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S I M P L I F I E D T E N S I O N
C A L C U L A T I O N I N
O F
S U S P E N S I O N
C A B L E
B R I D G E S
by
KENNETH
MARVIN
B.A.Sc. The
A
University
THESIS THE
of
SUBMITTED
in
accept
the
B r i t i s h
Columbia,
this
1959
PARTIAL
FULFILLMENT
FOR
DEGREE
THE
OF A P P L I E D
the
required
THE
Eng.)
IN
OF
SCIENCE
Department
. CIVIL
We
(Civil
REQUIREMENTS MASTER
RICHMOND
of
ENGINEERING
thesis
as
conforming
standard
UNIVERSITY
OF
BRITISH
September,
1963
COLUMBIA
to
OF
In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t
of
the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t
freely
a v a i l a b l e for reference
per-
and study.
I f u r t h e r agree
mission for extensive copying of t h i s
t h e s i s for
that
scholarly
purposes may be granted by the Head of my Department or by h i s representativeso
It
i s understood that copying, or p u b l i -
c a t i o n of t h i s t h e s i s for f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n .
Department The U n i v e r s i t y of B r i t i s h Columbia-, Vancouver 8 , Canada.
ii
ABSTRACT
This
thesis
d e t e r m i n a t i o n of the set of
p r e s e n t s a method w h i c h f a c i l i t a t e s
rapid
cable tension i n suspension bridges.
A
of t a b l e s and c u r v e s the
method.
or t r u s s e s
either' hinged at
bridges the
supports
s u p e r p o s i t i o n method i s d i s c u s s e d a n d
of i n f l u e n c e l i n e s f o r c a b l e
sion bridges
is
tension i n non-linear
the
suspen-
demonstrated.
A d e r i v a t i o n of the
suspension bridge equations
i n c l u d e d and v a r i o u s r e f i n e m e n t s
i n the
A computer program to a n a l y s e w r i t t e n as a n a i d i n t h e the
i n the a p p l i c a t i o n
continuous. A modified
use
i n c l u d e d f o r use
The method i s v a l i d f o r s u s p e n s i o n
with stiffening girders or
is
research
m a n u a l method p r o p o s e d .
included along with
Its
theory are
discussed.
suspension bridges
and f o r
is
the purpose
of
was testing
A d e s c r i p t i o n of the program i s
Fortran
listing.
vii
. ACKNOWLEDGEMENTS
The a u t h o r i s i n d e b t e d
t o D r . R. P. H o o l e y f o r t h e
a s s i s t a n c e , guidance and encouragement g i v e n and i n t h e p r e p a r a t i o n
of this
thesis.
g r a t e f u l t o the N a t i o n a l Research Council money a v a i l a b l e f o r a r e s e a r c h Columbia E l e c t r i c
during
Also,
the research
the author i s
o f Canada f o r m a k i n g
a s s i s t a n t s h i p , and t o t h e B r i t i s h
Company f o r t h e d o n a t i o n o f $500 i n t h e f o r m
of a s c h o l a r s h i p .
K. M. R.
September
1 6 ,
±9&3
Vancouver, B r i t i s h
Columbia
iii
TABLE OF CONTENTS Page CHAPTER 1.
INTRODUCTION
1
CHAPTER 2.
THEORY AND REFINEMENTS
5
General Cable Equation Girder Equation S o l u t i o n of E q u a t i o n s E f f e c t of Refinements CHAPTER 3.
COMPUTER PROGRAM
S o l u t i o n of the I n t e g r a t i o n of Program Linkage Input Data f o r F i n a l N o t e s on CHAPTER 4.
( l l ) and ( 3 5 ) i n T h e o r y on A c c u r a c y .
Girder Equation the Cable E q u a t i o n the the
DETERMINATION
Program Computer P r o g r a m
OF H
General S u p e r p o s i t i o n of P a r t i a l L o a d i n g Cases S i n g l e Span Three-Span Bridge w i t h Hinged Supports Three-Span Bridge w i t h Continuous G i r d e r Variable EI CHAPTER 5. APPENDIX 1 .
CONCLUSIONS
5 8 12 19 21 25 27 32 33 3^ 36 37 37 38 39 44 45 50 52
BLOCK DIAGRAM AND FORTRAN L I S T I N G FOR COMPUTER PROGRAM
55
APPENDIX 2 .
TABLES OF CONSTANTS
60
APPENDIX 3.
NUMERICAL EXAMPLES OF CALCULATION OF H
68
BIBLIOGRAPHY
8l
iv
TABLE OF SYMBOLS
Geometry L
=
Length of
span
B
=
Difference
x
=
A b s c i s s a of u n d e f l e c t e d
cable
y
=
Ordinate
c a b l e measured
i n e l e v a t i o n of c a b l e
of u n d e f l e c t e d
ing undeflected dx
=
Increment
in x
dy
=
Increment
in y
ds
= .Incremental
L
T
=
I f 1^-1 Jo
L
e
=
IJ
d
^|j
v efle= D c t i oVnes r t i c a l
3
d
x
x
corresponding
join-
t o dx a n d dy
f o r a l l spans
f o r a l l spans
d e f l e c t i o n o f c a b l e and
h
=
H o r i z o n t a l d e f l e c t i o n of
h&
-
H o r i z o n t a l d e f l e c t i o n of l e f t
.hg
='
H o r i z o n t a l d e f l e c t i o n of r i g h t
A
=
E q u i v a l e n t support (includes effect.of cable)
from chord
supports
l e n g t h of cable
Wj
L
cable
supports
girder
cable cable
displacement
cable for
support support inextensible
t e m p e r a t u r e and s t r e s s
cable
e l o n g a t i o n of
V
Forces w
=
U n i f o r m l y d i s t r i b u t e d dead l o a d of
p
=
Distributed live
q
=
Distributed load equivalent
=
G i r d e r support
r e a c t i o n at
left
Rj3
=
G i r d e r support
r e a c t i o n at
r i g h t end o f
H
=
T o t a l h o r i z o n t a l component
Hp
=
H o r i z o n t a l component
o f dead l o a d c a b l e
H-^
=
H o r i z o n t a l component
of cable
l o a d on b r i d g e
t e m p e r a t u r e change H ' L
=
bridge
to suspender
of cable
of cable
span span
tension tension
t e n s i o n due t o l i v e
and s u p p o r t
H o r i z o n t a l component
end o f
forces
load,
displacement t e n s i o n due t o
on e q u i v a l e n t b r i d g e w i t h i n e x t e n s i b l e
liwe
load
c a b l e and i m m o v a b l e
supports 8H
=
C o r r e c t i o n t o H-^' t o a c c o u n t support
for
e x t e n s i o n of cable
movement
B e n d i n g Moments =
B e n d i n g moment i n g i r d e r
=
B e n d i n g moment i n g i r d e r a t
left
Mg
=
B e n d i n g moment i n g i r d e r a t
right
M'
=
B e n d i n g moment i n e q u i v a l e n t g i r d e r w i t h no
M'i
E l a s t i c and T h e r m a l
support support cable
Properties
6
=
C o e f f i c i e n t of t h e r m a l expansion f o r
t
=
Temperature
A.
=
C r o s s - s e c t i o n a l area
E
=
Y o u n g ' s Modulus
I
=
Moment o f i n e r t i a o f
m
=
C o e f f i c i e n t of shear d i s t o r t i o n f o r g i r d e r or
cable
rise of
cable
girder truss
and
vi A
=
C r o s s - s e c t i o n a l area
G
=
Shear
A^
=
C r o s s - s e c t i o n a l area
0
=
A n g l e measured
w
of g i r d e r
web
modulus of t r u s s
from t r u s s
diagonal(s)
v e r t i c a l to
diagonal(s)
equations
approximating
Computer Program a-A
=
C o e f f i c i e n t s of d i f f e r e n c e
girder
equation D
F
P
f o r D e f l e c t i o n Theory s o l u t i o n
=1
h
s
=
0 f o r E l a s t i c Theory s o l u t i o n
=
1 to i n c l u d e e f f e c t
=
0 to d e l e t e
effect
= " 1 t o i n c l u d e change 0. t o d e l e t e
=
effect
of h o r i z o n t a l d e f l e c t i o n of h o r i z o n t a l d e f l e c t i o n i n cable
slope
of c a b l e
slope
i n cable
equation
change
Miscellaneous
a
=
E_ E II E R a t i o of side
b
=
f sf
V
L
7
EI EI
span l e n g t h ' L
g
to main span l e n g t h L
S I M P L I F I E D CALCULATION OF CABLE TENSION I N SUSPENSION BRIDGES
. CHAPTER 1 INTRODUCTION
This new t h o u g h t s subject
thesis
adds a f e w new w o r d s ,
to an a r e a
o f s t u d y w h i c h has
of a c o n s i d e r a b l e
analysis
amount
of s u s p e n s i o n b r i d g e s
from the u s u a l ' p r o b l e m s a n d somewhat differences
and t h e
i s a p r o b l e m somewhat by the
to s o l v e .
d i f f i c u l t i e s that
It
the
structural
engineer
i s because of
the
been
i n v o l v e d and t o
d i f f i c u l t i e s i n a n a l y s i n g and d e s i g n i n g
The
different
so much w o r k has
b o t h t o e x p l o r e e x t e n s i v e l y the problems come t h e
a l r e a d y been
o f s t u d y and l i t e r a t u r e .
encountered
more d i f f i c u l t
and p e r h a p s a f e w
done
over-
suspension
bridges. The p r o b l e m i n a n a l y s i s result
of t h e i r r e l a t i v e f l e x i b i l i t y
to d e f l e c t in
the
of s u s p e n s i o n b r i d g e s
i n s u c h a manner as
stiffening girder.
and t h e i r
to m i n i m i z e the
D o u b l i n g the
tures.
suspension bridges
are
desirable bending
the b e n d i n g
s a i d t o be n o n - l i n e a r
stresses sus-
moments. struc-
That i s , , t h e r e i s a n o n - l i n e a r r e l a t i o n s h i p between 1
a
ability
load a p p l i e d to a
p e n s i o n b r i d g e does n o t n e c e s s a r i l y d o u b l e Therefore,
Is
load
2
and r e s u l t a n t is
that
A direct
s u p e r p o s i t i o n of r e s u l t s
methods able
stresses.
of a n a l y s i s
i n the
result
of t h i s
of p a r t i a l
non-linearity
loadings,
d e p e n d e n t on s u p e r p o s i t i o n a r e
analysis
of suspension b r i d g e s .
It
and
not
applic-
w i l l be
shown
h e r e t h a t a m o d i f i e d s u p e r p o s i t i o n method c a n be a d a p t e d solution
of s u s p e n s i o n b r i d g e Investigation
by t w o . o b j e c t i v e s developed.
least
theory
lems,
so i n the
exact
theory
of a n a l y s i s .
analysis
been the
solutions
Theory, which takes account
vior
of s u s p e n s i o n b r i d g e s .
been
the
simplification
been
development
of the
use
of suspension b r i d g e
theory
l a b o r r e q u i r e d f o r a n a l y s i s and d e s i g n . the
a l i n e a r r e l a t i o n s h i p between
the
depending
on t h e
give different flexibility
i n order
. Chapter 2 i s devoted t i o n Theory or forms accepted in
the
standard
field
of i t .
in
geo-'
under
and t h e
As m i g h t be
results
usual
expected,
which can v a r y
widely
bridge.
to a development
of the
Deflec-.
T h e r e seems t o be no u n i v e r s a l l y
D e f l e c t i o n Theory.
favors
to
A result
changes
l o a d and s t r e s s
of the
has
Thus', t h e E l a s t i c T h e o r y
o f s u p e r p o s i t i o n c a n be u s e d .
two t h e o r i e s
the
beha-
Another g o a l of i n v e s t i g a t o r s
l o a d and t e m p e r a t u r e c h a n g e s .
methods
of
non-linear
metry r e s u l t i n g from d e f l e c t i o n , of a suspension b r i d g e
is
be
However,
c a n be o b t a i n e d b y t h e
been the E l a s t i c T h e o r y , w h i c h i g n o r e s
live
prob-
a completely
t o use f o r d e s i g n p u r p o s e s .
Deflection
has
have
inspired
i m p o s s i b l e t o d e v e l o p and w o u l d
accurate
the
been
As i n most e n g i n e e r i n g
reasonably
reduce
has
of s u s p e n s i o n b r i d g e s ,
is virtually
e x t r e m e l y cumbersome
has
two m a i n t h e o r i e s
One g o a l o f i n v e s t i g a t o r s
of an exact
the
problems.
of s u s p e n s i o n b r i d g e s
and a t
to
Each of the
a slightly different
version.
many
experts
Various
3 refinements
i n the
r a c y of the
calculated results
equations
may t a k e
desired. effect
different
and thus the
on t h e
equations
i s shown.
v e r s i o n o f the
is a quantitative
the
tion
is
D e f l e c t i o n Theory...,
f o r the
s h o u l d be n o t e d
is
shown in-
on a c c u r a c y
or n e g l e c t
t o be f o u n d
following
that
in provide
throughout
this
work c o n s i d e r a loadings.
i s g i v e n h e r e t o t h e more c o m p l e x c o n s i d e r a t i o n s
It
will
be s e e n
of
chapters.
c o n d i t i o n s and s t a t i c
d y n a m i c l o a d i n g s on s u s p e n s i o n
solution
Also
effects
of i n c l u s i o n
. No new t h e o r y
confined to s t a t i c
attention
the
development has been i n c l u d e d h e r e t o
a framework of r e f e r e n c e It
Theory
accuracy
d i s c u s s e d and
i n d i c a t i o n of the
refinements.
Chapter 2 but
are
accu-
The E l a s t i c T h e o r y i s
w h i c h m i g h t be e x p e c t e d as a r e s u l t some o f t h e
Deflection
f o r m s d e p e n d i n g on t h e
Some o f t h e s e r e f i n e m e n t s
as a s i m p l i f i e d cluded
t h e o r y may be i n c l u d e d t o i m p r o v e t h e
No of
bridges.
i n development
of the
theory
of a s u s p e n s i o n b r i d g e p r o b l e m i n v o l v e s the
that
simultaneous
s o l u t i o n o f a d i f f e r e n t i a l e q u a t i o n and a n i n t e g r a l
equation.
In the
necessary
more g e n e r a l and more e x a c t
to r e s o r t
t o n u m e r i c a l methods
equations.
The s i m u l t a n e o u s
t r y method.
solutions,
f o r the
s o l u t i o n of each of
these
and
H e n c e , s o l u t i o n o f a n u m e r i c a l e x a m p l e c a n become
Fortunately, i s no l o n g e r n e c e s s a r y
procedure
because of the
on a c o m p u t e r .
was w r i t t e n f o r t h e
by hand c a l c u l a t i o n s .
existence
of computers
t o p e r f o r m a l l c a l c u l a t i o n s by hand.
The p r o b l e m o f s u s p e n s i o n b r i d g e a n a l y s i s
investigate
is
s o l u t i o n i s found by a c u t
a n e x t r e m e l y l e n g t h y and t e d i o u s
solution
it
is well
Chapter 3 describes
I B M 1620
digital
suited
a program which
computer I n o r d e r
suspension bridge a n a l y s i s .
It
for
to
is believed
that
it
4 the
methods e m p l o y e d I n t h e p r o g r a m a r e
analysis.
For that reason,
a listing
has
been i n c l u d e d i n the hopes t h a t
the
preparation
of s u s p e n s i o n b r i d g e
The k e y t o a s i m p l i f i e d problems
is a rapid determination
tension.
it
I n the
more e x a c t
a c u t and t r y m e t h o d .
well
suited for
of the
Fortran
may s e r v e
the
of the
v a l u e o f the
cable
dimensionless examples
the
ratios.
use
These are
i l l u s t r a t i n g the
the
total
value
t o be d e t e r m i n e d a n d i s quired
to i n i t i a t e
therefore
calculations.
or curves
bridges
with continuous
either
hinged at
the
supports.
of c a b l e
tension the
Use o f
relating
certain
included, along with numerical
which i s
method.
The method
shown t o be
valid,
Since H is
the
unknown, an e s t i m a t e The i n i t i a l
improved by a r a p i d l y c o n v e r g i n g i t e r a t i v e of H .
believed
method i s d e v e l o p e d .
of H i s known.
accurate value
cable
a method,
a p p l i c a t i o n of the
employs a form of s u p e r p o s i t i o n , providing
bridge
The p r i n c i p l e s u p o n w h i c h
of t a b l e s
in
t e n s i o n i s found by
Chapter 4 describes
shown a n d t h e
method r e q u i r e s
as a g u i d e
programs.
t o be new, w h e r e b y H , t h e h o r i z o n t a l component
method depends a r e
program
s o l u t i o n to suspension
methods,
c a n be f o u n d e x t r e m e l y q u i c k l y .
computer
value
is.re-
e s t i m a t e of H i s
procedure
The method may be a p p l i e d t o stiffening girders
to g i v e an suspension
or w i t h
girders
5
CHAPTER 2 THEORY AND REFINEMENTS
General The f o l l o w i n g of
a loaded g i r d e r ,
suspended is
concerned with
or e q u i v a l e n t plane
truss,
the
tower tops or anchorages.
simplifying
assumptions
are
I n the
The s u s p e n d e r s a r e
Inextensible.
2.
The s u s p e n d e r s a r e
so c l o s e
together
may be r e p l a c e d b y a c o n t i n u o u s The d e a d the 4.
is
dead l o a d a l o n e ,
cable
is
Less exact
u s e u s u a l l y make t h e
they
along
are
forms
following
usual for
be
neglected.
no b e n d i n g
action moment.
and
hence
the
so-called
of suspension
bridge
D e f l e c t i o n T h e o r y i n common
additional
s m a l l compared w i t h
the
f o r each span,
theory
of the
under
parabolic.
The h o r i z o n t a l d e f l e c t i o n s
can
the
fastening.
straight
constant
" D e f l e c t i o n T h e o r y " o r more e x a c t
6.
that
is distributed
and c a r r i e s
initially
The above a s s u m p t i o n s
analysis.
bridge
initially
The d e a d l o a d i s the
analysis,
girders.
The g i r d e r of
5.
l o a d of the
which
made:
1.
3.
case
o f known r i g i d i t y ,
by v e r t i c a l suspenders from a p e r f e c t l y c a b l e ,
anchored at
following
derivation is
the
assumptions: of the
cable are
very
vertical deflections,
and
6
7.
Deflections of the cable are very small compared with cable ordinates, and their effect on cable slope can be neglected in calculation of cable extension.
8.
Shear deflections in the girders are very small compared with bending deflection and can be neglected .
Assumptions 6, 1 and 8 may be excluded with l i t t l e difficulty in the derivation, and may even be excluded in an analysis by digital computer.
Therefore, the effects of hori-
zontal deflections, cable slope change, and shear deflection are included here and discussed briefly.
It is'not to be thought
that their inclusion results in a complete theory, but perhaps these are some of the more important refinements which can be made.
Others* have discussed the effect of the above refine-
ments,- and in addition have introduced, or at least mentioned, other refinements such as tower horizontal force, tower shortening cable lock at midspan, effect of loads between hangers, temperature differentials between girder flanges, finite hanger spacing, weight of cable and hangers, variation of horizontal component of cable tension with hanger inclination, and so forth. The Deflection Theory of suspension bridge analysis results in a non-linear relationship between forces and deflections and hence the principle of superposition and methods dependent on superposition are not applicable in the usual manner.
In order to simplify the force-deflection relationship
"Into a linear one, i t is necessary:to make a further simplifying * Reference ( 1 2 )
7 assumption. for
the
It
is
"Elastic
errors
which
it
for
not
the
small
as
to
metry
of
the
ness. the
is
Much
end
to
here
cable
as
relating This
on
the
geo-
moment
arm
of
long
passed
out
has
that
by
been
cable
yield
of
the
of
is
two
at
and
loaded
end
reactions
and
end
moments
applied
support.
In q
a
Rg, to
of
design.
attempts
of
high
results
of
to
i n Were
useful-
simplify
accuracy
with
Theory;
and
i t
is
to
Theory
consists
of
the
a
and
is
to
end
hinged
addition,
equivalent
Is
an
that
d i f f e r e n t i a l and
as
cable
the
tension.
deflections
girder
are
i n i t i a l l y
dead
moments
the
or
girder
the
the
load M^ a n d the is
a
Mg,
with
are
posi-
supported
span w,
equation.
bridge
and
to
to
equation
girder
suspension
equal
referred
deflections
forces
girder
to
f i r s t
cable
loads here
constant
and
the
span
distance a
The
relates
girder
single
cable
with
is
to
distances,
by
girder
and
equation
a
separated
The
in
equations.
referred
1 shows
Both
load
Deflection
deflection
shown.
economy
since
E l a s t i c
Theory
devoted.
second
as
E l a s t i c
results
equation,
equation
the
expended
to
the
solution
The
have
is
tive
distributed
hence
the
force.
would
A l l
the
and
on
so
calculations,
loads.
at
cable
effect
are
Theory
applied
B
negligible
girder
Deflection
Figure
A
and
of
girder
second
a
cable
follows:
lengthiness
thesis
loads.
the
as
basis
satisfy
Theory
the
have
stated
the
to
Solution simultaneous
of
be
is
large
too
energy
this
may
which
that
approaching that
assumption
known
Theory
Deflection
ease
cable
well
are
the
E l a s t i c
It
deflections
the It
further
Theory".
The
9.
this
length live which
L.
load
p,
may
be
result
of
continuity
subject
to
the
suspender
forces.
The
TO
Figure
2.
FOLLOW
PAGE
cable is connected to the girder by vertical suspenders and carries the distributed load q.
The cable is in tension, the
horizontal component of which is constant and is equal to H. At the supports, the vertical components of the cable tension are y ' and Vg'. A
Under the action of live load and temperature
changes, the cable and girder deflect from the positions shown in solid lines to the positions indicated by dashed lines. The cable supports deflect horizontally the distances h^ and hg. The original cable position is given by co-ordinates x and y measured horizontally from A and vertically from the chord joining the undeflected cable supports at A and B.
A point P on
the cable deflects from its initial position to a point P' horizontally a distance h and vertically a distance v. A point Q, on the girder deflects from Its initial position vertically below P to a position Q,' vertically a distance v. Cable Equation Figure 2 shows an elemental length of the cable at point P.
Its undeflected position is shown as a solid line,
while its deflected position is shown as a dashed line.
The
length of the element in the undeflected position is given by (ds)
(dx)
2
•~
(dy
2
+
\
Bdx\
-I
2
J
-•
(
1
>
Under the action of live loads the cable deflects as shown and the length of the same element of cable in the deflected position is given by (ds + Sds)
2
(dx + dh)
/dy
2
+
Bdx
dv\
2
+
...
(2)
Subtracting (l) from (2) and rearranging terms, i t is found that ds S'ds -
1 6ds
fl
—
+
-
dh
—
1
1
dx
dh\
dv /dy
2 dx/
B
1
-
dx Vdx
L
+
dv ]
2 dx.
. . .
(3)
dx dx \ 2 dx Since ^ds and ^h both extremely small compared with unity, dx dx they may be dropped from the terms 1 + i ^ and 1 + 1 ^ 1 . 2 dx 2 dx The term i is generally small compared with - ^ over 2 dx dx li most of the span, but may be significant, especially in very a r e
s
flat cables. ds 6ds dh
Expression ( 3 ) then reduces to dv /dy B 1 dv'
dx
dx \dx
dx
dx
L
2 dx>
The extension of the cable 8ds as caused by temperature expansion and stress is given by 1 8ds
6tds
dx where:
2
ds
dx
(5)
AE dx
6
= coefficient of thermal expansion
t
= temperature rise
. H-^ = change in horizontal component of cable tension due to application of live load, temperature changes, support movement, etc. A = cross-sectional area of cable E
= Young's Modulus for cable material
Again, since fUl is extremely small compared with unity, i t may /
d X
\2
be deleted from- the term ( 1 + ^±\ . Then, the binomial theorem V bracketed dx/ All but the first can be applied to expand the term, <
two terms can be neglected, giving 8 d.s dx
£ tds dx
H
L
ds
AE dx
1
1
/dy
2 Vdx
B' L,
+
dv /dy
B
1
dv\
dx Vdx
L
2 dx/
(6)
10
or, since' ds
1
dx
2
1
/dy B>
2
\dx L,
(7)
then 6ds
etds
dx
dx
H ds
ds
L
+
AE dx dx
dv / dy
B
dx
L
V dx
1 2
dv>
. (8)
dxy
A combination of equation (4) representing cable geometry and equation (8) representing Hooke's Law gives the cable equation as dh dx
et
/ds\
H fas\
2
3
*~
T
+
^dxy
—
AE VdxJ
HL
ds)
AE Vdxi
2
dv / dy
B
dx I dx
L
dv
A
1 2
dx, (9)
The above cable equation may be simplified significantly i f i t is observed that the term [OS- - — + In expression (5) is \dx L dx/ normally less than .2, and £LY_ is generally small compared with dx dy - I L Hence, dv_--j_- ^ y significant in the total expresdx L dx sion and can reasonably be neglected. Since — has already dx s n
o
v e r
been neglected compared with unity in the same expression, this amounts, to neglect of the effect of deflections on cable slope 6ds etds 'dsVbecomes H (5) and expression dx
-
-H T
+
(5a)
dx AE \dx/ When (5a) is combined with ( 4 ) , the simplified cable equation is dh
et /ds\
dx
idxy
2
H +
L
AE
/ds\ \dx>
3
dv (dy dx \dx
B
I d v' L 2 dx,
(9a)
It can be seen that neglect of the change of cable slope is reflected in expression (9) by neglect of the term ^ (4J AE \dxI
11 _J± is usually of the order .001 and — AE Is normally not much larger than 1, so a term of order .001 has
compared with unity.
d x
been neglected compared with 1. argues that i t is negligible.
On this basis, Timoshenko However, i t is not difficult to
see that a given percentage error in one term of expression (9) could be magnified by subtracting that term from another of similar magnitude to give a larger percentage error in ^Jl. Expression (9) can be further simplified i f — — is 2.Mx
neglected compared with
- 5. in expression ( 4 ) . dx L
Then the
cable equation becomes dh et /ds\ H Ids\ 3 dv /dy B\ — = — + dx \dx/ AE \dx/ dx \dx L/ This final expression gives a linear relationship between hori2
T
(
9
b
)
zontal and vertical deflections. It should be noted that the above linear relationship between horizontal and vertical deflections does not imply that the structure is linear.
The cable equation has been reduced
to a linear equation, but a non-linear relationship can and does s t i l l exist between stresses and applied loads. If the cable equation is integrated over the span length and the horizontal displacements of the supports are inserted as constants of integration, the following expression results: h
B
-h
r
L
A
J
r
L
o
Gt /ds\ dx 2
o HL
ds
AE Vdx,
dx
(10)
12 |ds_^ dx
L
or, i f
denoted by L, and
2
i s
L
f _^.^ d
3
d x
is denoted
_^ o \ y
by L. , then Q
h
B "A h
€
t
L
=
C
t
+
L
%
e AE L
H /ds
v
2
L
AE \dx',
dv /dy
B
1
dv
dx \dx
L
2 dx
dx
(li)}
If the change in cable slope can be neglected, then - h
G
n
tL
D may be e i t h e r
theory,
I n a d d i t i o n P ^ may be one o r z e r o t o
one o r
zero,
as p r e v i o u s l y d e s c r i b e d .
Include or d e l e t e
the
effect
. 30 of h o r i z o n t a l d e f l e c t i o n
of the
cable.
The e f f e c t
of
shear
d e f o r m a t i o n may be d e l e t e d - s i m p l y b y a s s u m i n g a z e r o v a l u e m, t h e
shear f l e x i b i l i t y
flexural
rigidity
differences,
as
of the
of the
girder,
In equations
constant E I , derivatives In order equation,
it
is
(44).
I n the
For
case of a g i r d e r
to express (45).
c a s e o f z e r o moment a t
following M
equation,
differential
boundary
in
The c a s e o f z e r o d e f l e c t i o n
at
i n the
form
If
the
*
1 v.
2 v.
"2
.,
-"2
t\
+
(47)
If
the
equal p a r t i a l girder,
involving
it
the
girder
lengths,
Note t h a t the exterior
supports
the
difference
2
1
-2
each of the
"...(49)
dx / i 2
.
the
p o i n t under
on e a c h s i d e for
N - 1 i n t e r i o r points
t o w r i t e an e q u a t i o n
the
i n v o l v e the
f i c t i t i o u s points
Is p o s s i b l e
at
d e f l e c t i o n at
equation
(48)
2
i s d i v i d e d i n t o a f i n i t e number N o f
is. possible
two a d j a c e n t p o i n t s
(31).
V
i , then
1 + DHm \
d
the
( p + w + Hd y \
:
L\
d
a point
-m =
~2
(29a) and
dx
zero at
1 v . ,-,
&
d
the
\
b e n d i n g moment i s becomes
i s made o f
2
/
equation
use
Elm /p + w + Hd y
2
2
a point,
d e r i v e d from equations
d v ( E l ( l + DHm)\ dx
of
conditions
obviously expressed
the
the
...
the
the
finite
= o
Vi
at
may be r e p l a c e d by
to o b t a i n a s o l u t i o n to
i s necessary
supports
D e r i v a t i v e s of E I ,
of E I v a n i s h .
s i m i l a r form to equation the
girder.
for
o f the
p o i n t under
deflections the
(45)
at
the
supports.
The f o u r a d d i t i o n a l
at
consideration.
nearest
the
supports Therefore,
to write N - 1 t y p i c a l i n t e r i o r equations
N + 3 unknown d e f l e c t i o n s .
form
c o n s i d e r a t i o n and
i n t e r i o r points
exterior, to
of the
on
and it
involving
equations
31 required that
a r e p r o v i d e d a c c o r d i n g t o the boundary
I s , z e r o d e f l e c t i o n at. e a c h e x t e r i o r
bending
moment a t e a c h e x t e r i o r
tinuous girder, support. point that be
support.
I n the case
the t y p i c a l i n t e r i o r
point.
An.array f o r a girder
Coefficients
w i t h no i n t e r i o r
there i s a
equation at
d i f f e r e n t from
supports
that
of
the form
equation.
t h i r d and f o u r t h a multiple
The f i r s t
Then, t h e f i r s t equations
determined
determined
t o z e r o by
This process the l a s t
of the
of
subtracting elimination
equation i s
a s i n g l e unknown
deflection
Then, a l l o t h e r d e f l e c t i o n s
by s u b s t i t u t i o n
into preceding
a multiple
non-zero c o e f f i c i e n t s i n the
of the second e q u a t i o n .
which i s r e a d i l y determined.
i s evident
c o e f f i c i e n t of the t h i r d
c a n be r e d u c e d
t o an- e q u a t i o n i n v o l v i n g
quickly
the equations
t o z e r o by s u b t r a c t i n g
o f c o e f f i c i e n t s c a n be c o n t i n u e d u n t i l reduced
b y a n x.
Condensed
of the array.
e q u a t i o n c a n be r e d u c e d
equations.
can
of v a l u e s as they a r e The p r o c e d u r e
described
a b o v e i s known a s t r i a n g u l a r i z a t i o n and b a c k s u b s t i t u t i o n . relatively
simple.and
to
i s indicated
zero are indicated
- A s y s t e m a t i c method o f s o l v i n g
is
con-
of c o e f f i c i e n t s f o r a s e t of equations
Coefficients
be
of a
i s s i m p l y r e p l a c e d by a n e q u a t i o n f o r z e r o d e f l e c t i o n a t
below.
first
s u p p o r t , and z e r o
a t one o r more o f t h e i n t e r i o r p o i n t s
I n that, case,
written
from
conditionsj
f a s t here
since
t h e r e i s a band
It
width
32 of
only f i v e
that
the
non-zero c o e f f i c i e n t s .
values
of the
coefficients
a r r a y remain zero throughout ming,
use
allotted storage
i s made o f t h i s I n the
i n the
shown t o t h e
necessary
computer,
the
of the
solution.
it
to
note
of
the a r r a y full
i s condensed
the
In program-
k n o w l e d g e , a n d no memory s p a c e
to the
is For
form
array.
value of H .
times,
The e q u a t i o n s
i s worthwhile to f a c t o r
c o e f f i c i e n t s a^j
compute and s t o r e
values
v a l u e . of H , the
and b ^ .
out terms Then i t
of constants
once
is
for
(46) f o r a_ ..
t i m e - c o n s u m i n g c a l c u l a t i o n s , e v e n on a
each of the
as
entire
shaded p a r t
to s o l v e the g i r d e r e q u a t i o n . s e v e r a l
Therefore,
new t r i a l
i n the
important
s o l u t i o n of a s u s p e n s i o n b r i d g e problem, i t
e a c h e a c h new t r i a l represent
the
is
computer f o r these z e r o c o e f f i c i e n t s .
right
In
It
computer. involving H in
is possible
ab. ., such that f o r
c o e f f i c i e n t s a^^ a n d bj_ a r e
to each
computed
follows:
a . ,1 = a b .i . i (1 + DHm) I v
a . „ = ab . „ + DHab . .. i2 l3 14 a
i3
=
a b
i5
+
D H a b
i6
a . . = ab._ + D H a b . i4 17 10 a^ - ab^g ( l + DHm)
...
Q
b.
= a b . . , ^ + Hab.-,-, + D H a b . + DHH ab . ^ llO i l l il2 L 113
The
c o m p o s i t i o n of the
1
of
n o
equations
Integration
(4.6)
of the
terms a b . .
T
is
n
obvious from an e x a m i n a t i o n
a n d n e e d n o t be w r i t t e n
here.
Cable Equation
Integration
of the
applying Simpson's Rule,
(50)
c a b l e e q u a t i o n i s performed by
w h i c h may be s t a t e d
as
follows:
33 B
r
f(x)
dx
.
A
E
f. _ + 4 f . + f. , _ i-l i i+i
1=2,4/6
where, of
N
I n the
case
o f the
i n t e g r a t i o n are
the
... ( 5 1 )
dh i s — and the l i m i t s dx span under c o n s i d e r a t i o n .
cable equation,
two ends o f t h e
f
dh
I n the program, dh
H
dx
AE V d x /
L
D
+
/ds\
equation for — dx
G t ( where I i s t h e g i r d e r moment o f i n e r t i a a t m i d -
The a b o v e v a l u e s e a c h o c c u p y t h e
cards
Hinged
EI,
a-:-set o f c a r d s
is
the this read
36 giving
the
v a l u e of the
live
l o a d on t h e
One c a r d i s r e a d f o r e a c h p o i n t anchorage
to the
F6.4
format.
Final
Notes
g i r d e r as a r a t i o
on t h e g i r d e r f r o m . t h e
r i g h t anchorage.
The r a t i o s
are
P. w
left
given i n
I n A p p e n d i x 1 t h e r e i s a c o m p l e t e b l o c k d i a g r a m and a listing tions
of the
for
this
p r o g r a m as thesis.
it
was w r i t t e n a n d u s e d
No c l a i m i s made t h a t
the b e s t
one t h a t c o u l d h a v e b e e n w r i t t e n f o r
taken.
However, i t
did give satisfactory
a c c u r a c y was g o o d and t h e
amount
it
was n o t
ticated
value
the program the
studies
results.
It
In i t s
may be t h a t
it
some v a l u e as a g u i d e
of
s i m i l a r programs,
and f o r
to others
that reason
i n the
will
therefore
to the
i t has
the
the program
form u s u a l l y r e q u i r e d of a l i b r a r y program.
may have
under-
The
p r e s e n t f o r m , and
considered worthwhile to r e v i s e
it
is
of i n f o r m a t i o n y i e l d e d by
p r o g r a m was e n t i r e l y , a d e q u a t e . be o f l i t t l e f u r t h e r
in investiga-
more
sophis-
However, preparation
been
preserved
here. The m a j o r p o r t i o n o f t h e s o l v e the
set
of equations
Running time f o r each t r i a l minutes.
Since three
Deflection
the
girder
equation.
s o l u t i o n was a p p r o x i m a t e l y two were r e q u i r e d f o r a n E l a s t i c
or f i v e
Theory s o l u t i o n ,
approximately fourteen
representing
trials
T h e o r y s o l u t i o n and f o u r
c o m p u t i n g t i m e was t a k e n
the
trials total
to s i x t e e n
were r e q u i r e d f o r
computing time
minutes.
was
a
to
37
CHAPTER 4 DETERMINATION OF H
General It involves Since the
the
the
h a s b e e n shown t h a t a n a l y s i s o f s u s p e n s i o n simultaneous
method o f s o l u t i o n must be a t r i a l
c a l c u l a t i o n s are
determined,, i t deflections, further
of
lengthy.
bending•moments
H.
t o use
( l i b ) and
are
repeated
procedure,
value of H i s
t o compute a l l
i n the g i r d e r .
f o r most p u r p o s e s ,
equations
The e q u a t i o n s
and s h e a r s
i n H and v .
and e r r o r
H o w e v e r , once t h e
i s a s t r a i g h t f o r w a r d matter
b e e n shown t h a t ,
accurate
s o l u t i o n o f two e q u a t i o n s
bridges
it
(4-3) i n t h e
is
It
has
sufficiently
determination
here f o r convenience of
reference. h
B
- h
e tL
A
EId v
Hv
2
t
+
Hy
H
l
L
L
r
dv / d y
B \ dx
dx V dx
L
...
e
j o
AE M'
-
—p
dx^
... (43) It
w i l l be n o t e d
c a b l e due t o t e m p e r a t u r e
i m m e d i a t e l y t h a t e x t e n s i o n of
r i s e and s t r e s s
e l o n g a t i o n has
effect
exactly equivalent
to a s m a l l r e l a t i v e support
If
term A
as
the h-o
B
(lib)
hA A
is defined
6 t L t.
HL T
e
the an
movement.
/ s
AE then
A
may be t h o u g h t
o f as
the
equivalent
support
displacements
38 of
an i n e x t e n s i b l e c a b l e . r
dv / d y
L
Jo
to
B \ dx
dx \ d x
... (54)
L J
A method b a s e d here,
Equation ( l i b ) reduces
whereby a d e s i g n e r
on e q u a t i o n s can determine
(43) a n d
(54) i s
v e r y q u i c k l y the
of H f o r a s i n g l e span or f o r a m u l t i p l e span b r i d g e hinged at
the
supports
Superposition
or
(43) i s a l i n e a r d i f f e r e n t i a l
v a r i o u s r i g h t hand s i d e
permissible
to replace
then equation
Eld^v
H y
dx
2
VQ
=
x
+
The s o l u t i o n t o V
H y
Q
a number
expressions..
H on t h e
b y H Q + H-pHv
either
continuous.
i s p e r m i s s i b l e to superimpose
for
value
of P a r t i a l Loading C o n d i t i o n s
Since equation it
presented
of s o l u t i o n s
to (43)
In p a r t i c u l a r ,
r i g h t hand s i d e
(43) may be
equation,
of the
it
is
equation
written
M' ... (.55)
(55) may be g i v e n b y
V]_
+
where EId v 2
Q
Hv
H y
0
Q
M'
... ( 5 7 )
dx" Hv-
EId v2
dx
n
... ( 5 8 )
£
Then A
may be r e p l a c e d b y . A
0 o A
H y
dv,0
'dy
B \ dx
dx
\ dx
L
/ dy
B \ dx
r L dv o
dx
1
\ dx
+ A ^ i n equation
(54) t o
give
... ( 5 9 )
... (60)
39 The p h y s i c a l s i g n i f i c a n c e o f e q u a t i o n s is
difficult
to describe
since
the
a s u p e r p o s i t i o n of mathematical physical
states.
superposition
Then t h e
l o a d moments by H Q .
Q
deflections
v
The d e f l e c t i o n s
exists
It
and t h e the will
e'qual t o z e r o . from the
A
v ^ and
A c a n
two p a r t i a l
loading
be shown t h a t the
it
of cable
H-^ a r e
still
stretch
functions
value
of the
T h e n , H-^ i s
When
compatisum o f
tension
total
A
result-
inextensible
the p o r t i o n of
be r e m e m b e r e d ,
Is
displacement.
Both H
value of H , but
it will
sensitive
cable
to e r r o r
the and
Q
be
i n an
of H .
i f a p p l i e d i n the
tension. the
the
Span
make u s e
of
it will
of H^ i s not
Since superposition valid
to
i s a d v a n t a g e o u s t o make
and s u p p o r t
shown t h a t d e t e r m i n a t i o n
applied
cases.
a p p l i e d l o a d a c t i n g on a b r i d g e w i t h
effect
force
represented
g i v e n by the
p o r t i o n of c a b l e
t e n s i o n r e s u l t i n g from A , which,
Single
tension
be a t t r i b u t e d
s o l u t i o n f o r v Is
Then H Q i s
of the
and A ]_ t o t a l A , t h e n
0
two
to t h i n k of
by a c o n s t a n t
a result
cable
sum o f
t e n s i o n r e p r e s e n t e d by H ^ .
c a b l e and immovable s u p p o r t s .
estimated
are
Q
M' and a p o r t i o n of the
V Q and V ] _ f o r
ing
and
Q
a n d H ^ t o t a l H and when A
bility
the
H o w e v e r , i t m i g h t be c o n v e n i e n t
a c t i o n o f the p a r t i a l c a b l e H
is r e a l l y only
s o l u t i o n s and n o t
a b r i d g e w i t h movable anchorages r e s t r a i n e d H.
(57) t o (60)
manner
of r e s u l t s outlined,
has
b e e n shown t o
be
it
is permissible
to
of the R e c i p r o c a l Theorem i n d e t e r m i n a t i o n F i g u r e 8 shows, t h e
theorem.
attributable
two c a s e s r e q u i r e d f o r
Case 1 i l l u s t r a t e s
to the p a r t i a l c a b l e
the
deflections
tension H
1 #
This
of
cable
application v-^ and A ]_ corresponds
TO
CASE
I
Figure
8.
FOLLOW
PAGE
4o t o the live
s o l u t i o n of equations
load
2 the
tension
( 5 8 ) and
due t o a u n i t
(60).
load
on t h e
This
(57) and
case of a s i n g l e u n i t
(59) f o r t h e
where H Q i s
the
sum o f H
According
1
In
L
corresponds
and the
to the
case from
to a s o l u t i o n of
equations
l o a d on t h e
dead l o a d
the
span,
tension H ^ .
r e c i p r o c a l theorem,
the
equation of
i n f l u e n c e l i n e f o r H£ i s g i v e n b y
V v
span
c a b l e i s assumed t o be i n e x t e n s i b l e and s u s p e n d e d
immovable s u p p o r t s .
the
Case 2 shows
v-
A H fL
2
x
(61) Equation
(58) c a n be r e a d i l y s o l v e d 2
4
E I (CL)
2
x -
L
L
2 +
,
(CL)'
2((l-e-
C L
)e
(CL)^(e
to give +(e
C x
U i j
-
e
C L
-l)e-
- U i j
C x
)
) (62)
.. ( 6 3 ) Equation v
1
(62) c a n be w r i t t e n i n a s i m p l e r f o r m
H fL
as
2
x
. (64)
EI where v.
4
x \
(CL)'
2
L, When t h e
stitution
+
+
(CL)'
H rE I L
A
1
(CL) (e 2
C L
-e-
e
C L
C x
) (65)
)
e x p r e s s i o n f o r v ^ i s d i f f e r e n t i a t e d a n d sub-
i s made i n e q u a t i o n 2
v
2((i-e-CL)eCx ( CL-i)e-
x
(60), i t
Is found
that
... ( 6 6 )
41 where
64
•1
(CL)'
12
T
±
Then t h e H
L
3
of the
and
is,
t i v e p o s i t i o n on t h e
are
-e"
-e-
C L
C L
(CL+2) ... ( 6 7 )
)
i s g i v e n by
A ^ are
d i m e n s i o n l e s s and a r e
func-
d i m e n s i o n l e s s q u a n t i t y C L , where C i s d e f i n e d by
(63).
influence
C L
line for H £
Note t h a t v
equation
(CL-2)
\
±
tions
C L
(CL) (e
influence
L / v
-
4 + e
line for
tabulated
of course,
span.
a l s o a f u n c t i o n of the
V a l u e s of -
V
l , representing
on a s i n g l e s p a n i n d i m e n s i o n l e s s
i n T a b l e 1 o f A p p e n d i x 2.
shows i n f l u e n c e l i n e s f o r v a l u e s
of
rela-
Also,
the
form
Figure 10 '•
( C L ) = 1 and 2
:
( C L ) = 100. 2
To f a c i l i t a t e d e t e r m i n a t i o n o f H f o r d i s t r i b u t e d l o a d s ,
the
area under
been
plotted
i n the
A-^ t o t h e A
f
n 1
= J o
one o f t h e
x
left
v ~
n
i n f l u e n c e curves
same f i g u r e . of p o i n t x ,
The c u r v e shows t h e
area
where ... ( 6 9 )
A 1
By t h e u s e
a l s o i n c l u d e d l i n T a b l e 1. of the
i s p o s s i b l e to determine
curves
or t a b l e s d e s c r i b e d above
H £ and hence H Q ,
i n an i n e x t e n s i b l e c a b l e w i t h
and s u p p o r t
displacement.
effect
2
Equation
tension that
of c a b l e
it
would Then
stretch
F i g u r e 13 shows a p l o t o f A ^
( C L ) w h i c h c a n be u s e d t o f i n d
Tabulated values are
the
immovable s u p p o r t s .
a c o r r e c t i o n 8H must be a d d e d f o r t h e
against
partial
dx
Tabulated values are
exist
i n F i g u r e 10 h a s
the
c o r r e c t i o n 8H.
a l s o i n c l u d e d i n T a b l e 1.
(53) shows t h a t A
i s a f u n c t i o n of H
the
42 unknown l i v e this
value
load tension.
initially
•
It
is possible
to a v o i d
by r e w r i t i n g (53) i n the
estimating
form
L. e
6H
A = A
(70)
AE
where hB
hA
€ tin
H
L'^e
.., ( 7 1 )
AE The c o r r e c t i o n S r l ' I s ,
SH
i n the
case of a s i n g l e
span
AH. (72)
where
i t h a s b e e n shown i n e q u a t i o n
(66) t h a t
1
H-
A
X
,2 r-L
(73).
—
T A
EI Equations
( 7 3 ) and
( 7 l ) c a n be s u b s t i t u t e d
i n equation
(72) t o
give
• 8H L
A -
8H
C
_
AE_
f
... ( 7 4 )
A~l
2 L
EI Equation
(74) c a n t h e n be s o l v e d f o r :8'H t o
8H
give
A!f L 2
A
1 1
EI In order
+
. (75)
L
-1
AE
to determine
f o r H-^ a n d e s t i m a t e
oE,
the
It
value
i s necessary
t o compute
another
once
o f H t o f i n d TJT^ f r o m F i g u r e 1 3 .
Then 8H and h e n c e H c a n be computed f r o m e q u a t i o n computed v a l u e
A '
o f H does n o t a g r e e w i t h
the
(75).
estimated
c o m p u t a t i o n must be made w i t h a d i f f e r e n t
If
value,
value of
the
43 N u m e r i c a l examples •converges
i n A p p e n d i x 3 show t h a t
i
of H and so a n i t e r a t i v e p r o c e d u r e
the
of H£.
l i n e s are
CL.
This
CL.
A t one e x t r e m e
is
+
At
the
is
so f l e x i b l e
extreme as
c a n be shown t h a t
V_
L 3
x
f 4
L
influence line
estimate
the
determina-
as
zero,
x
extreme the
value of
values
elastic
of
theory
i s g i v e n by ... ( 7 6 )
L CL becomes
infinitely
t o o f f e r no r e s i s t a n c e the
of the
because
4'
3
2 / x
L
other
r e l a t i v e l y independent
as CL a p p r o a c h e s
v a l i d and t h e
8
implied for
i l l u s t r a t e d by a s t u d y of the
x f
Is
on a n i n i t i a l
However, i t e r a t i o n i s u s u a l l y unnecessary
influence
becomes
iteration
rapidly. D e t e r m i n a t i o n o f H£ d e p e n d e d
tion
the
influence line
large,--the
girder
t o d e f l e c t i o n , and
it
equation'is
2' (77)
F i g u r e 9 shows i n f l u e n c e l i n e s f o r H ' f o r
the
extreme
Li
values line
of C L .
F u r t h e r i n v e s t i g a t i o n shows t h a t
ordinates
by the
extreme
intersect.
It
f o r a l l values values except i s apparent
be i n t r o d u c e d b y i n a c c u r a t e
of CL l i e w i t h i n i n the
the
the range
r e g i o n where t h e
t h a t no s i g n i f i c a n t e r r o r
defined
curves i n Hi^ w i l l
value of H . For T-2 ca l l v a l u e s o f CL t h e a r e a u n d e r t h e c u r v e must be — . This 87' p o i n t becomes c l e a r e r when i t i s r e a l i z e d t h a t a u n i f o r m l o a d p covering results pL 8f
the
entire
estimates
influence
of the
s p a n i n t r o d u c e d no g i r d e r b e n d i n g moment and
i n a c a b l e t e n s i o n w i t h a h o r i z o n t a l component e q u a l
to
TO
FOLLOW
.2 CL = 0 —
«
C L = cx -•
/
.2
.4
.6
.8
x
17 X L
CL= 0
CL= o o
.05 .10 .1 5 .20 .25 .30 .35 .40 . .45 .50
.0311 .0613 .0899 .1160 .1392 .1588 .1745 .I860 .1926 .1953
.0356 .0675 .0956 .1200 . 1406 .1575 .1706 .1800 .1856 .1875
Figure
9.
PAGE 43
44 Three-Span Bridge w i t h Hinged It
i s a simple matter to extend
method d e s c r i b e d a b o v e hinges
at
the g i r d e r
solved
to f i n d
a total
a = ratio 2 s
£2
values
of
spans which
with
(60) must
be
is ... ( 7 8 )
span l e n g t h L
to main span l e n g t h L
g
EI
b C L
) of
b
("^ l) s against
(CL) for
selected
2
b. line
f o r H£ i n t h e
main span i s
then
X
v-
L'
(58) a n d
A
shows c u r v e s
The I n f l u e n c e H
Equations
bridge
the
EX -
14(a)
case of a t h r e e - s p a n
A ^ f o r a l l the
of s i d e
( A l l s = "S ! ( Figure
a p p l i c a t i o n of
2a b ( A . ) 1 s
+
L
the
3
EI
b
to the
supports.
1
where:
Supports
(79)
f where X 1
+
2a3b(A
... ( 8 0 )
)
L
- a
The i n f l u e n c e L f
line
f o r H£ i n t h e
s i d e span
Is
v-
(81) 1/s
The s u b s c r i p t
s i n the
term
-
v
l
indicates
that
the
influence
A 1/s
curve f o r main span, X f o r the g
the
side
span i s g e n e r a l l y d i f f e r e n t
due t o t h e d i f f e r e n t v a l u e o f C L . s i d e span Is g i v e n by
from that f o r The
multiplier
the
45
2 a b ( A ]_) x
s
..: 1
A~l
g
(79) a n d the
( A 1^s
b
Figures
1
" > x
... ( 8 2 )
'•
and sag f o r against
t
2a3b(Ai).
+
In equations
X
s
(8l),
L and f a r e
the
main span.
F i g u r e 14(b)
for
values
selected
and X , the
multipliers for
the
of span
shows X and X
of a.
and 3.4(b) be u s e d t o g e t h e r
14(a)
values
It
is
plotted
intended
i n order
influence ^line
g
to
length
that
determine
ordinates
from
F i g u r e 10. I n the shown t h a t
the
. / IL
SH
case
of the
m u l t i p l e span b r i d g e ,
it
c a n be
e q u a t i o n f o r 8H i s g i v e n b y
\ 1 ... ( 8 3 )
A l I
X
As i n t h e
case
equations
( 7 l ) and
f o r 8H t o
give
8H
2
f
L
EI Values
A
s i n g l e span,
s u b s t i t u t i o n c a n be made
1
a
b
from
(73) a n d t h e r e s u l t i n g e q u a t i o n c a n be s o l v e d
(84)
A ,1 , L e — +
X
of
of the
AE
( ^ 1^s
are
tabulated
i n T a b l e 2,
A p p e n d i x 2,
and
A~l Appendix 3 contains
a n u m e r i c a l e x a m p l e s h o w i n g how t h e
o r F i g u r e s 10 a n d 14 c a n be u s e d t o d e t e r m i n e span b r i d g e w i t h h i n g e d s u p p o r t s . procedure
is
the
same as
is
that for a single
Three-Span Bridge with Continuous Up t o t h i s
It
point,
H for a
clear
that
tables three-
the
general
span.
Girder
c o n s i d e r a t i o n has
been
restricted
46 to
s i n g l e and m u l t i p l e span b r i d g e s ' w i t h h i n g e s
The p r o b l e m becomes continuous
at
the
supports,
manner.
Equation
equations
(85) to
Eld^v-L
Hv
somewhat
at
all
more c o m p l i c a t e d i f t h e but
it
supports.
girder
may be s o l v e d i n a s i m i l a r
(43) c a n be r e p l a c e d
by e q u a t i o n
(57) and by
(87) below:
Hjy
1
(85)
dx' Eld
v
Hv
2
MgX
2
dx 2
. (86)
L
E I d,2^ v
HV3
3
M (L-x) 3
dx^ (85) to
l o a d moments action
but
of the
(87) r e p r e s e n t a s i n g l e w i t h end moments
partial
The t h r e e e q u a t i o n s equation
It
i s necessary
exists.
single 1
Equation
span and i t lijfL
on a c o n t i n u o u s
equivalent
span.
to
the
In a n a l y s i s
to e q u a l i z e
( 8 5 ) has
been
c a n be shown t h a t
from
of a
continuous
where
solved for
the
slopes
girder.
single
end s l o p e s
the
the
contin-
case of a
are
/ dv > 1
EI
dx
together are
s p a n w i t h no a p p l i e d
Mg and M^ r e s u l t i n g
t e n s i o n H]_ a c t i n g
(58) f o r a s i n g l e
structure, uity
... ( 8 7 )
L
Equations
dv
is
... ( 8 8 )
\dx
where
4
dv 1
(CL)3
dx / o Equation
4 .+ e
( 8 6 ) c a n be s o l v e d t o
C L
(CL-2)
e
CL
-CL - e
e"
C L
(CL+2) (89)
give
o — v
2
M
2
Ir
EI where
V
2
. (90)
47 x
Vr
(CL) Then the
2
dx
C x
-
e
L " e
C L
-
e~
end s l o p e s
Mg L
dVg
e
_ C x
... ( 9 1 )
C L
c a n be f o u n d
from
dVg
. (92)
E I dx
where /dV \
1
1
CL
CL
1
1
CL
CL
2
•o /dv^ \dx
L
j
e e
q
=
x
(v ) _ 2
L
CL
. (93)
-CL - e +
e
-CL ... ( 9 4 )
~CL ^CL e - e
F r o m symmetry o f t h e (v )
OL
girder,
it
is
clear
that . (95)
x
(96)
... ( 9 7 ) I n the
case of a s y m m e t r i c a l t h r e e - s p a n
l o a d moment,
t h e b e n d i n g moments a t
the
bridge, towers
w i t h no a p p l i e d are
equal.
unknown moment M c a n be f o u n d b y e q u a l i z i n g end s l o p e s towers
at
The the
and i s g i v e n b y ab / dv-
M
H
l
,dx / L s
f dv^ dx
It
to
+
a Vdx j Ls
c a n be shown t h a t
where
the
elastic
... ( 9 8 )
b / dv 2
i n the
theory
case of an e x t r e m e l y s t i f f
is valid,
equation
(98) c a n be
girder reduced
48
H
l
1 + ab:
f
...
3
b
2
a
In
the
(99)
— + —
will
be n e c e s s a r y
shown t h a t of
i t e r a t i v e p r o c e d u r e r e q u i r e d t o compute H ,
it
t o compute M a number o f t i m e s , a n d i t w i l l
i s advantageous
H a n d c o r r e c t b y means
m i n e d f o r e a c h new t r i a l
t o compute M w h i c h i s e
value of H .
Values of K are
these parameters
p u t e d and K i s found from the
b and
(CL) .
i n F i g u r e 16.
tables
or c u r v e s ,
A
s o l u t i o n s to equations
(86) a n d
also
When M i s
com-
M i s - f o u n d from
(87), i t
M .. f L
X
3
i s found
that
... ( 1 0 1 )
EI
A3
K is
... ( 1 0 0 )
Mg f IL X 2
2
deter-
tabulated
g
M = K JVL From the
be
independent
o f a m u l t i p l i e r K w h i c h must be
i n Appendix 2 f o r s e l e c t e d values of a, plotted against
it
3
... ( 1 0 2 )
EI where
X
3
(CL) The to the
4 + e (CL-2) -
4
~A~
2
C L
"3
C L
(103)
-CL
,CL e
t o t a l v a l u e of the
s o l u t i o n of equations
e" (CL+2)
s u p p o r t movement c o r r e s p o n d i n g
(85) to
(87) f o r a l l t h r e e s p a n s
g i v e n by
A
A 2.
H-^f L T
t
...
EI
(10-4)
where T
1
3
1
A t
M
2a b(A ),
+
2A
2ab(A ). 2
2
+
A 1
:
... ( 1 0 5 )
is
49 where M
1 + ab
K
... ( 1 0 6 )
3 b — + — a
2 Values
of
b
( ^ l ^ s are
three-span dix
the
same as
bridge with hinged supports
2 and F i g u r e 1 4 ( a ) .
also
and a r e
A 2 and
V a l u e s , of
b
(CL)
i n F i g u r e 15 f o r
the
case of a
found i n Appen-
^ A 2^s a r e
A
,2
against
those used f o r
plotted
A l
selected
values
of b and
are
i n c l u d e d i n A p p e n d i x 2. Influence
l i n e s f o r H' i n a continuous
suspension
li b r i d g e must be f o u n d b y s u p e r i m p o s i n g two i n f l u e n c e first
is
the
same i n f l u e n c e
second i s a c o r r e c t i o n f o r m a i n s pL a n X t hve I n f l u e n c e 1 2 3 Y
v
A i
The
l i n e used f o r hinged g i r d e r s . continuity.
I n the
case of
The the
l i n e , i s g i v e n by
+ v
+
f
lines.
A 2
+
(107)
A3
where X
1
... ( 1 0 8 )
T Y
2X2
M
(109)
T Curves of 2 v
A~
values are
+
v
3
are
+ A~
2
shown p l o t t e d
i n F i g u r e 12 a n d
tabulated
3
f o u n d i n A p p e n d i x 2. For
the
left
side
span,
the
influence
l i n e f o r H-^ i s
g i v e n by L
X
s /
V
l A
Y
+ ii
sf
^2 \ A 2/
... ( 1 1 0 )
50 where ...
(ill)
T
Y
S
=
Mab
Figure for
( Ao) ^
s
11 shows c u r v e s
selected
values
of 2
of course
but
hand.
a n d A p p e n d i x 2 has
v
of
s i d e span i s opposite
(CL) .
i t e r a t i o n procedure of the
tables
(112)
tabulated
values
The i n f l u e n c e l i n e f o r t h e
2
similar
to t h a t f o r
the
left
side
The n u m e r i c a l e x a m p l e i n A p p e n d i x 3 shows t h a t
use
,
...
f o r d e t e r m i n a t i o n of H converges
o r c u r v e s makes t h e
right
span
the
r a p i d l y and
calculations simple.
V a r i a b l e EI... It (86) and
c a n be s e e n t h a t
(87) a r e g i v e n f o r t h e
r i g i d i t y E I w i t h i n the constants constant
the
tabulated
span i s
is possible
numerically,
for
and t a b u l a t e
similar
to equations
s p e c i a l case constant.
(58),
i n which the
Therefore,
i n A p p e n d i x 2 depends
E I w i t h i n each It
solutions
on t h e
use
t o s o l v e the
equations,
other p a r t i c u l a r v a r i a t i o n s d a t a f o r use
at
least
in girder
in analysis.
rigidity
A suitable
a t y p i c a l mode o f v a r i a t i o n o f
s u c h t h a t most o r a l l s u s p e n s i o n b r i d g e
h a v e ,a s t i f f n e s s
v a r i a t i o n which l i e s w i t h i n a range data.
Analysis
d e t e r m i n e d by i n t e r p o l a t i o n between However,
it
Is
the
assumption of
girder stiffness
of t a b u l a t e d
of
span.
a p p r o a c h . m i g h t be t o d e t e r m i n e
two s e t s
girder
suggested
the that
constants tabulated
girders
d e f i n e d by
might then
be
values.
the a s s u m p t i o n of
constant
51 girder rigidity culations.
i s a reasonable
and d e s i r a b l e
Some n u m e r i c a l e x a m p l e s were s o l v e d u s i n g
computer program d e s c r i b e d
i n the p r e c e d i n g
s p a n b r i d g e was a n a l y s e d as a c o n t i n u o u s at
the
supports.
minimum o f
one f o r hand
The m a i n e p a n g i r d e r
.5 t i m e s
the
maximum o f 1.5 t i m e s
A three-
g i r d e r and w i t h
midspan s t i f f n e s s
the.mid-span
the
chapter.
stiffness at
stiffness
the
towers
at. the
e q u a l to the average
study
that a reasonably accurate assuming an average H-^ e n c o u n t e r e d recommended
value f o r
were l e s s
order
the
of H
girder stiffness.
than 2 per
cent.
Therefore,
to determine
the
average
indicated by
Errors it
in
is
girder
v a l u e be assumed f o r e a c h
value f o r H.
points.
rigidity
c a n be d e t e r m i n e d
t h a t f o r a l l hand c a l c u l a t i o n s , a c o n s t a n t
r i g i d i t y E I e q u a l to the in
value
of the
to a
quarter
girder
The r e s u l t s
hinges
v a r i e d from a
The same b r i d g e was a n a l y s e d a s s u m i n g a c o n s t a n t value.
cal-
span
TO
FOLLOW
PAGE
51
Figure
13
52
CHAPTER 5 CONCLUSIONS
It
i s d o u b t f u l t h a t D e f l e c t i o n Theory a n a l y s i s of
suspension bridges procedure
b y hand c a l c u l a t i o n s w i l l
o r one t h a t
enthusiastically.
solutions
does
structural
engineer w i l l
approach
However, i t has been f o u n d , t h a t
Theory s o l u t i o n s are i n many c a s e s .
the
e v e r be a s i m p l e
too i n a c c u r a t e
Therefore, exist.
It
Elastic
even f o r p r e l i m i n a r y d e s i g n
the need f o r D e f l e c t i o n w o u l d be e x p e d i e n t
Theory
to t u r n the
o v e r t o a computer and a v o i d a l l hand c a l c u l a t i o n s , but not always a p r a c t i c a l procedure. a design, tions,
there w i l l
and i t
w i t h the
ments
that
(43) a n d
that
the
i s hoped t h a t t h e s e w i l l
was a p p a r e n t
the
i n the
simpler Deflection
chapter
Theory represented
Reference
(l)
It
Refine-
by is
equations question-
by f u r t h e r is
of the D e f l e c t i o n
computer a n a l y s i s .
be f o u n d t a b u l a t e d
calculaeasier
on T h e o r y a n d
the a d d i t i o n a l a c c u r a c y a t t a i n e d
f o r use
some h a n d
of h i g h a c c u r a c y .
more r e f i n e d v e r s i o n s
attractive
is
e a r l y stages of
be made somewhat
j u s t i f i e d i n hand c a l c u l a t i o n s , and i t
reserved for
that
here.
( l i b ) gives results
a b l e whether ment i s
a l w a y s be a n e c e s s i t y f o r
methods p r e s e n t e d It
D u r i n g the
task
refine-
recommended T h e o r y be
E q u a t i o n (43) i s r e l a t i v e l y
i n hand c a l c u l a t i o n s s i n c e s o l u t i o n s a r e i n Steinman's
text*
on s u s p e n s i o n
to
bridges.
53 Once t h e
cable
tension for a total
possible
to superimpose
solutions for p a r t i a l
as g i v e n i n S t e i n m a n ' s Equations method p r e s e n t e d
the
of d e t e r m i n i n g inherent
c a n be s e e n
Appendix 3 that the
( l i b ) f o r m the the
i n the
the
total
the
more t h a n
girder presents
p a r t l y p a i d f o r by e f f o r t
theory for
of cable
hinged at
the
but
A
no
C o n t i n u i t y i n any s t r u c t u r e
is
in.analysis.
s t e p t o w a r d an a c c u r a t e ,
simplified
tension.
the
cable
It
i n A p p e n d i x 3 meets t h e the
method
simple
is believed
method d e v e l o p e d i n C h a p t e r 3 a n d i l l u s t r a t e d i n
Therefore,
especially
supports.
some a d d i t i o n a l d i f f i c u l t y
method o f d e t e r m i n i n g
simplicity.
for
calculated
to a p p l y ,
must be a n a c c u r a t e ,
calculations
tension.
i s made
tension
of a n a l y s i n g s u s p e n s i o n b r i d g e s
the
the
calculations given i n
e x t r e m e l y easy
s h o u l d be e x p e c t e d .
The f i r s t
conditions
accuracy.
sample
case of a b r i d g e w i t h g i r d e r s
continuous
value
v a l u e of c a b l e
i n the
method i s
basic
above e q u a t i o n s
method has a r e l a t i v e l y h i g h It
in
and
method g i v e n a n d h e n c e
by t h i s
loading
is
text.
(43)
No a p p r o x i m a t i o n n o t
l o a d i n g c a s e i s known i t
objectives
that
sample
o f a c c u r a c y and
method s h o u l d be u s e f u l as p a r t
of •
a t o t a l method o f a n a l y s i s . One a p p r o a c h
to a s i m p l i f i e d
be a d e t e r m i n a t i o n a n d t a b u l a t i o n factors his
i n a manner
thesis
similar
on n o n - l i n e a r whatever
might
o f i n f o r m a t i o n on a m p l i f i c a t i o n
to t h a t d e s c r i b e d by A . F r a n k l i n
in
arches.
methods
to recognize
method o f a n a l y s i s
are
is
important
in
suspension bridge a n a l y s i s ,
of
suspension bridges.
used
to complete
t h a t methods despite
So l o n g as
the
the a n a l y s i s ,
of s u p e r p o s i t i o n are the
non-linear
total.value
it
valid
behavior
of the
cable
54
tension is known and applied in the equations for the partial loadings, the bending moments and deflections for the partial loading cases may be superimposed to give the total values. The key, then, is the determination of the total cable tension, and a simple, accurate method of determining the cable tension has been presented In this work.
55
APPENDIX 1 COMPUTER PROGRAM
C C C C C C
KEN RICHMOND CIVIL ENGINEERING THESIS PROGRAM TO ANALYSE SUSPENSION BRIDGES AND COMPUTE MAGNIFICATION FACTORS. MAIN LINE PROGRAM
43 1 2 3 4 5
6 7
8 10
9
11 12
14 13 15
DIMENSION E l ( 5 3 ) , P(53) , A B ( 1 3 , 5 3 ) , A ( 5 3 , 5 ) DIMENSION E(53) , B ( 5 3 ) , BM(2,53) READ 1 , KODE , C , R , S FORMAT ( 12 / ( E l 4 . 7 ) ) READ 2 , F , SIDE , RISE , T , SLIP FORMAT ( F 6.4 / F 6 . 4 / F 6 . 4 / ( E 1 4 . 7 ) ) PRINT 3 , KODE , R , S FORMAT(//6H KODE= 14 ,3H R= E14.7 ,3H S= E14.7 / ) PRINT 4 , F , SIDE , RISE FORMAT ( 3H F= F 7 . 4 , 6H SIDE= F11.4 , 6H RISE= F l 1 . 4 / PRINT 5 , T , SLIP FORMAT ( 3H T= El 7.7 , 6H SLIP= El 7.7 ) EIO =1.0 / (8.0 * F * C ) IA= 17.0 - SIDE * 20.0 ID= 37 + 17 - IA IF ( KODE - 10 ) 8 , 8 , 6 KODE = KODE - 10 DO 7 I = IA , ID , 1 E l ( I ) = EIO GO TO 11 DO 9 I = IA, ID, 1 READ 10 , E l ( I ) FORMAT ( F 7 . 4 ) E l ( I ) = EIO * E l ( I ) PRINT 12 FORMAT(/23H I P El / ) DO 13 I = I A , I D , 1 READ 14 , P ( I ) FORMAT ( F 8 . 4 ) PRINT 1 5 , 1 , P ( D , E l ( l ) FORMAT ( 13 , F 9 . 4 , E17.7 ) DF = 0.0 SF = 0.0 GO TO ( 1 6 , 1 7 , 1 8 , 1 9 , 1 6 , 1 7 , 1 8 , 1 9 ), KODE
)
56
16 DF = 1.0 17 SF = 1.0 GO TO 19 18 DF =1.0 19 N2 = 1 GO TO 200 20 D = 0.0 HD = 0.125 / F HL = 0.1 DO 36 N = 1,2 K = 0 21 HT = HL + HD IF (KODE - k ) 22,22,23 22 Hk = 2 I I = IA IE = ID GO TO 400 23 Hk = 1 N2 = 3 GO TO 200 2k ERROR = 0.0 N2 = 2 GO TO 200 25 ERROR = ERROR + SLIP PRINT 26 , HL , ERROR 26 FORMAT ( A H HL= E17.7 , 7H ERROR= El 7.7 ) IF(K-I) 27,28,27 27 HL1 = HL ERR = ERROR HL = HL + .1 * HD K = 1 GO TO 21 28 I F ( A B S ( E R R 0 R ) - 1 . 0 E - 5 ) 3 0 , 3 0 , 2 9 29 DELH = ERROR *(HL - HL1)/(ERROR - ERR) ERR = ERROR HL1 = HL HL = HL - DELH GO TO 21 30 E ( I A - 1 ) = -E(IA+1) DO 31 l = I A , I D , 1 BM(N,I) = ( 400.0*(2.0*E(I)-E(I-1)-E(I+1)))*(1.0+D*HT*SM) 31 BM(N,I) = (BM(N,I)-SM*(P(I)+1.0+HT*D2Y))*EI(I) BM(N,IA) = 0.0 BM(N,ID) = 0.0 IF ( KODE -k ) 3 3 , 3 3 , 3 2 32 BM(N,17) = 0.0 BM(N,37) = 0.0 33 SUM = 0 Q = ID - IA + 1 DO 3k I = IA,ID,1 3k SUM = SUM + E l ( I )
57
35 36 41
39 38 37 42 C C
c
AVG = SUM / Q CDL = HD / AVG CLL = HL / AVG CTOT = HT / A V G PRINT 35 , CDL , CLL , CTOT FORMAT (/ 5H CDL= E l 7.7 , 5H CLL= E l 7 . 7 , 6H CTOT= E l 7.7) D = 1.0 PRINT 41 F0RMAT(/48H I DEFLECTION BM ELASTIC BM PHI DO 37 I = I A , I D , 1 IF ( B M ( 1 , U ) 3 8 , 3 9 , 3 8 PHI = 1.0 GO TO 37 PHI = B M ( 2 , I ) / BM(1,1) PRINT 42 , I , B M ( 2 , I ) , B M ( 1 , I ) , P H I , E ( I ) FORMAT(I3#F14.8,F17.8,F17.8,E20.7 ) GO TO 43
/)
SUBROUTINE 1 100
101 C C C
E I ( I A - I ) = EI(IA+1) EI(ID+1) = E l ( I D - 1 ) SM = S / EIO D2Y = - 8 . 0 * F RAE = R * R / EIO DO 101 I =1 I , I E , 1 DEI = 10.0 * ( E l O + 1 ) EK - 1 ) ) D2EI = 4 0 0 . 0 * ( E l ( 1 - 1 ) - 2, 0 * El (I) + E l ( 1 + 1 ) ) X = I - II X = .05 * X DY = 4 . 0 * F * (AL - 2 . 0 * X ) - BL / AL DS = S Q R ( 1 . 0 + DY * DY ) AB(1, = 1.6E5 * E l ( I ) - 8 . 0 E 3 * DEI AB(3, = - 6 . 4 E 5 * E l ( l ) + 1.6E4 * DEI + 4 0 0 . 0 * D2EI AB(4, = SM*AB(3,D + 400.0*(-1.0-DF*DY*DY)+20.0*DF*DY*D2Y AB(5, = 9 . 6 E 5 * E l ( I ) - 8 0 0 . 0 * D2EI AB(6, = S M * A B ( 5 , D + 8 0 0 . 0 * ( 1 . 0 + DF*DY*DY ) AB(7, = - 6 . 4 E 5 * E l ( I ) - 1.6E4 * DEI + 4 0 0 . 0 * D2EI AB(8, = SM*AB(7,I) + 400.0 * (-1,0-DF*DY*DY)-20.0*DF*DY*D2Y AB(9, = 1.6E5 * E l ( I ) + 8 . 0 E 3 * DEI AB(10,I = ( P ( l ) + 1 . 0 ) * ( 1 . 0 - SM * D2EI) AB(11,1 = D2Y * ( 1 . 0 - SM * D2EI) AB(12,1 = - D F * T * D 2 Y * ( 3 . 0 * D Y * D Y + 1.0 ) = - D F * R A E * D 2 Y * D S * ( 4 . 0 * D Y * D Y + 1.0 ) AB(13,I GO TO 201,202,203),N1 SUBROUTINE 2
200
II = IA IE = 17 AL = SIDE
201
202
C C C
204 203
BL N1 GO II IE AL BL N1 GO II IE AL BL N1 GO GO
= RISE = 1 TO 204 = 17 = 37 = 1.0 = 0.0 = 2 TO 204 = 37 = ID = SIDE = -RISE = 3 TO ( 1 0 0 , 3 0 0 , 4 0 0 ) , N 2 TO ( 2 0 , 2 5 , 2 4 ) , N 2
SUBROUTINE 3 300 DO 301 I = I I , IE , 1 X = I - II X = .05 * X DY = 4 . 0 * F * (AL - 2 . 0 * X ) - BL / AL DS = S Q R ( 1 . 0 + DY * DY) RAE = R * R / EIO IF(I - I I ) 302 , 302 , 303 302 DE = 2 0 . 0 * E ( l + 1 ) GO TO 307 303 IF (I - IE) 305 , 304 , 304 304 DE = - 2 0 . 0 * E ( l - 1 ) GO TO 307 305 DE = 10.0 * ( E ( l + 1 ) - E ( l - 1 ) ) 307 B ( I ) = H L * R A E * S F * D S * D S * ( D Y * D E + . 5 * D E * D E ) - . 5 * D E * D E 301 B ( l ) = D*B(I)+RAE*HL*DS**3+T*DS*DS-DY*DE IS = IE - 2 DO 306 I = I I ,IS , 2 306 ERROR = ERROR +(.05/3.0) * ( B ( l ) + 4 . 0 * B ( l + 1 ) + B ( l + 2 ) ) GO TO ( 2 0 1 , 202 , 2 0 3 ) , N1
C C C
SUBROUTINE 4 400 DO 401 I = I I , IE , 1 A ( l , 1 ) = A B ( 1 , I ) * ( 1.0 + SM*D*HT ) A ( l , 2 ) = AB(3,D + D * HT * A B ( 4 , 1 ) A ( l , 3 ) = AB(5,D + D * HT * A B ( 6 , I ) A ( l , 4 ) = AB(7,D + D * HT * A B ( 8 , 1 ) A ( l , 5 ) = AB(9,D * ( 1 . 0 + SM*D*HT ) 401 B ( l ) = A B ( 1 0 , I ) + H T * A B ( 1 1 , 1 ) + D * H T * A B ( 1 2 , 1 ) + D * H T * H L * A B ( 1 3 , 1 ) IM = II -1 IN = IE +1 B(IM) = - ( P ( l l + 1 ) + 1.0 + HT *D2Y) * ( SM / ( 1 . 0 + HT * SM
oo #
O A
_
,—^
s—s
in
+
o
<
U J
•
<
S
'
—
1— - — .
'—*
—•
>CM
O
*
»—•
+
—>
—
—
X
+
CM
L U
* - * 4
< * N * — CM C O
•
*—
+
—
' — s C A O A
«»
a:
a:
I
I
t—-
.
1
->v. L—U
.—,
< U J — *
—_
*
MD
<
•—•> C Q ' — -
o
—
CM
ctL
5
—+
+
< * —
1—
U J
—
CM
—>
—^
<
1
CO
0
N
L A
K
U J
-5
—
—
'
'
« — L U
o
-4-4 —
»— r—
O
I «• LU L A
—
Cu
0
•
O 0
O
O
^
0
0
-
•O O •O 0 • • O •
•»< < * + O A + I S Z - ^ ^ - J - — >— » — C M ^ .
II
• • • I
I 4 4 I 4 N II
CM
OA < I —
»•»
r- CM II
II II II II II II II II II II II II II II I — +
—
II CO
+
^-.^-s^^^-,^^^^^^->^->^^Z
II OQ
" J
*———* ^—v
—CO
Z O Q ^ ^
OO I - 4 CM
C Q
- 3 w || ||
»'—* L A
-4
I
» w CO
OA
O •CM
— LU
—» l l
^*>_ - ,
CM - 3 - > " J —) 0 A 0 A O A 0 A O A - 4 L A * — CM O A — O A <
C
» Z
l ^ - 0
^
'—- L U
< C — ^ - - Z —
— ' II
—5 ' — II
3
|| || || || ||
» w + CM"—' +
— —
• O OO O O 0 0 O
O O O O ' - ' - ^ ^ J -
+
» C M
k/ft.
f \M r >
< „
2 5 0
.
'Je
2 5 Q
t\
f\
k/ft
' .1
1000'
Given: Length of span L = 1,000 ft.. Sag of cable f = 100 ft.. EI = 1..5 ( 1 0 ) K ft. /girder 8
2
AE of cable = 7 - 0 (lO ) K 5
*L = e
1,082
ft.
*L .= 1,054 f t . t
Dead load w = 1.0 K/ft. Live load p = .4 K/ft. distributed where shown + 25 K at quarter point as shown et
= 3.25 ( 1 0 ~ ) 4
Support displacement hg - h^ = -.5 f t . Formulae given in References (l) and (6)
69
Step 1. H L
Compute
1.0 (IOOO)
8f
8 (100)
1250 K
2
(1000)
2
EI L
wL
2
D
and design constants.
.OO667
2
1.5 ( 1 0 ) 8
1082
e
.0015
AE
7 (10 )
f L
(100)
5
2
EI
2
1.5
(IOOO)
.0667
(IO ) 8
Step 2. Compute H-^ . 1
Here, i t is necessary to estimate H in order to select an influence line. H = H.
Then
D
HL
2
It is sufficiently accurate to use
= 1250 (.00667) = 8.3'
EI H^' is found from the influence lines plotted in Figure 10. The influence line ordinate at the location of the point load is .139 L . is
The area under the curve from .25 to .50
(.0625 - .Ol85)L
.
Therefore
f H'
2 5 (.139) (1000)
L
=
.4 (.0625 - .0185) H
100
(1000)
100
= 35 + 176 = 2 1 1 K
Step 3. Compute A '•• A
' = B *A h
h
" f
t L
H ^e 'L t " "L". AE
='- .5 - 3.25 ( 1 0 = -1.17
ft.
T
- 4
)
€
(1054) - 2 1 1 (.0015)
2
70
Step 4.
Compute SH.
Here, another estimate of H must be made in order to determine . A - ^ .
It is sufficiently accurate to estimate
H =H +H .
Then
1
D
HL
2
L
= ( 1 2 5 0 + 2 1 1 ) .OO667 = 9.75
EI Figure 13 is used to determine A-, for • EI H T
1
found that ^ SH
A
f L 2
= .277.
" EI Step 5.
-1.17 L
E
-59K
. 2 7 7 ' ( . 0 6 6 7 ) + .0015
AE
Compute H.
H = H
D
+ H ' + 8H L
= 1250 + 2 1 1 - 59 = 1402
Step 6.
= 9.75 and i t is
Then 8H can be found from
<
A.}
2
K
Compare the value of H' computed in step 5 with the
value estimated in step 4, and repeat steps 4 and 5 until they are the same. HL
2
= 1402 (.00667)
=9.37
EI From'Figure 13, i ^ = .284 8H
-1.17 .284
-58
K
(.0667) + .0015
H = 1403 K
In the case of a single span, one repetition of steps 4 and 5 should give a sufficiently accurate value of H.
Example 2.
Three-Span Bridge with Hinged Supports
Q 25
k
fA
k/ft
i
1
f\
750'
Given: Main span length L = 1000 f t . Main span sag f = 100 f t . Side span length L„ = 500 f t . EI main span = 5.0 (lO?) K ft. /girder 2
EI side span = 2 . 5 ( 1 0 ? ) K ft. /girder 2
AE of cable = 7 . 5 (lO ) K 5
Side span rise = 112.1 f t . L
e
= 2 l 8 l ft.
L = t
2116
ft.
Dead load 1 K/ft. Live load .4 K/ft. on main span as shown + 25 K on side span where shown et = 3.25 (IO ) -4
Support displacement H
B
-h
A
=0
k/ft
72
Step 1.
Compute H and design constants D
Prom equation ( 3 9 ) H
1.0
D
(1000)
8 L
EI
5.0
a
500
K
(100)
(1000)
2
. 1250
2
.020
2
-(10?) .5
1000
b
500
2
5.0
(io?)
2.5
(10 )
1000
f L 2
(100)
EI
7
(1000) . . 2 0
(lO ) 7
5
Step 2 .
2
.5
Compute H-^'
Estimate H = H = 1250 K D
HL
=
2
1250
(.020)
=25.0
EI Figure 14(a) shows values of ( A i ) plotted against Mil— EI Al for selected values of b. For HL> = 2 5 . 0 and b = . 5 , the EI ordinate '-^^ is . 7 8 0 . Figure 14(b) shows values of "Al the multipliers X and X plotted as abscissae against the ordinate ( A j) for selected values of a. For a =• . 5 - — . • "A~l and ( A i ) = . 7 8 0 , the multipliers are: b
s
,
—
N
s
s
10
s
b
s
x x
"Al
.836
= a
=
.082
The main span influence line area from . 0 0 to . 7 5 is found 2
from Figure 10 to be .IO63 — X. Therefore the contribuf is. tio'n to H-j^ from the main span 1
H<
.1063 ( 1 0 0 0 )
L
(.836) (.4)
2
356 K
100
The influence line ordinate for the point load on the side span is . 1 9 4 1 X .
Therefore, the contribution to H '
G
L
from the side span is H» L
.194 (1000) (.082) ( 2 5 K)
4 K
100
The total H ' = 356 + 4 = 360 K. L
Step 3.
Compute A '.
From equation (71) A' = 0 - 3.25 (IO ) ( 2 1 1 6 ) - 360 (.0029) -4
= -1.72
Step 4.
ft.
Compute 8H.
Estimate H = H + H < = D
HL
2
L
1250 + 360 = l 6 l 0
K
= 1610 (.020) = 3 2 . 2
EI 2
Figure 13 shows "A" plotted against h= ±
. EI
1
2
For
H L
= 32.2,
EI
.126
From equation (75) '8H
-52 K
-1.72 .126 (.20)
.0029
.832
Step 5.
Compute H.
H = 1250 + 360 - 52 = 1558 K
Step 6.
Compare the value of H computed in step 5 with the
value estimated in step 4. convergence. HL
EI
2
= 1558 (.020) = 31.2
Repeat steps 4 and 5 until
74
From Figure 13, .A ^ = .130 6H
-1.72 .130
(.20) +
-50
K
.0029
.833
H = 1250
+ 360 - 50 = 1560
K
Compare the value of H computed at the end of step 6
Step 7.
with the value estimated in step 2.
Repeat steps 2 and 5
to convergence. From Figure 14 X = x
s
H
.833 =
.083
< L
=
.1063 .
(1000)
2
(.833) (.4)
.194
( 1 0 0 0 ) (.083) (25)
+
100 = 354 + 4 = 358
100
K '
H = 1250 + 358 - 50 = 1 5 5 8 K
Step 7 will.seldom produce any significant improvement in the accuracy of H.
For most suspension bridges a is
usually less than .5 and b is usually less than .5.
There-
fore, X
Since
0
is small and not sensitive to changes in H.
X is equal to 1-2 X , i t also is not sensitive to changes • in X.
Example
Continuous
3.
400'
100'
Suspension
Bridge
250' •
750'
<
>
Given: Main
span
length
Main
span
sag
Side
span
length
Main
span
EI
=
1.5
(lO )
K
f t .
2
Side
span
EI
=
7.5
(io?)
K
f t .
2
AE
of
Side
L
e
L
t
cable span
= 2l8l
f
-
L
100
ft.
=
500
a
ft.
1000
8
(105)
ft.
K ft.
112.1
ft.
2116
ft.
Dead
load
=1.0
Live
load
=
=
=
=7.5
rise
L =
K/ft. K/ft.
.4 +
25
K at
distributed main
span
-4
displacement
hg
-
h^ =
shown
quarter
et = 3.25 ( i o ) Support
as
0
point
76
Step
1.
Compute
H
(1000)
D
and d e s i g n (1.0)
2
constants.
1250 K
8 (100) L
(1000)
2
EI L
.00667
2
(IO )
1.5
8
2l8l
e _
AE
(IO )
7.5
5
(100)
f L 2
EI '
(1000)
2
500
.0667
(IO )
1.5
a
.0029
_
8
.5
1000
b
500
2
1000
M
e
Step
1.5
(lO )
7.5
(io )
.1 + ( . 5 )
7
(.5)
f
1.5 + . 5 / . 5
2.
Compute H ^ '
Estimate H L
2
.5
8
.500
= 1 2 5 0 (.OO667) = 8 . 3 3
EI Figure of H
L
1 6 shows K p l o t t e d
a and b .
against
TJT
2
f o r selected values EI . F o r a = . 5 and b = . 5 ( F i g u r e 1 6 ( d ) ) a n d
= 8 . 3 3 , K i s f o u n d t o be
.836.
EI (106) M = .836 (.500)
= .418
From F i g u r e 1 4 , ( A i ) b
From F i g u r e " 1 5 ,
s
=
.623
A2 = -.630
"Al b(~~A ) 2
"Al
s
= -.405
Then f r o m
equation
77
By equation (105) T = 1 + 2 (.5 ) (.623) + .418 2 (-.630) + 2 (.5) (-.405) 3
= 1 + 2
T=
(.078) - .526 - 2 (.085)
.460
The 'multipliers X and Y for the main span influence line ordinates are found from equations (108) and (109) .X
1
2.17
.460
Y
.526 -1.14 .460 ~
The multipliers X and Y for the side span influence line g
g
ordinates are found from equations ( i l l ) and (112) X
s
.078
.170
.460 ~
Y
s
.085 -.185 .460 ~
The main span influence line is made up by superimposing curves from Figure 10 and Figure 12 in accordance with equation
(107).
The contribution to H-^' from the main
span is H< L
2.17 (1000)
.4 (1000)
(.106) + .139 ( 2 5 )
.4 (1000)
(.106) + .142 ( 2 5 )
100 1.14 (1000) 100
H< = L
997 - .524 = 4 7 3
K
The side span influence line is made up by superimposing curves from Figure 10 and Figure 11 in accordance with equation (llO). span is
The contribution to H-^' from the side
H' L
.170 ( 1 0 0 0 )
.4 (.125 - .019)
2
100 • .185 ( 1 0 0 0 )
.4 (.125 - .014)
2
100
H ' = 77 - 82 = -5 K L
The total value of H^' from main and side spans is H < = 473 - 5 = 468 K L
Step 3.
Compute A ' .
Prom equation ( 7 l ) A'
= 0 - 3.25
= -2.03
Step 4.
(IO ) -4
(2116) - 468
(.0029)
ft.
Compute 8H.
Estimate H = 1250 + 468 = 1718 K HL
2
= 1718 (.00667) = 11.-4
EI From Figure 13, "/T^ = .252 By equation (104) 8H
-2.03
-191 K
.252 (.460) (.0667) + .0029'
Step 5.
Compute H.
H = 1250 + 468 - 191.= 1527 K
Step 6.
Compare the value of H computed in step 5 with the
value estimated in step 4 and repeat steps 4 and 5 until convergence. HL
2
= 1527 (.00667) = 1 0 . 1
EI From Figure 13, 8H
= .270 -2.03
.270
(.460) (.0667) + .0029
-181 K
79
H =
1250 + 468 - 181 = 1537 K
Step 7.
Compare the value of H computed at the end of step 6
with the value estimated at the beginning of step 2 and repeat steps 2 to 6 until convergence. HL
= 1537 (.OO667) = 1 0 . 2
2
EI From Figure 16(d), K = . 8 l 2 M = .812 (.500) = ,406
From Figure 14, ( ^ l ^ s = .636 b
From Figure 1 5 , - A 2 = -.632 b("A" ) . = -.420 2
s
T = 1 + 2 ( . 5 ) (.636) +• .403 3
2 (-.632) + 2 (..5) (-.420)
= 1 + 2 ,(.080) - .509 - 2 .(.O85)
T =
.481
X =:
1
2.08
.481 ~
Y
.509
-1.06
.481 ~
X
•• .080
s
.166
.481 ~
Y c
o
H
.085
-.177
.481 ~ 1
L
2.08 (1000)
.4 (1000) (.106) + .139 ( 2 5 )
100 - 1.06 (1000) 100
.4 (1000) (.106) + .142 ( 2 5 )
8o
.166
+
(1000)
2
(.4)
(.111)
2
(.4)
(.111)
100 .177
(1000) 100
= 945 -
A'
487 + 74
= 0 - 3.25 =
-2.05
(IO ) -4
-
69 = 473 (2116)
-
K 473
(.0029)
ft.
From Figure 1 3 "A^ = . 2 7 0 8H
-2.05 .270
(.481)
(.0667)
+
.0029
= -178 K
H =
1250 + 473 -
178 = 1545
K
Here, the improvement in accuracy from 1537-K to
1545
K
represents a relatively large improvement compared with what might usually be expected.
For shorter or more rigid side
spans, the Improvement will be less significant and step 1 might reasonably be omitted In many cases.
81
BIBLIOGRAPHY 1.
Steinman, D. B., A Practical Treatise on Suspension Bridges, .2nd Ed., London, Wiley, 1929.
2.
Pugsley, Sir Alfred Grenville, The Theory of Suspension Bridges, London, Edward Arnold, 1957.
3.
Johnson, Bryan and Tourniere, Theory and Practice of Modern Framed Structures, 1 0 t h Ed., New York, Wiley, 1928.
4.
Priester, G. C., "Application of Trigonometric Series to Cable Stress Analysis in Suspension Bridges", Engineering Research Bulletin No. 12, Ann Arbor, Michigan, University of Michigan, 1929.
5.
Timoshenko, S., "Theory of Suspension Bridges", Journal of the Franklin Institute, Vol.
6.
235
(March
1943).
Timoshenko, S., "The Stiffness of Suspension Bridges", Transactions, A.S.C.E., Vol. 95 (1930).
7.
Steinman, B. D., "A Generalized Deflection Theory for Suspension Bridges", Transactions, A.S.C.E., Vol.
8.
100,
(1935).
Shortridge-Hardesty and Wessman, H. E., "Preliminary design of Suspension Bridges", Transactions, A.S.C.E., Vol. 101 (1936).
9.
Westergaard, H. M., "On the Method of Complementary Energy and its Applications to Structures Stressed Beyond the Proportional Limit, "to Buckling and Vibrations, and to Suspension Bridges'.', Transactions, A.S.C.E., Vol.
107
(1942).
82
10.
Tsien, Ling-Hi, "A Simplified Method of Analysing Suspension Bridges", Transactions, A.S.C.E., Vol.
11.
(1949).
Gavarini,.C., "Considerations on Suspension Bridges", AcierStahl-Steel, Vol.
12.
114
26,
No.
2
(March
196l),
Brussels, Belgium.
Szidarovszky, J., "Corrected Deflection Theory of Suspension Bridges", Proceedings, A.S.C.E., Vol. 86, No. st 11 (Nov. I960).
13.
Szidarovszky, J., "Practical Solution for Stiffened Suspension Bridges of Variable Inertia Moment and its Application to Influence Line Analysis", Acta Technica, Vol. 19, No.
14.
3-4,
Budapest
Heilug, R., "Eine Bernerkung zur Haengebruechentheorie", Der Stahlbau, Vol.
15.
(1958).
26,
No.
2,
Berlin (Feb.
1957).
Muller-Breslau, "Theorie der durcheinen Balken versteiften Kette", Zeitschrift der Arch un Ing, Vereins zu Hannover, 1881.
16.
Rode, H. H.,, "New Deflection Theory", Kgl. Norske Viden skabers Selskabs Skrifter, Oslo, 1930.
17.
Atkinson, R. J. and Southwell, R. V., "On the Problem of Stiffened Suspension Bridges and its Treatment by Relaxation Methods", Journal, Inst. Civ. Engrs., Longon, 1939.
18.
Crosthwaite, C. D., "The Corrected Theory of the Stiffened Suspension Bridge", Journal, Inst. Civ. Engrs., London, 1947.
19.
Crosthwaite, C. D., "Shear Deflections", Publications,. • I.A.B.S., Vol.
20.
12,
Zurich
(1952).
Selberg, A., "Suspension Bridges", Publications, I.A.B.S., Vol.
8,
Zurich
(1947).
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