ANALYTICAL SOLUTION FOR
IN

PLANE DISPLACEMENT OF MULTI

SPAN
CURVED BRIDGE
X
. F.
Li
1
,
F
.
Li
u
2
and
Y
. H.
Zhao
3
1
Institute of Road and Bridge
Engineering
, Dalian Maritime Univ
ersity
,
116026
,
Dalian, Liaoning, China
;
PH
(
+
86

411)
81127318
;
e

mail:
leorenrb
@
163.com
2
Institute of Road and Bridge
Engineering
, Dalian Maritime
Univ
ersity
,
116026
,
Dalian, Liaoning, China
;
PH
(
+
86

411)8472
4285;
e

mail:
vwff
@
hotmail
.
com
3
Institute o
f Road and Bridge
Engineering
, Dalian Maritime
Univ
ersity
,
116026
,
Dalian, Liaoning, China
;
PH
(
+
86

411)
84725602
;
e

mail:
yhzhao@
yahoo.com
Abstract
:
Based on the theory of virtual work and principle of thermal
ela
sticity
,
exact
solutions
for in

plane displacements of curved beams
with pinned

pinned
ends
are derived explicitly
.
I
n the case of infinite limit of radius
,
these equations coincide
with that of the straight beams
.
C
ompared with the results of FEM, t
he
ana
lytical
solutions
by the proposed
formulae
are accurate.
A real multi

span curved bridge
subjected to concentrated loads
caused by the friction force on the top of piers
and
thermo load
due to
temperature difference
is analyzed by using the newly derived
e
quations as well as FEM
.
The agreement
further suggest
s
the
practicability
of the
proposed theory
.
The
analytical solutions obtained in this paper would provide a
scientific base for further study and design of the curved
bridge
s
.
Key
w
ords:
curved beam
,
c
urved bridge
,
displacement
,
point load
,
thermo load
1
Introduction
Curved beam
, as a common structure form, is
widely used
in many fields,
such
as
civil
and mechanical
engineering.
A
bundance of research references
can be found
on mechanical behavior of cu
rved beams
(
Aucillo
,
1994
,
Zhao
et al
, 2006,
and
Yao
,
1989
)
, especially the out

of

plane behaviors.
Natural frequenc
y
for out

of

plane
vibrations of curved beams
was s
tudies by
Wang
et al (1982
)
.
Wu
(2003)
and
Howson
(1999)
work
ed on o
ut

of

plane responses
of curved Timoshenko beam
.
As
for the in

plane
mechanical analysis
, more are related to the finite element results
(
Raveendranath
et al
200
0,
Ozturk
et al
2006
)
.
The constitutive equation and the
numerical procedure proposed
by
Padovani
et al
(2000)
have
been
used in a masonry
arch subjected to a
uniform temperature distribution
.
So far few references
concern
the analyt
ica
l solutions of in

plane deformation
of the statically
indeterminate
curved
beams
(
L
i 2004)
.
O
n the other hand, much less research has
oc
curred
for situations
involving effects of curved beam under temperature variation.
T
he aim of this paper is
to
provide
explicit
equations
for
in

plane
displacement
s
of
statically indeterminate
curved beam
s
with pinned

pinned ends
.
The b
eams are
designed
to subject a
concentrated force in radial direction
or
undergo
a temperature
variation
.
A
ll equations
are simplified to the solutions of
correspond
ing
straight
ICTE 2007
10
22
beams
at
the infinite limit of curvature radius
. Numerical solutions are developed
to
compare
th
e present results with the finite element data
. The agree
ments further
suggest
the correctness of the proposed theory.
Also
a
real
curved bridge is analyzed
by using the newly derived equations as well as FEM.
T
he
consistence
of
the
solutions again
indicat
e
s the engineering
significance
of this research.
2
Displacements under C
oncentrated load
s
Figure
1
.
Curved beam model
A schematic curved beam is shown in
Figure
1. Polar coordinate system
(
r
,
)
is
accepted.
Following
the principle of virtual work,
d
isplacement
is generally
expressed as
rd
EI
M
M
rd
EI
M
M
rd
EI
M
M
rd
EA
N
N
rd
EA
N
N
rd
EA
N
N
PR
R
PL
R
PL
L
PR
R
PL
R
PL
L
PL
1
1
0
1
1
1
0
1
0
(
1
)
rd
EI
M
M
rd
EI
M
M
rd
EI
M
M
rd
EA
N
N
rd
EA
N
N
rd
EA
N
N
PR
R
PR
L
PL
L
PR
R
PR
L
PL
L
PR
1
1
0
1
1
1
0
1
in terms of Young
’
s modulus
E
,
moment of inertia of the section about the neutral
axis
I
and
a
rea of cross

section
A
.
N
PL
,N
PR
,M
PL
,M
PR
the ax
ial force
s
and the bending
moment
s
at section
, and
N
1
L
,N
1
R
,M
1
L
,M
1
R
the corresponding force
s
and moment
induced by the
unit load applied in the
primary structure
.
T
he In

place
displacement
of
curved beam
with pinned

pinned ends is obtained
as
0
cos
1
sin
sin
1
cos
cos
sin
sin
cos
cos
sin
sin
2
3
1
2
1
2
3
EI
r
PC
C
C
C
EI
r
EA
r
P
PL
P
r
p
N
p
M
ICTE 2007
1023
cos
1
sin
sin
1
cos
sin
cos
cos
sin
sin
cos
sin
cos
sin
sin
2
3
1
2
2
1
3
EI
r
PC
C
C
C
EI
r
EA
r
P
PR
(
2
)
where
2
2
2
2
2
1
3
sin
2
]
cos
2
[
2
1
cos
cos
cos
2
2
sin
sin
Ar
I
Ar
Ar
I
Ar
Ar
I
C
sin
cos
1
sin
1
2
C
C
(
3
)
Setting
r
to infinity, the
Eq
.
2
is readily reduce to the solution of straight
beam as
)
0
(
)
2
)(
(
6
lim
2
2
a
x
a
x
al
a
l
EIl
Px
t
PL
r
)
(
)
2
)(
(
6
lim
2
2
l
x
a
a
x
xl
x
l
EIl
Pa
t
PR
r
(
4
)
At this point it is instructive to validate the above formulae. Parameters of
curved beam shown in
Figure 2
are calculated by current theory and FEM.
Figure
2.
FE
model of
curved beam
Figure
3
presents the displacement solutions
of
curved beam
which
subject
s
an
concentrated force at
section
4
/
.
F
our
dimension
s of
subtended angle
–
o
o
o
o
120
,
90
,
60
,
30
–
are studie
d and
the horizontal ordinate means
/
.
A quick
glance reveals that the
analytical solution
from current study
is
in
a good agreement
with FE result
in all
curved
beams.
3
Displacements Induced by
T
emperature variation
T
hermal effects
are commonly found in both mechanical and structural systems.
A
s a general rule, thermal effects
in curved beam are much more
important
for the
design of statically indeterminate structures than statically determinate ones.
I
n this
b
a
l
r
P
N
P
m
b
m
a
m
A
m
I
v
m
N
E
1000
,
2
.
0
,
5
.
0
,
1
.
0
,
00208333
.
0
,
18
.
0
,
/
10
0
.
2
2
4
2
10
(
a)
curved beam model (b)
material and
geometrical
properties
1024
ICTE 2007
section, the
analytical
solution
for in

place
displacements
caused by temperature
change is
derived.
T
o investigate the deflections due to
t
emperature variation
, consider an element
of length
ds
cut out from the curved beam.
T
he curved element
ds
will rotate
through an angle
ds
t
and extend an length
ds
t
.
B
as
ed
on the principle of virtual
work,
the
displacement of curved beam
under temperature
variation is written as
ds
M
ds
N
t
t
T
(
5
)
w
here
N
,
M
are respectively the axial force and the bending moment
of
curved
beam
with pinned

pinned
ends
at cross section
induced by the
unit load.
T
he
displacement equation can be obtained from (
5
).
T
r
C
T
3
(
6
)
w
here
2
2
2
2
2
3
3
sin
2
]
cos
2
[
2
]
1
cos
cos
[cos
2
2
]
sin
sin
[
Ar
I
Ar
Ar
I
Ar
Ar
I
C
(
7
)
a
nd
α
is the factor of thermal expansion,
T
is the uniform
increment
of
temperature
Curved beam model is analyzed
showed in Figure. 2
again
,
by substituting
P
with a temperature load
20
T
.
Figure
4
show
s
the numerical so
lutions of
displacements
.
0.0
0.2
0.4
0.6
0.8
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
90
o
120
o
30
o
Finite element method
displacement (m)
Subtended angle
60
o
Analytic solution
0.0
0.2
0.4
0.6
0.8
1.0
1.0
0.8
0.6
0.4
0.2
0.0
0.2
90
o
120
o
30
o
Finite element method
displacement (m)
Subtended angle
60
o
Analytic solution
W
e can readily find a
tendency
in
comm
on that
the
transversely displacements
increase
with the
decrease
of
curved beam subtended angle.
T
he correctness of the
proposed
equations is shown by the consistency of
analytical solution
curves and the
dots from FEM.
1
N
2
N
1
V
2
V
p
T
1
N
2
N
1
V
2
V
Figure
3.
D
isplacement of curved beam
under concentrated load
Figure
4
.
Displacement of curved beam
under
temperature variation
ICTE 2007
1025
4
A
pplic
ation
s
M
ulti

span curved
beam
is widely used in modern bridge structure
s
.
In view of
the fact that a continuous curved
beam
bridge
(
Figure
5
)
suddenly di
splaced
transversely
after operation for two years,
its
in

plane
displacement are analy
zed
based on the proposed method in this paper
.
And a
FE simulation
(
Figure
6
)
is also
developed for reference.
T
he bridge is simplified as a
plane
curved
beam
with
box
cross section
showed
in
Figure
7
.
The loads
1
P
,
2
P
……
6
P
are centripetal frictional resistance
induced by
rubber pads
on the top
s
of pier
s and
with
position angle
s are
61
,
47
,
38
,
28
,
19
,
5
,
re
spectively
. Data of
geometr
y
and material
properties are provided.
Figure
7.
Analytic
al
model of
curved bridge
In order to
illustrate the accuracy of the present study, finite element model
is
established
,
w
hich
is a 3

D simulating model
with
box beam, pier
s
and
rubber
supports in
between
.
V
alues of transverse displacement
in six piers and mid

span
computed from analytical equation and two FE models are listed in
T
able
1
. This
comparison again suggests a sati
sfactory result.
Table
1
.
Comparison between analytic solutions and FEM of curved bridge
5
Conclusion
s
pier
1
pier
2
pier
3
mid

span
pier
4
p
ier
5
pier
6
Analytical solution
(m)
0
.
016
4
0
.
042
8
0
.
067
4
0
.
0693
0
.
067
4
0
.
0428
0
.
016
4
FEM
solution
(m)
0
.
0164
0
.
0
40
2
0
.
068
4
0
.
0697
0
.
0684
0
.
0
40
2
0
.
0164
pier 1
pier2
pier 3
mid

span
pier 4
pier 5
pier 6
Figure
5
.
C
urved concrete box bridge
Figure
6
.
Finite element model of
cu
rved bridge
(a)
curved bridge model
(
b
)
material and
geometrical
prope
r
ties
r
r
T
1
P
6
P
5
P
4
P
3
P
2
P
.
20
,
102139
,
167800
,
138617
,
138617
,
167800
,
102139
,
225
,
825
.
31
,
12
.
7
,
66
,
00001
.
0
,
18
.
0
,
/
10
794
.
3
6
5
4
3
2
1
4
2
2
10
C
T
N
P
N
P
N
P
N
P
N
P
N
P
m
r
m
I
m
A
v
m
N
E
1026
ICTE 2007
C
onclusions drawn from this study are as follows:
(
1
)
T
wo equations for
in

plane displacement
s
of the curved beam
s
with
pinned

pinned
ends
under concentrated load and temperature variation
are
derived
based on fo
rce
approach of structural analysis
.
(
2
)
Double validations are developed for proving the correctness of the
equations. Firstly, all e
quations are proved to reduce to the solutions of the straight
beams at the infinite value of curvature radius r. Then cur
ved beam model
is
computed both by the current theory and FEM. The comparison of the two kinds of
results shows good agreements.
(3)
Finally the displacement solution for pinned

pinned ends under one point
load is generalized to analyze
a real continuous c
urved bridge. Comparing result
with that of FEM illustrates a satisfactory consistency.
The conclusion implies that
the analytical solutions obtained in this paper
would provide a scientific base for further study and design of the curved beams.
6
Acknowl
edge
The work
was supported by the National Natural Science Foundation of China
,
No.
50578021
Reference
s
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,
N.M.
, and
Derosa
,
M.A.
(1994).
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ree vibrations of circular arches: a
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”
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.
Sound
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Vib
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(4), 433

458
.
Howson
.
W.P.,
and
Jemah
.
A.
K
. (1999)
,
“
Exact out

of

plane natural frequencies of
curved Timoshenko beams
.
”
J
.
E
ng.
M
ech.
,
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,
19

25.
Li, X. F., and Zhao, Y. H. (
2004
).
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Ozturk
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, Yesilyurt
,
I
.
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and
Sabuncu
,
M
.
(2006)
,
“In

plane stability analysis of
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uniform cross

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”
J
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Vib
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,
296
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291
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and
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,
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”
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590.
Wang
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T.
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and
Brannen
,
W.
F.
(1982)
,
“
Natural frequencies for out

of

plane
vibrations of curved beams on elastic foundations
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”
J
.
Sound
.
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Wu
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(2003)
,
“
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”
Int
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J
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,
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7448
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A
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ICTE 2007
1027
curved beams
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”
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