • 1.'7
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I' 2 - .a7eG9 0 .9mJ 0 .~., 0Uil1G· • ..... ,..;x .0 1 • .0IIll00 ~ClllDIllI .iIll§GD Al!DJ ...
I' 2 • .0IllIlIll .ll:l/IIIO ~CIIllIlt3 olmllllll ...,0 a - .Clllllllll .lIlllllllI .Cl!lIllllI ocm!nll .-
-~10 1 .. .SIUO'1+ , ..JtoaJ + 1 .'esl8 .. I .'15'Il 0 .~'·4
I' 2 • .179'/11" 1 •'2Ila1 • 1 .7"" 0 ...J6tll& 0 ._-4I' 2 - .1797)' , .~,., 041171 • t .72t116 • , .1lIGIl4 • a
~I' , , .~1 0 .1,- 0 .1• 0 ~., .n.,.· •II a ,+ 0347'1'1 0 .1IIS63 0 .';U~ 0 .617J1) .. , oJIII'I. 5
I' a · 024'lll11 .'0 J10fA 0 J7J:l,l) 0 o6JM • ...,. 0
"
8'1IL.GII +.- .00 O."~ 0.50 o.?S 1,00
.1 ,I 1 • .A:I4lIi • 1 .~.I .1lJ177 - I .829?0 - 1 ,azffl· IPii
I' 2- • ,,&;nllQ - " .tlCH71 ~ 1 ,~-1 .777)0 ~ , .16224 • 1: ' .
.a6Il~ - 1,I·.;z. · $181 ~ I .-'i64 ~', .a'11~ - 1 ,'11)6) - I
~(I 1 • ,,~~. 1 .~ 0 ,'1~)ll 0 .Y)1II! 0 .'1:046 0
I' :t • .9"Ool/$ .. 0 .IlllJ19 0 .~1'!6 0 .82679 ~ .~911 0."II 2 · .~?JlJO 1/ ,W11.D (/ .1\lSJo. 1 .11658 • 1 .1))57' + 1
-;.. II I • ,..1(\ll8y O-~-I -.19567 - ~ •90761 - 2 oJOO7lI- - ,
., :z. • .79~ - 1 .~-1 .~-1 •)21711 .. I .J2'1J) • ,
I' c - .79:nf - 1 .107111 - 1 .1l1'l2J - 1 .•6J!iOJ9 ~ I • .1865$ - 4
_'JI I' 1 • .QlIJD7 • , 03'~+ 1 .18978 • 1 .7"30 .0 -.18/,41) 0
I' 2- • .m17., .1Clti~. 1 .~ 0 .11066 o -.;Dt\42 0
I' 2 - .17219' I .11IbIJaI • 1 JIJtTn ., .eli75/. " .14l9?'1 + 2
~t l I • .0lIll00 •17a!Plt • a ~·l .'7670 - II :,~-a
t' 2 + .1S13? 0 .1771& 0 ,17))2 0 .17070 0 .1€1#'l1ll 0
I' 'a - .tB1J8 0 .1a".s 0 .181\6) 0 .18'180 0 •las" 0
•
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.1 " •• "A15"'..... , .lU186 - ,. .82S ~ 1 .tJ2tJtT1 - I
~ ;,.1: ' ...... _~CO 2 • .....#1Im - 1 .'I9CO' - I .,..10 -I .74326 - , •72171 -,
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t .1GI64' I .'0108 • 1 .9S024. 0 .91~9 0 .avm 0
"It 2, ,+ .llJffl) 0 ~ 0 .8OIlJ7 1/ ,7'1m 0 .~9 0
~, l '- oS9'I'1S 0 -.•9'1160 0 .10'l8J + 1 .12i087 t 1 .1SlB1 • I'-:-'---"~T ~-~- -
-~cl 1 , • ,..'Zl44 ' 0 _.uz~ - , 08JJ86 - J .'lOO67 - 2 .sUllO - 6
. '
.9880S • 6-10 .. " :.4720J • I 041926 ,. 1 J)lY7 - , .1'195',. I
" .~-, • '0ll70 - 4I' 2 · '04721l6 _ 1 04'1413 • , 046m - ,- J»18 • •71YSl o -04947J • 1
-~ I' I , •tJJrJ#I + 1 I .1?27S • ,!
8' 2 + .1'lQ61:. + I .110166. 1 .674034 0 .lSS2/l o -.'1'1260 • I
I' ii - .17041 + I .2SJ68 + 1 ~~'6t '+.~'".~+2
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IIt' 2 ~ oZlTI? 0 .'lSTIO':; 1/ ... 0 1.'161ii 0 .171J7 0
••:2 - ~ 0 ..w.14 0 .26116 0 Io1JR19 0 .1llWilO It
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Ii all 1 • .--99 -. .a»t6 •• .831'10 •• ~ .. , ..".. ,.. a .. 082752 - • ~-, ••." .. 1 076ln1" • .ftIla ...
sl 2 - .82752. t o85ll26 - , .lI'74'D .. 1 o~'" ...,... ,f--- r---- ..
;a SO , +1 .1CU401 + 1 :-mStZ 0 .~ 0 .~ 0 .JtIt1 0
.0 2 +1 .91\15) 0 ..~ 0 .Q86 0 .~ 0 .'IiD'I 0I
i
10 2 - .919Sl 0 .'Yi86? 0' .10616" 1 .t1ll6S. 1 •.,..,. t
-~1 0 1 + .1. O-~'" ro.,27m .. 2 .-... -.--,I' 2 .! olIl41Q1- 1 .'W?1 - 1 .""'iIi! .. , oJ2'DO - , ..._..,I' 2
I084163 - 1 .~-1 .Ila5Oll- , • 'I6ll11 - 1~-.-
-~I' 1 + '-+1 .J61J8 • 1 .1'W19. , .." 0-"" •U' 2 + .~+1 ...,., 0 04'" 0 .)U6J. a.....aO l - .144a4" 1 ""+1 .JS!IIIh • 1 .-., .~.a
~
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•
-88-
~ 0<. ~ -1.00 -0.75 -0.50 -0.25 0 +0.25 +0.50 +0.75 +1.00
0' 1 0.618 0.663 0.813 1.163 2.435 1.163 0.813 0.663 0.6182 4.585 2.024 1.272 0.870 0.518 0.455 0.354 0.301 0.280
0.1 1 0.649 0.71'03 0.!'372 , 1.182 1.996 1.182 0.872 0.719 0.6491
0.643 0.484 0.381 0.324 0.2912 4.2f-.37 2.008 1.291: 'J.893
0.5 0.5 1 0.710 0.788 0.913 1.154 1.697 1.154 0.913 0.788 0.7102 3.728 1.922 1.296 0.938 0.704 0.547 0.436 0.366 0.317
1.0 1 0.740 0.810 0.921 1.132 1.645 1.132 0.921 0.810 0.7402 3.479 1.880 1.298 0.962 0.740 0.578 0.463 0.380 0.314
0 1 (').723 0.757 0.871 1.132 1.975 1.132 0.871 0.751 0.7232 :3.514 1.812 1.243 0.926 0.723 0.588 0.502 0.452 0.437
o. ~~,75 0.1 1 O.74E3 0.804 0.916 1.137 1.646 1.137 0.916 0.804 0.7482 3.272 1.784 1.247 0.941 0.744 0.611 0.525 0.472 0.441
0.5 1 U.SOI 0.854 0.944 1.112 1.449 1.112 0.944 0 • .954 0.8012, 2.864 1. 700 1.~;41 0.975 0.796 O.SGg 0.579 0.514 0.464
0 1 0.827 0.849 0.926 1.093 1.548 1.093 0.926 0.849 0.8272 2.470 1.551 1.17~ 0.965 0.827 0.732,0.672 0.637 0.625
.:. I0.1 1 C.D49 0.884 0.952 1.089 1.342 1.089 iO.952 0.884 0.849
2 2.314 1. :523 1.H30 0.979 O.F.347 0.754 0.692 0.651 0.622
o t"'r;: 0.5 1 ~.i .832 U• 91:5 (J .. 969 1.072 1.228 1.072 0.969 0.913 0.882.<:::0
2 12.058 1.457 1.169 0.998 0.880 0.795 0.731 0.682 0.640
1.0 1 0.895 0.921 0.971 1.062 1.229 1.062 0.971 0.921 0.8952 2.025 1.432 1.165 1.004 0.893 0.810 0.748 0.690 0.640
0 1 0.928 0.938 0.9'74 1.047 1.190 1.047 0.974 0.938 0.9282 1.552 1.2.59 L096 1.g96 0.928 0.881 0.851 0.834 0.828
:).1 1 0.941 0.957 0.987 1.039 1.106 1.039 0.987 0.957 0.941fa 1.480 1.239 1.096 1.004 0.940 0.895 0.862 0.839 0.820
0.125 0.5 1 0.954 0.969 0.992 1.029 1.065 1.029 0.992 0.969 0.9542 1.382 1.206 1.089 LOI0 0.954 0.913 0.881 0.855 0.832
1.0 1 0 •.962 0.974 0.997 1.020 1.069 1.020 0.997 0.974 0.9622 1.317 1.194 1.092 1.014 0.962 0.920 0.889 0.861 0.837
Table XXVIII Coefficients of Lateral LoadD1str1but1on(a/h = 1.7)
•
•
.'
-89-
~ ~ ~ -1.00 -0.75 -0.50 '-0.25 0 +0.25 +0.50 +0.75 +1.00
0 1 0.528 0.581 0.762 1.185 2.863 1.185 0.762 0.581 0.5282 .5.544 2.169 1.274 0.806 0.528 0.355 0.254 0.202 0.181
0.1 1 0.579 0.674 0.855 1.211 2.157 1.211 0.855 0.674 0.5792 4.947 2.146 1.309 0.854 0.569 0.394 0.289 0'.228 0.191
0.5 0.5 1 0.689 0.776 0.913 1.162 1.743 1.162 0.913 0.776 0.6892 3.971 1.984 1.316 0.930 0.680 0.508 0.393 0.316 0.258
1.0 1 0.740 0.810 0.921 1.132 1.649 1.132 0.921 0.810 0.7402 3.479 1 0 880 1 0 298 0.962 0.740 0.579 0.463 0.380 0.314
0 1 0.651 0.694 0.832 1.153 2.281 1.153 0.832 0.694 0.6512 4.213 1.951 1.262 0.887 0.649 0.494 0.399 0.345 0.326
0.375 0.1 1 0.704 0.776 0.907 1.159 1.760 1.159 0.907 0.776 0.7042 3.781 1 0920 1.287 0.929 0.698 0.540 0.441 0.377 0.336
0.5 1 0.785 0.847 0.941 1.118 1.477 1.118 0.941 0.847 0.7852 3.046 1.759 1.263 0.975 0.780 0.644 0.542 0.467 0.409
0 1 0.780 0.809 00912 1.111 1.730 1.111 0.912 0.809 0.7802 2.924 1.671 1.210 0.949 0.780 00667 0.593 0.551 0.539
0.1 1 0.822 0.867 0.948 1.104 1.402 1.104 0.948 0.867 0.8222 2.620 1.622 1.213 0.977 0.818 0.709 0.630 0.576 0.535
0.25 0.5 1 0.874 0.909 0.967 1.077 1.244 1.077 0.967 0.909 0.8742 2.165 1.498 10186 0.998 0.872 0.777 0.707 0.649 0.599
1.0 1 0.895 0.921 0.971 1.062 1.229 1.062 0.971 0.921 0.8952 2.025 1.432 1.165 1.004 0.893 0.810 0.748 0.690 0.640
0 1 0.906 0.919 0.964 1.058 1.262 1.058 0.964 0.919 0.9062 1.738 1.324 1.116 0.989 00905 0.848 0.811 0.790 0.783
0.1 1 0.930 0.948 0.983 1.047 1.124 10047 0.983 0.948 0.9302 1.610 1.292 1.115 1.004 0.929 0.873 0.831 0.800 0.774
0.125 0.5 1 0.952 0.966 00992 1.032 1.070 1.032 0.992 0.966 0.9522 0.424 1 0228 1.098 1.012 0 0 952 0.904 0.868 0.838 0.812
1.0 1 0.962 0.974 0.997 1.020 1.069 1.020 0.997 0.974 0.9622 1 0317 1.194 1.092 1.014 0.962 0.920 0.889 0.8f)1 0.837
Table XXIX Coefficients of Lateral Load Distribution(a!h = 1.0)
.. -90-
o
~ t Loterol Prestress
zFIG. I Multi-Beam Bridge With Lateral Prestress
..
• ---- ----
,...--Lateral· Prestressing Rod
----
FIG.2 Schematic Cross-Section of a Multi-Beam Bridge
FIG. 3 Corrugated Steer Sheet as Orthotroplc Plate
..y
..
=91·~
stringer. Moment of Inertia Ix
BecJm: Moment of Inert.la Iy
c
-92-
*=1.7 0.6*= 1.5• *=1.2
~=VCX-----• *=1.0 0.4~ = 3(K + 0.3330()/ --_.-
/f3 = 3 K(I - 0(12 )+ 0(
//
• // a0.2 I- .,
// {O K = f (t>I
00 0.2 0.4 0.6 0.8 1.00<
Fig. 6 Assumed Relations Between 0< and ~
"..
II
•
•
-93-
FIG. 7 Deformation Produced by Constant TWisting Moments
1" A;: + (Mxy +Myx)
-A.-(Mtty+My~ "'A~ -(Mxy+ My~
!tA.tIMxy+My~l
FIG, 8 Equivalent Corner Loading
-94~
•
• rI
C!.)R. i pI u 'p=-_1-... ._ 2c
Y zSimple support
....-
I
Ip
I)IL.
!! FIG. 9 Plate Strip of Infinite Width Under Line Loading
•
.. -b b
e"b - e'b-- v ~.
pt}c--.
u
II___00_ ..
o 01
e'+e"=2V" b(l - e')
--.__e._
YY.
FIG. 10 Bridge of Finite Width
-95-
45°-+-- --- --f--JJ--- -- -- -----~- . - ~-
1_ 2c =(8+h) Inches ~
--C'J
E• 0
OJCO
.. Y-o
..c.
..a-C-O)
0II
..c
FIG.11 Assumption for the Longitudinal Distributionof a Wheel Load
3.0
~.o .co~
:::J-0
~ 1.0 E====+===::::::::f~~o
lath = 1.00 I
-0o3 0.0 '-__---l- ..l--__~ ..J..______l_ _.J.._____l. _l
I\00'I
too
Ia Ih = 1.70 I
0.25 0.50 0.150.0
b/L = 0.500.3750.2.50.12.5
-0.50 -0.25-0.75
0.0 '--__-'- ...!.-__--L- ....L-__---l- --L-__--L -J
E -1.00
'0c..~o-J 2.0CJo
FIG.12 DislTibution of Load Applied at Center for oc= a.' . ..."
•~;
Ia/h =1.00 IbtL = 0.500.3750.250.125
2.0
co
...L-
~1.0~~~~~~~f~~i:::'~-1--..::::::r~""""i~~~~~~~o-g 0.0 ~__--'- ---'---__---' ---i- -'--__---'- ---'--__-'
.soL
E~ 2.0 I a Ih =1. 70 I
1.000.750.500.250.0-0.25-0.50-0.75
0.0~__--L.. -!.-__--I'--__-.L- -!--__---'- ---'--__----J
£ -1.00
FIG.13 Dis~ribu~ion of Load Applied at Center for ex: = 0.1
•-
Ia/h =1.00 IbIt = 0.500.3750.250.12.5
2.0
c
i1.0~~=~===:=::~i"....-.:::a'O=~-1----'=r""--=='~~~~~~3Co
..LV)
Q 0.0 '---__--L...,---__--'--__--JI.--__---L- ....l.....-__----1 --'--__-----'
co'-
2ro 2.0-JC&-o
0.500.3750.250.125
Ia/h =1.70 I
I\0OJ
•1.000.150.500.250.0- 0.2.5"-0.50- 0.75
0.0 --L.. --'-- L..--__~ ....L-__---' -.&.-__-----I
E -1.00
FIG. 14 Distribution of Load Applied at Center for 0(= 0.5
1.000.150.500.250.0
bIt =0.500.250.125
-0.50 -0.25-0.75
2.0 -
0.0 L-__-L L-__-l..-__--' --L-__---l -L-__---l
e -1.00
I\0\0I
FIG. 15 Distribution of Load Applied at Center for Isotropie Plate ex = 1.0
6.0
5.0 J 5.0
c c0 0.- :.L:-"- :J::J
. ..0 -0.
-c::: 4.0 Ia/h = 1.70 I Iajh =1.00 I4.0 'C:
..L-...L- V)\n .-Cl Cl
-0b/L = 0.50 b/L=0.50
"U0 00 3.0 3.0 0.....J 0.375 O.~75 . .....J
0 0.25 0.250L 0.125 0.125 '-.E OJ
...L-a 0--J2..0 2.0 --'
u... e.-o 0
-"-....l.- ec~.JOJ0-
u 1.0 1.0 ~u... u-CL.Q)QJ00
u u
0.0 0.0 I......0e -1.00 -0.75 -0.50 -0.25 0.0 0.25 0.50 0.75 1.00 0I
FIG. 16 Distri bution of Load Applied at Edge for ex:;:: 0
5.0
c:o
':.L::3
~4.0..1Ul
a'"0oo :3.0
.-:J
Ialh= 1.70IbIt : O~50
0 ..3750.250.125
.. ,
I alh :: 1.00 1
.. bit = 0.500.3750.2.50,.12.5
, .~
5D. co
..1-
:J...0
4.01:U)
o\j\)
. . 0-3.0--1
0.0 L....-__---I.. --L- ~___..L. ...L_ L..____.L______' O~O
E -1.00 - 0.15 -0.50 -0.2.5 0.0 G.lS 0,50 0.75 1.00t.....o.....J
FJG..."11 Distribution of Load Applied at Edge for 0. :: 0.1
..
c c00
4.0 4.0~:.L: Ia/h = 1.70 I Iajh = 1.00 I::l..0 -0'C:: C-
..L-..L- (J)en --0 b/L = o.so b/L= 0.50 £:)
-0 3.0 0.375 0.375 3.0 -00.u 0.25 0.25 00 -J-J 0.125 0.125
0 0C-c.. (1)
.E 2.0 2.0-00-l ---J
e....- Ci.-a 0
..L- .J-
C1.0
cQJ 1.0 QJ
!J Ut.L-. CJ-CL- CL..OJ (lJ0 0u c..J
0.0 0,0..
C -1.00 -0.75 -0.50 -0.25 0.0 0.2.5 0.50 0.75 1.00
It-'0NI
FIG. 18 Distribution of Load Applied at Edge for ex. =0.5
..
1.000.750.500.250.0-0.25-0.50- 0.75
4.0c:0
:i:::J
..0c..
...L-V) 3,,0a-000~
0 2.0 bit 0.50r.. =OJ....-..0.250
--I 0.125t.&-o
C 1.0OJu.-
c.&-C&-alo
U 0.0 '--__-----I --1.. ---l... --1- -l.-__----'~______l....____---J
e -1.00
FIG.19 Distribution of Load Applied at Edge for Isotropie Plate ex =1.0
I.....oWI
.' •
p:. 47.700 # a~ Midspan
C p. Jj ~(
I I II I
a II I I I I II I
I II I I I I
II I I I I
I I II I, r
I I I II t
- f I I I20 I I I I
0 - I I I r I I~
0 "'- I I I I..J
0 I I~
I - - II I I I
"..../ rI -~
t II ....-r""" I
c I I ....... I oc = O.S" I /' t....... I I /' I-- ........
40 I oc = 0.1 I ./
C I0 OC=O
"/
..£- , /uC1J
, /
- \ I /c.L- \. I /QJ 60 \\
I
0 v" ""- /
" - I ./ ""-.J...--80 - ----Experimen~ar DePlecHons (Cenrerport Bridge)
---Theare~ic.al Deflec~ions (with E= 6.68 •106 psi)
FlG.20 Comparison Between Experimental and TheoreHcal Deflectionsfor Load Applied in Bridge Axis
I.....o.p-I
."
o
'.
p= 41~700# a~ Midspan
[1~fl, f?
--- -~Experiment-al DePlecHons (Cenrerport' Bridge)---Theore~ical DePlecHons (wit-h E=8.68 -10spsi)
20
-a
~ 00~
40c
c0--..L- 60uOJ-u-(J)
C)
80
100
I/"Y
//
/I /I ./L.../
;',./
--J_---
0< = 0.5oc = 0.1DC:: 0
IIIII'-----'
'----I,I---
I~
oVII
FIG.21 Comparison Between Experimental and Theoretical Deflectionsfor Load Applied at Edge
.. •
•.....o0',
981
Experimental Results (based on Deflec~ions)
Theoretical Resul~s for oc = 0 :(based on DePlecHons)(based on Moments)
4 5 6Beam Number
. 32
EoQJ
a::J.c.~ 2.4 t------;----+----+--\-----,
lLJ
~
. ..a 20 t------;----+----+--+------1---.,----r----.,A----~--____r-0
CIJ·C~ 16 r------+----+---------i;;;............-:,,-=---j----=~~~e:.---+---_+_--____t
U
"'0g 12 1-------+-----'-'--'--+-::-4''-----:zlfII'=-t-----'=::::.....-----I------'-==-+--='''''~~-::+----_+_--___I
--ItL- 9.3o B ~~::;;::~=__-~~_~---~---+--~:#---~~:::==........j8.0ID 83'---,W U u
-E 4 t-----+----+----+-----1I-----t-----+-------f------IOJ~~ 0 '--__....1-__--L.-__--.L-__---J~__~__.....L-__--L..__-.t
1
FIG. 22 Comparison Between Experimental and Theoretical Load Distrib\Jt~·v~
. Coefficients for Load at Bridge Axis
..
E}(perimen~al Resulf-s Cbased on DefiecHons1
Theoret-ical Results roroc: 0 :.(based on Deflec~ions)
(based on Momenrs)
6.'35.1
65 s..r -. 5.44.9 4.6
3.5-- .... - 2.9
I
4 5t-'
6 7 8 9 0'-l
Beam NumberI
32o
1
OJQ")
2 8 1-----+-------+---_f------------"""""""=--o7"-...=::::a,-=l--'-'~ .----1-=-----+-----.4
C(]Ju~ 40-
Eccu
a:l
-5 2B tt-2B_.3__-l-__-+-__--J ~__~_____L .l_______J
t3~
..c 24 t----J~--+----+--------l
-0cu
ot 20 ~~~::-I-1-9-.6--+----_____tyC
U
~ 16 ~-----+---~,_____;_--+--_'_-I___--__.__--__r__~______;,L..._,_---.____--____ta-J
c::; 12. I---~--+--~~~~:--+-__.r_r~--------+-
L
FIG. 23 Comparison Between Experimental and TheoreHcal Load DistributionCoefficients for Load at Edge
p= wheel loadall 2' 6'
..
4' 6' 3'
c tee 3'...1
0.:.c 4.0::J..0-c:...L..(f)
0
'-0 3~O00
--J
0£...
Jl2 2.0c.....JC&-o
..L.C 1.0Q).-u
(£:e-OJ0
CJ Q
I2.71 ftI f II l IIf't
I J II It I II II I II f IJ t II I I I I,
InPluence Line Par load DisrribtJ~ion
I CoePPicien~ for' Edge Beam:(cx.=Oj bIL=o.31s j o/h=1r70)
I I I II I J I
I II
f Areas:II A -, 2.73 +1.60 :1 - 6 50[1 - 2. .~ -. .
:. A2. =1.18 1°.93 ·3 =, 3..16'
t A" =.-0.85 +0.69 •~ ;; 2, ~'1'13 ~ ~ ~~,
t A4:;0.58 ~ 0.51 •. 3 = 1.64-
! 13..61I * 13.61 . ,al F'Pr S -= 3.9."10Q: 50.3 100 '
IIIIII
It-'ocoI
FrG.24 Determination of Load Carried by Edge Beam
.. •
1.00.5
l=5~1108
b) Edge Beam
/
L=~/36
a/h =1.70 Bri dge formed of 9. Beams
- L= 27 L/4-';36 108II /
"
I--
a!h=1-00 Bridge formed of 15 Beams
~-0
QJ
t.. 60c-oo
-0oo
...J 40
100 r
EoQJ
a:l
i 80,- r- __ .!.Pr.:sent:-SpecifkaHon~_LU
OJQJ. -c3:
Beams
Beams
C&-
o 20 r--'-CQ,)
Fbridge in n. ~OJ0-
o0.5 1.0 ()( 0 0.1
Coerricien~' of Lateral Bending Stiffness oc0.1
..--
a) Cen~er Beam
f---~__ LPresen~SpeciPi~tion~_
_a/h =1.70 Bridge Pormed oP 9
f-- L=27 l= 54"/36 "!JOB
1/
I--
a/h=1.00 Bridge formed or 15
L= 2.~! L=5~/r- 36 - 108
t = Span 0
.1oOC 0
OJQJ
-Co~
c....0
2.0-'-cOJ\oJc...oJ
a...
E 100oQJ
CD
'QJ
-'-
~ BOC,.)
~-0
Q).-t 60o
U
-0ao~ 40
FIG.25 Percent of Wheel Loads Carried by Center - and Edge Beams in a 21ft. Wide Bridgeof Variable Span
Ito-'o\0I
=110=
NOTATIONS
Roman Alpha.betc~,~"~ _~'._""--__
.i
\',;-
,Ii
•
..
£,,,,
Am~A*m2b
'j\' !iiiBm~Bm,Brn2cCx,CyCx.,CyC~Cm
D,Dx~Dyn DC"'~, ill
eJ "e ,e
EEx
EY
GhHiilm,i4rnifm' i.tmII x
K
Width of the be~us
Constants of integJ:'ationWidth of the bridgeConstants of illtegx:'ationLength of the applied line load (Fig. 10)Spacing of Btringer and beams (Fig. 4)Torsional rigidity of stringers and beams (Fig. 6)Constan.ts of integrationFle:x:ural ri.gidities of isotropic &: orthotropic platesConstants of integrationBase of natural logarithmEccentricity of applied load (Fig 0 10)Modulus of elasticityModulus of elasticity of material parallel to
x=di:r.ectionModulus of elasticity of material parallel to
y-directionShearing modulusDepth of plate or b~ams
Coefficient in equation 1Beam number i or·n beamsFu.nctions defined by equation 58Functions defined by equ.ation 62Moment of i.nertia per:' unit width of pla,teM~ment of inertia per unit width for a, section of
the orthotropic plate perpendicular to the xdire.ctionand with respect to the y=axis
Moment of inertia per unit width for a section ofthe orthotropic plate perpendicular to the ydirection and with respect to the x~axis.
Functions defined by e.quation 58Functions defined by equation 62Modified roots of characteristic equa.tion, defined
by equations 37,460Cross=secti.on factor for torsional rigidity of
rectangular beams. .
Span of bridgeTerm of Fourier=seriesBending moments per unit length of sections of a
plate perpendicular to' x- and y-axes, respectively
-111-
NOTATIONS (continued)
t
•
•
Mxy,Myx
np (x,y)
rRX,RySxy
S~iS.J.m ,S~m
Slm'S4mU,vwWowlx,y,zYm
Twisting moments per unit length of sections ofa plate perpendicular to x- and y-axes,respectively 0
Number of beams in a multi=beam bridgeIntensity of the load at the point (x,y)
parallel to zTotal load uniformly distributed over the length 2cShear forces parallel to z-axis per unit length
of sections of a plate perpendacular to x- andy-axes, respectively 0
Roots of characteristic equations given by equation 36Boupdary shear forces defined by equation 260Coefficient of lateral load distribution defineq
by equation 68Percent 9f load carried by beam iFunctions defined by equation 54Functions defined by equation 63Coordinates of applied line load (Fig. 10)Vertical deflection of plateSolution of homogenous differential equationSolution of infinitely wide plate stripRectangular coordinates .Function of y, equation 33
~.
Greek Alphabet
a Ratio of bending stiffnesses (equation 3)
f3 Coefficient of torsional rigidity (equation 2)
f3x ,f3y Constants in equation 14
f30 Coefficient of torsional rigidity for a = 0
(} Abbreviation defined by equation 32
£ Abbreviation defined by equation 50
q,f"l, Abbreviation defined by equation 50
A Abbreviation defined by equation 50
-112-
NOTATIONS (continued)
~ ~pefficient of static friction
11 Poisson's ratio. Abbreviation defined by equation 3Z!i ' '"
, I
iI
~
/o
A9brevi,ation defined by equation 32
Parameter of torsion
Normal stress
I•
..
i" I1
-c Shear stress
.
•
"
~l13-
LIST OF REFERENCES.
1 0 "Standa.rd Specifications for Highway Bridges"(MSHO Specifications) Washington; the American
o Association of State Highway Officials, 1953
2. A. Roesli, A. Srnislova, C.E.Ekberg and W.J.o Eney"Field Tests on a Prestressed Concrete Multi-Beam Bridge"Lehigh University, Bethlehem, Pa., 1955 (In preparation)'
3. C. Massounett"Contributton au Calcul des Ponts a Poutres Multiples"Bruxelles; Annales des Travaux Publics de Belgique,1950, June, October. and Decernber$ page 377.
4. Y. GuyonIleale.ul des Ponts Larges A Poutres Multiples solidariseespar des Ent:cetoises"paris, Annales des Ponts et Chaussees, 1946, September,October, page 553.
5. C. Massonnet"Methode de Calcul des 'Ponts a poutres Multiples Tenantcompte de leur Resistance a la Torsion"Zurich, Publications, International Association forBridge and Structural Engineering, 1950, Vol.10, pp. 147-182.
6. P.B. Morice and G. LittlelII..oad Distribution in Prestressed Concrete Bridge Systems"London, Structural Engineer, 1954, Vol.32, Nc. 3, Ma+"ch,page 83-111.
7. E. Seydel"Ueber das Ausbeulen VOrl rechteckigen isotropen oderorthogonal anisotropen Platten bei Schubbeanspruchung"Berlin, Ingenieur Archiv, 1933, No.4, page 169.
8. S. Timoshenko"Theory of Plates a.nd Shells"New York, McGraw-Hill, 1940
9 • M.T 0 Huber"Die Theorie der kreuz\veise be~vehrten Eisenbetonplattenebst Anwendungen auf mehrere bautechnisch wichtigeAu£gaben Uber Rechteckplattenl!Berlin, Der Bauingenieur, 1923, page 354
-114-
10. M. T. Huber"Ueber die Biegung ei.ner Rechteckplatte von ung1eicherBiegungssteifigkeit in der L!ings - und Querrichtung"Ber.lin~ Der. Bauingenieur~ 1924~ page 2590
11. Ko rGirkmann"FUichentragwerke"Wi.en~ Springer Ver1.ag~ 1948 s Second Edition
120 S. Ti.moshenko"The Theor~y of Elastici.ty"New York, McGraw=Hill, 1951, Second Edition
13. A. Puche.r"Lehrbuch des Stahlbeton=bauses"Wien , Spri.nger Ver.lag~ 1949~ p.164.
140 A. Nadai"Elasti.sche pLatte.n"Berlin, Springer Verlag~ 1925
15. A. Pucher"Rechteck.p1atten mit zwei eingespannten RHndern"Berlin, Ingenieur Archiv, 1943, No. 14, pp. 246
16. H. Olsen and Fo Reinitzhuber"Die Zweiseitig gelagerte platte"Berlin ll Verlag Wilhelm Ern·st and Son ll 1950, Vol. I and II.
--------~
'.
•..... -... •. - .-.
.... - ~- "7"'"
, 1
/' \t~\tHfUNH!ERSj;'j
JAN ~a 1969A
~,!eJR£~,H.'l ~,,/.~.•---'