Analytical Solution for In-plane Displacements of Curved Beams

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Nov 15, 2013 (4 years and 1 month ago)

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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


Aucillo
,

N.M.
, and

Derosa
,

M.A.

(1994).


F
ree vibrations of circular arches: a
review
.


J
.

Sound
.

Vib
.,

176
(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.
,

125
,

19
-
25.

Li, X. F., and Zhao, Y. H. (
2004
).


Analytical solution on Transverse displacement of
Hyperstatic curved beam.


Proceedings of
The Thirteenth National

Conference on Structural Engineering
, NoII,

513
-
516
.

Ozturk
,
H
.
, Yesilyurt
,
I
.
,

and

Sabuncu
,
M
.

(2006)
,

“In
-
plane stability analysis of
non
-
uniform cross
-
sectioned curved beams
.


J
.

Sound
.

Vib
.
,

296
,

277
-
291
.

Padovani
,

C
.
, Pasquinelli
,

G
.
,
and
Zani
,

N
.

(2
000)
,


A numerical method for solving
equilibrium problems of

no
-
tension solids subjected to thermal loads
.


Comput. Method
.

Appl
.
M.
,

190
,
55
-
73
.

Raveendranath
,

P., Singh
,

G
.
,
and
Pradhan
,

B.

(2000)
,


Free vibration of arches using
a curved beam element b
ased on a coupled polynomial displacement field
.


Comput
.

Struct
.
,
78
,
583
-
590.

Wang
,

T.

M.,
and
Brannen
,

W.

F.

(1982)
,


Natural frequencies for out
-
of
-
plane
vibrations of curved beams on elastic foundations
.


J
.

Sound
.

Vib
.,

84
(2),
241
-
246.

Wu
,
J
.
S
.
,
and

Chiang
,

L
.
K
.

(2003)
,


Out
-
of
-
plane responses of a circular curved
Timoshenko beam due to a moving load
.


Int
.

J
.

Solids
.
Struct
.
,

40
,

7425
-
7448
.

Yao
,

L
. S.
(19
89
)
. “Curved beams
.


China

Communications Press
.
,
Beijing
.

Zhao
,

Y
.Y
.
,
Kang
,

H
. J
.
,
Feng
,

R
.
,
a
nd
Lao W
. Q
.

(2006).

A
dvances of research on
ICTE 2007

1027

curved beams
.

A
dvances in
M
echanics
.
, 36(2)
,
170
-
186
.