Thin-Walled Structures 49 (2011) 534–542

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Thin-Walled Structures
journal homepage: www.elsevier.com/locate/tws

Natural frequency for torsional vibration of simply supported steel I-girders
with intermediate bracings
Canh Tuan Nguyen a, Jiho Moon b, Van Nam Le c, Hak-Eun Lee a,n
a

Civil, Environmental & Architectural Engineering, Korea University, 5-1 Anam-dong, Sungbuk-gu, Seoul 136-701, South Korea
Civil & Environmental Engineering, University of Washington, Seattle, WA 98195-2700, USA
c
Bridge & Highway Division, Department of Civil Engineering, Ho Chi Minh City University of Technology, Ho Chi Minh City, Viet Nam
b

a r t i c l e i n f o

abstracts

Article history:
Received 31 May 2010
Received in revised form
19 November 2010
Accepted 1 December 2010
Available online 8 January 2011

Natural frequency is essential information required to perform the dynamic analysis and it is crucial to
thoroughly understand natural frequency in order to study the noise and vibration induced by cars or

trains, which is the major disadvantage of the steel I-girder bridge. In this study, analytical solutions for
the natural frequencies and the required stiffness for torsional vibration of the I-girder with intermediate
bracings are derived. The derived equations have simple closed forms and they can be applied to an
arbitrary number of bracing points. The proposed equations are then verified by comparing them with the
results of finite element analyses. From the results, it is found that the proposed equations provide good
prediction of natural frequency for torsional vibration of the I-girder with intermediate bracings. Finally,
the derived equations are applied to a twin I-girder system as an example of a practical civil engineering
application and a series of parametric studies is conducted to investigate the effects of a number of bracing
points and total torsional stiffness on torsional vibration.
& 2010 Elsevier Ltd. All rights reserved.

Keywords:
Natural frequency
Torsional vibration
Stiffness requirement
Steel I-girder
Torsional bracing

1. Introduction
Natural frequency is important information for the design of the
steel I-girder bridge because a steel I-girder section is composed of
a thin-walled element, and noise and vibration is therefore the
major design consideration of such bridges. Generally, each I-girder
is connected by cross beams or other types of bracing systems, as
shown in Fig. 1, because the strength of the I-girder is considerably
enhanced using intermediate bracings [1]. In this case, these
intermediate bracings are modeled as torsional springs. Thus, it
is crucial to thoroughly understand natural frequency for torsional
vibration in order to analyse such I-girder systems with a good
degree of accuracy.

A considerable number of studies have been conducted on the
torsional stiffness and torsional vibrations of a beam. The torsional
effects on short-span highway bridges were investigated by Meng
and Lui [2]. They reported that the torsional effect should be
considered in the seismic design of bridges. Zhang and Chen [3]
presented a new method for thin-walled beams with constrained
torsional vibration based on the differential equations including
the effect of cross-sectional warping. Eisenberger [4] proposed the
exact solution for the torsional vibration frequencies of the
symmetric variable and an open cross-section bar. Mohri et al. [5]

n

Corresponding author. Tel.: + 82 2 3290 3315; fax: + 82 2 928 5217.
E-mail address: (H.-E. Lee).

0263-8231/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.tws.2010.12.001

derived an equation for torsional natural frequency for simply
supported beams with open sections based on reduced differential
equations. The torsional responses of a composite beam were
investigated by Sapountzakis [6] and Vo et al. [7]. Sapountzakis [6]
presented numerical examples on torsional vibration of composite
bars with arbitrary variable cross-section. Vo et al. [7] extended the
theory of Mohri et al. [5] to the free vibrations of axially loaded thinwalled composite beams. Recently, several studies on the coupled
bending and torsion vibration of a beam were conducted [8–10].
Dokumaci [8] developed a closed form solution for the coupled
bending–torsional vibrations of mono-symmetric beams, neglecting the effect of warping. Bishop et al. [9] extended the theory to
allow the warping of the beam cross section. Banerjee et al. [10]

provided an exact dynamic stiffness matrix of a bending–torsion
coupled beam including warping. However, their studies are
limited to beams without intermediate bracings. This case is not
typical of practical civil engineering practice because the beams (or
girders) are connected to each other by cross beams or other types
of bracing systems and these bracing points can be modeled as
intermediate support with proper lateral or torsional springs.
The lateral vibration behavior of a beam with intermediate
supports was investigated by Albarracin et al. [11] and Wang et al.
[12]. They reported that the natural frequency is significantly
affected by the stiffness of the intermediate spring. Gokdag and
Kopmaz [13] proposed an analysis model for the coupled bending
and torsion vibration of a beam with intermediate bracings.
However, they consider intermediate supports as linear springs


C.T. Nguyen et al. / Thin-Walled Structures 49 (2011) 534–542

Nomenclature
E
G
Ig
Ip
Iw
J
k
L
n
q
R*

Rm
Rn
RT
RFEM
T
T
V
W
x,y,z

Young’s modulus
shear modulus of elasticity
mass moment of inertia about centroidal axis
polar moment about centroidal axis
warping constant
pure torsional constant
order of bracing points
span length of I-girders
number of torsional bracings
time-dependent generalized function
summation of required stiffness Rm
required stiffness that changes the mode shape from
mth to (m+ 1)th
required stiffness that changes the mode shape from
nth to (n+ 1)th
total torsional stiffness requirement from theory
total torsional stiffness requirement from FEM
kinetic energy
potential energy
torsional slenderness

principle coordinate axes

that prevent lateral displacement. This differs from the case where
torsional bracings are described as a cross beam or X-bracing
system as shown in Fig. 1.
The mode shape of an I-girder with intermediate torsional
bracings differs depending on the stiffness and number of bracings.
Fig. 2 shows the mode shape of the I-girder with central torsional
bracings. The total required stiffness RT is defined as the minimum
stiffness of the bracing that acts as a full support. For example,
the combination of the first and second modes occurs when the
torsional stiffness of the bracing R is smaller than RT, while the
second mode is generated when R is larger than RT as shown in Fig. 2.
Thus, stiffness and number of bracings are the major parameters
that affect the natural frequency of torsional vibration.

f

r
o
oFEM
oo
oFEM
o
om
om
on
on
on + 1
c

ck

535

admissible shape function
material density
fundamental natural frequency of braced I-girder
from theory
fundamental natural frequency braced I-girder
from FEM
fundamental natural frequency of unbraced I-girder
from theory
fundamental natural frequency unbraced I-girder
from FEM
natural frequency corresponding to the mth
mode shape
intermediate natural frequency between the mth and
(m+ 1)th mode shapes
natural frequency corresponding to the nth
mode shape
intermediate natural frequency between the nth and
(n +1)th mode shapes
natural frequency corresponding to the (n+ 1)th
mode shape
twisting angle of the cross section of main girder
twisting angle of the cross section of main girder at
bracing points

This study focuses on the natural frequency for the torsional
vibration of an I-girder with intermediate torsional bracings.

I-girders are considered to be simply supported in flexure and
torsion and have doubly symmetric cross sections so that the
natural frequency for bending and torsion can be evaluated
separately. A simple analytical solution for natural frequency
and stiffness requirement for torsional vibration are derived for
an arbitrary number of bracing points. Then, the derived equations
are successfully verified by comparing them with the results of
finite element analysis. In this study, as practical examples of
derived equations, the popular twin I-girder systems with cross
beam, which are adopted based on the actual bridge dimension, are
analyzed, and a series of parametric studies is performed. The main
parameters are the total torsional stiffness of the bracings and the
number of bracing points. Finally, the effects of total torsional
stiffness and number of bracing points are discussed.

2. Natural frequency and stiffness requirement for torsional
vibration of I-girder with intermediate bracings
2.1. Natural frequency
Fig. 1. Types of intermediate torsional bracings.

Fig. 2. Mode shapes of I-girder with central torsional bracing.

The natural frequency for torsional vibration of an I-girder with
intermediate torsional bracings is derived using Lagrange’s equation [14] herein. This method can simply provide an acceptable
solution for the free vibration problem. The solution is derived for
an arbitrary number of torsional bracing points n. Torsion and
warping behaviors are taken into account with the following
assumptions: (a) the deformation of the member is small;
(b) the cross-section distortion is neglected; (c) the material
remains elastic; and (d) the elastic torsional restraints are attached

to the centroidal axis of the I-girder.
The I-girder with an intermediate torsional bracing system is
shown in Fig. 3. The girder is equally spaced by n number of
torsional bracings. The length between bracing points can be
defined as L/(n+ 1), where L is the span length of the girder. The
boundary conditions of the girder are simply supported in flexure
and torsion. A coordinate system is also shown in Fig. 3. The
principle axes x, y, and z represent the in-plane, out-of-plane, and


536

C.T. Nguyen et al. / Thin-Walled Structures 49 (2011) 534–542

Assuming that the time dependence of qj is harmonic, a system
of homogeneous equations, which represents the eigenvalue
problem, can be given as
n
þ2
X

ðkij Ào2 mij Þqj ¼ 0

for i ¼ 1, 2, :::, n þ2

ð8Þ

j¼1

Then, Eq. (8) can be expressed in a matrix form as

ð½KŠÀo2 ½MŠÞfqg ¼ f0g:

Fig. 3. I-girder with intermediate torsional bracings.

longitudinal directions, respectively. The rotation of the cross
section is represented by the twisting angle c, and the torsional
bracings are considered to be elastic rotational restraints represented by torsional stiffness R as shown in Fig. 3.
Since the cross section of the I-girder used in this study has
constant area and is doubly symmetric, the kinetic energy for
torsional motion of the I-girder can be expressed as
Z
1 L _2
T¼
Ig c dz
ð1Þ
2 0
where c is the twisting angle about z-axis; Ig is the mass moment of
inertia about centroidal axis defined as Ig ¼ rIp, where r is the mass
density of the material (M/L3); and Ip is the polar moment about the
centroidal axis.
The potential energy including the warping effect can be given
as
Z
n
1 L
1 X
V¼
ðEIw c00 2 þGJ cu2 Þdzþ R
c2
ð2Þ

2 0
2 k¼1 k
where E is Young’s modulus, G is the shear modulus of elasticity, Iw
is the warping constant, J is the pure torsional constant, and ck is
the twisting angle of the cross section at restrained points. The
beam is equally spaced with n number of torsional bracings so that
the location of the kth torsional bracing can be defined as z¼
(k/(n + 1))L, where (k¼1, 2, y, n).
The function of rotation c with respect to time t can be
expressed as a series of time-dependent generalized functions
qi(t) multiplied by admissible functions fi(z), which satisfy the
following geometric boundary conditions: (a) f(0)¼ f(L) ¼0 and
(b) f00 (0) ¼ f00 (L)¼0. In this study, to guarantee good degree of
accuracy of the solution, admissible function is considered up to the
(n+ 2) mode. Thus, function c can be defined as

cðz,tÞ ¼

n
þ2
X

fi ðzÞqi ðtÞ

ð3Þ

i¼1

where



fi ðzÞ ¼ sin

ipz
L



for i ¼ 1, 2, . . ., n þ2:

ð9Þ

Thus, the natural frequencies for torsional vibration can be
obtained from non-trivial solutions of Eq. (9) and they are given as
ffi
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!
u

2
u p 2 GJ
p
EI
R
w
om ¼ t
1þ
þðn þ 1Þ
Ig
Ig L

L=m
L=m
GJ
for m ¼ 1, 2, 3, . . ., nÀ1 when R r RÃ

ð10aÞ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0
1ffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u

2
u 2
LR
C
RL
u p GJ @
n
2
on ¼ t
An þ Bn W 2 À2Bn ð1 þ 2An W 2 Þ þ
þ 2 A
L Ig
2p2 GJ
p GJ
when RÃ o R rRT

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

!
u

2
uGJ
p
EIw
t
on þ 1 ¼
1þ
L=ðn þ 1Þ Ig
L=ðn þ 1Þ
GJ

p

ð10bÞ

when R 4RT
ð10cÞ

where
An ¼ ðn þ1Þ2 þ 1; Bn ¼ ðn þ 1Þ4 þ 6ðn þ1Þ2 þ 1; Cn ¼ n þ 1

In Eq. (10), om and on represent intermediate natural frequencies with contributions of torsional stiffness, (om r om r om þ 1
and on r on r on þ 1 ), where om, on, and on + 1 represent natural
frequencies corresponding to the mth, nth, and (n + 1)th mode
shapes,
respectively.
Torsional slenderness W is defined as

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
ðp=LÞ EIw =GJ and it represents the effects of warping.
It is noted that Eq. (10a) can only be used when the number of
bracing points n is larger than 1. For the I-girder with a central
torsional bracing (n¼ 1), Eqs. (10b) and (10c) can be used to
calculate the natural frequencies for symmetric and asymmetric
mode shapes, respectively. In this study, Rm and RT are the required
stiffness to change the mode shape from the mth mode to the
(m +1)th mode and the total stiffness requirement that provides full
bracing, respectively (refer Fig. 4). It is also noted that Eq. (10a) is
available when R r RÃ , where R* is the summation of the required
stiffness for the (n+ 1)th mode, and Eq. (10b) can be used to
calculate the natural frequency when RÃ o R rRT . Eq. (10c) can be
used to calculate the natural frequency when R4RT.

ð4Þ

Substituting Eq. (3) in Eqs. (1) and (2) with Lagrange’s equation,
an equation of motion can be obtained as
n
þ2
X

mij q€ j þ

j¼1

n
þ2

X

kij qj ¼ 0

for i ¼ 1, 2, :::, n þ2:

ð5Þ

j¼1

where
Z
mij ¼

L
0

Ig fi fj dz

ð6Þ

and
kij ¼

Z

L
0

ðEIw f00i f00j þ GJfiu fju Þdz þ R


n X
n
X
i¼1j¼1

fi fj

ð11Þ

ð7Þ
Fig. 4. Increments of frequency with increase in torsional stiffness.


C.T. Nguyen et al. / Thin-Walled Structures 49 (2011) 534–542

2.2. Stiffness requirement
The relationship between natural frequencies and stiffness is
shown in Fig. 4. From this figure, it can be seen that the natural
frequencies increase with increase in the stiffness of the restraint.
Each increment of natural frequency from the mth mode to
(m+ 1)th mode requires an amount of torsional stiffness that is
defined as Rm. Thus, the total stiffness that is required to obtain the
(n À 1)th mode shape is defined as RÃ . When R4RT, full bracing is
provided, and the natural frequency is equal to on + 1.
A torsional natural frequency corresponding to an arbitrary mth
mode shape is given as
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!
u


2
p u
EIw
tGJ 1 þ p
ð12Þ
om ¼
L=m Ig
L=m
GJ
The required stiffness Rm can be obtained in increments of
natural frequencies from om to om + 1 and are expressed as


Ig L
ðo2m þ 1 Ào2m Þ
ð13Þ
Rm ¼
n þ1
Thus, the total amount of stiffness RÃ to obtain the nth mode is
simplified as

 X
nÀ1
Ig L
ðo2 Ào2m Þ
ð14Þ
RÃ ¼ SRm ¼
nþ1 m ¼ 1 mþ1
and RÃ can be expressed in terms of o1 and on as follows by the

summation of right term in Eq. (14):


Ig L
ðo2n Ào21 Þ
ð15Þ
RÃ ¼
nþ1
Substituting Eq. (12) when m ¼1 and m¼n in Eq. (15) yields
RÃ ¼

p2 GJ
L

Â
Ã
ðnÀ1Þ 1 þðn2 þ1ÞW 2

ð16Þ

where RÃ is the summation of required stiffness Rm .
From Eqs. (10b) and (10c), the required stiffness Rn, which
changes the mode shape from the nth to (n+ 1)th mode, can be
similarly computed in the last increment of the natural frequency
from on to on + 1, as shown in Fig. 4. Thus, Rn is obtained as
Rn ¼

p2 GJ ðan W 4 þ bn W 2 þ wn Þ
2ðn þ 1Þð1 þ dn W 2 Þ


L

ð17Þ

where

an ¼ 16ðn þ 1Þ6 À4ðn þ 1Þ4 þ 4ðn þ1Þ2 À1;
bn ¼ 16ðn þ1Þ4 þ 4ðn þ 1Þ2 À2; and
wn ¼ 4ðn þ 1Þ2 À1; dn ¼ 6ðn þ1Þ2 þ 1

ð18Þ

Finally, the total stiffness requirement RT can be determined
by the summation of RÃ in Eq. (16) and Rn in Eq. (17), and it is
given by
RT ¼

p2 GJ
L

Fn

Thus, the natural frequency for the torsional vibration of the
I-girder with intermediate torsional bracings for an arbitrary
number of bracing points n can be calculated from Eq. (10) with
Eqs. (16)–(21) using the following procedure: (a) computing RÃ and
RT with Eqs. (16)–(21); (b) comparing R with RÃ and RT; and (c) the
corresponding natural frequency can then be calculated from
Eq. (10).


3. Verification of proposed equation
3.1. Description of finite element models
Frequency analyses are performed using the structural analysis
program ABAQUS [15] to verify the proposed equation for the
natural frequency for the torsional vibration of an I-girder with
intermediate torsional bracings. Four-node shell elements with
reduced integration (S4R) and spring elements are used to model
the I-girder and torsional bracings, respectively. The boundary
conditions are shown in Fig. 5. Point A is a hinged end where the
displacement in directions x,y,z and the rotation about z-axis are
restrained. Point B is a roller end where the displacement in
directions x,y and the rotation about z-axis are restrained. At the
supports, the x and y directions along the lines a and b are
restrained to prevent the premature local buckling of the web
and flange, respectively. The torsional braces are considered to be
rotational springs attached at the bracing points along the centroidal axis of the I-girder and transverse stiffeners are installed to
prevent the cross-section distortion at the bracing points. The
thickness of the transverse stiffeners at a bracing point is 15 mm.
Detailed profiles of the analysis models are listed in Table 1.
Convergence studies are conducted to obtain the refined
analysis model and the results are shown in Fig. 6. It can be found
that the ratio oFEM
=oo converges to 1.0 with increase in the
o
number of elements of the flange, where oo and oFEM
are the
o
natural frequencies of the I-girder without intermediate torsional
bracing (n ¼0) obtained from Eq. (10) and from finite element
analyses, respectively. The proper convergence can be obtained

when the number of the elements of the flange panel is larger than
6. In this case, the error between the theory and the finite element is
3.9%. Thus, six elements of the flange panel are used for the analysis
models.
Local deformations and distortions of cross sections may affect
natural frequencies. In order to investigate influences of such
behaviors on torsional natural frequencies, beam elements with
7 degrees of freedom (7DOFs) including warping are also adopted
in frequency analyses. A size of element is taken as L/400 of total
span length of an I-girder to provide a sufficient degree of accuracy.
The girder is simply supported in torsion and flexure with free

ð19Þ

where
Fn ¼

f2 ðnÞW 4 þ f3 ðnÞW 2 þ f4 ðnÞ
2f1 ðnÞð1 þ f5 ðnÞW 2 Þ

ð20Þ

and
f1 ðnÞ ¼ n þ 1
f2 ðnÞ ¼ 28ðn þ 1Þ6 À48ðn þ 1Þ5 þ 70ðn þ 1Þ4 À56ðn þ 1Þ3 þ 16ðn þ 1Þ2 À8ðn þ 1ÞÀ1
f3 ðnÞ ¼ 30ðn þ 1Þ4 À32ðn þ 1Þ3 þ 18ðn þ 1Þ2 À12ðn þ 1ÞÀ2
f4 ðnÞ ¼ 6ðn þ 1Þ2 À4ðn þ 1ÞÀ1
f5 ðnÞ ¼ 6ðn þ 1Þ2 þ 1

ð21Þ


537

Fig. 5. Boundary conditions of analysis model.


538

C.T. Nguyen et al. / Thin-Walled Structures 49 (2011) 534–542

Table 1
Profiles of analysis models.

bf (mm)
tf (mm)
tw (mm)
h (mm)
Lb (mm)

IG1

IG2

IG3

IG4

IG5

IG6


IG7

IG8

IG9

IG10

230
16
12
900
5500

240
18
15
950
6000

280
20
15
1100
6500

300
22
15

1200
7000

320
22
15
1250
7500

330
24
16
1300
8000

340
24
16
1350
8500

350
25
17
1400
9000

360
25
17

1450
9500

370
26
18
1500
10,000

Fig. 6. Results of convergence study (IG1).

warping at each end. Results from analyses using beam elements
are compared with those using shell elements.
3.2. Verification results
Fig. 7(a)–(c) show the comparisons of the natural frequency of
the I-girder with intermediate torsional bracing while the number
of bracing points n is equal to 1, 2, and 3. x- and y-axes represent the
non-dimensional frequency ratio o/oo or oFEM
=oo and the nono
dimensional torsional stiffness ratio R/RT, respectively, where o is
the natural frequency proposed in this study, oFEM is the natural
frequency obtained from finite element analyses, and oo is the
natural frequency I-girder without intermediate torsional bracing
(n¼0).
It is found that the natural frequency increases with increase in
torsional stiffness of bracing R. However, the natural frequency
remains as a constant when the torsional stiffness of the bracings R
reaches the total stiffness requirement RT. Thus, torsional stiffness
has no effect on natural frequency when full bracing is provided.
From Fig. 7(a)–(c), it can be seen that the proposed equations agree

well with finite element analysis and can provide a good prediction
of the natural frequency for the torsional vibration of the I-girder
with intermediate bracings. The results of beam-element analysis
is compared with those of shell-element analysis and the proposed
analytical solution. The results from beam-element analyses are
almost identical to those from shell-element analyses. Maximum
discrepancies between the two analysis methods are about 4.2%,
and this reveals that effects of local deformations and distortions of
cross sections are not large in the analysis models.
The validation of the proposed equation for the total stiffness
requirement RT is also examined. The verification results for RT are
shown in Fig. 8(a)–(c). Fig. 8(a)–(c) shows the comparison results of
the required stiffness for I-girders with various numbers of
intermediate torsional braces. x- and y-axes denote the nondimensional total stiffness requirement RFEM
=RT and torsional
T
slenderness W, respectively, where RT is the stiffness requirement
calculated using Eq. (19), while RFEM
is the stiffness requirement
T

Fig. 7. Comparisons of natural frequency of I-girder with intermediate torsional
bracings (IG1): (a) n ¼1, (b) n¼ 2, and (c) n¼ 3.

obtained from frequency analyses. It can be seen that the proposed
RT show a good agreement with those of the finite element analysis
regardless of the number of bracing points n and amount of


C.T. Nguyen et al. / Thin-Walled Structures 49 (2011) 534–542


539

Fig. 9. Typical finite element analysis models for twin I-girder bridge: (a) n¼1 and
(b) n ¼3.

Table 2
Profiles of twin girders and cross beams.

Girder
CB1
CB2
CB3
CB4

bf (mm)

tf (mm)

tw (mm)

h (mm)

Lb (mm)

800
400
300
180
140


30
14
12
10
10

17
12
10
8
8

2440
844
642
400
300

50,000
6000
6000
6000
6000

Fig. 8. Comparisons of required stiffness for I-girder with intermediate torsional
bracings: (a) n¼ 1, (b) n¼ 2, and (c) n ¼3.

torsional slenderness W. The maximum differences between the
results from this study and those from the finite element analyses

are 8.37%, 5.28%, and 5.07% for n is equal to 1, 2, and 3, respectively.
Fig. 10. Details of connection between main girder and cross beam.

4. Applications of proposed equations and parametric study
4.1. Description of analysis model and variables for parametric study
The proposed equations are applied to a practical civil engineering example and parametric studies are conducted to investigate the effects of torsional stiffness and the number of bracings
on natural frequencies for torsional vibration herein. The twin

I-girder system, which is widely used across the world, is adopted
for analysis. The profiles of the twin I-girder system are chosen
from actual bridge dimensions. Fig. 9(a) and (b) shows examples of
the twin I-girder systems with 1 and 3 cross beams, respectively.
Detailed dimensions of the main girder and cross beams are
given in Table 2. The length of the girder is 50,000 mm, the width of
the flange is 800 mm, the thickness of the flange is 30 mm, the
height of the girder is 2440 mm, and the thickness of the web is


540

C.T. Nguyen et al. / Thin-Walled Structures 49 (2011) 534–542

Fig. 11. Deformed shapes of cross beams under bending.

Table 3
Results of total torsional stiffness of bracings.
Torsional stiffness
times ( Â 106 N mm)

CB1


CB2

CB3

CB4

Lower bound
Upper bound

78,614.5
108,559.1

41,706.9
69,232.2

13,535.9
30,289.7

5605.3
14,707.6

17 mm. The distance between the girders is set as 6000 mm and the
girders are connected by cross beams. Four different cross beams
(CB1-4) are selected as shown in Table 2. Cross beams are assumed
to be connected to the main girders along the centroidal line. Oneside transverse stiffeners with full depth where the width is
250 mm and the thickness is 20 mm are used to prevent web
distortion at bracing locations as shown in Fig. 10. Also, the number
of bracing points varies from 1 to 4 and these are evenly distributed.
Thus, a total of 16 models are analyzed. The boundary conditions of

the twin I-girders are assumed to be a simply supported condition
in flexure and torsion.
To apply the proposed equations to the twin I-girder system, the
total torsional stiffness of the bracing system should be calculated
properly. The total torsional stiffness of the bracings is affected by
the stiffness of the girder, transverse stiffeners, webs, and by the
bracing itself [16]. Thus, it is a complex task to obtain accurate
values of the total torsional stiffness of the bracing with the
analytical method. In this study, the total torsional stiffness of
the bracing is obtained from finite element analysis [15]. Static
analysis is performed as shown in Fig. 11. Bending moments are
applied at each end of the cross beams to calculate the total
stiffness of the bracings. It can be found that the lower bound (LB)
and upper bound (UB) total torsional stiffnesses are obtained for
single curvature bending and double curvature bending of the cross
beams, respectively, as shown in Fig. 11. In Fig. 11, Mz is a bending
moment at each end of the cross beams, and c is a twisting angle of
the cross section of the main girders at mid-span. Thus, an elastic
total torsional stiffness is obtained as R¼ DMz/Dc, including the
bending stiffness of cross beams, stiffener stiffness, and the web
stiffness of main girder. The results of the lower bound and upper
bound total torsional stiffnesses of the bracings are given in Table 3.

4.2. Effects of total torsional stiffness of bracing and number of
bracing points

Fig. 12. Mode shapes of twin I-girders with intermediate cross beams for n¼3 and 4.

The analysis results for the twin I-girder systems are discussed
herein. Typical mode shapes of the twin I-girder systems are shown


Fig. 13. Variations of natural frequency with number of cross beams: (a) CB1, (b) CB2, (c) CB3, and (d) CB4.


C.T. Nguyen et al. / Thin-Walled Structures 49 (2011) 534–542

in Fig. 12. From the mode shapes, it is found that the cross beams
are deformed with a single curvature. Thus, it can be expected that
the low bound values for the total torsional stiffness of the bracing
provide a good prediction of natural frequency for torsional
vibration. Fig. 13(a)–(d) presents the variation of natural frequency
of the analysis model with the number of bracing points for CB1-4.
From Fig. 13, it can be seen that the gap between the upper and
lower bound solutions increases with increase in the number of
bracing points and decrease in the total torsional stiffness of the
bracing (CB1 and 4 have the largest and smallest total torsional
stiffness of the bracings in this analysis, respectively. Refer Table 3).
The results of the finite element analysis match well with those of
the lower bound solution. The average error between the theoretical values and the finite element analysis is 2.95%, while the
maximum error is 6.4% for the twin I-girder system with 4 bracing
points and the CB4 type cross beam. A comparatively large error is
observed when the number of bracing points increases. This is
caused by an excessive local deformation, which is generated with
an increase in the number of bracing points. Such deformations
lead to the distortion of the section. This then results in errors
occurring between the theory and finite element analysis. Fig. 14
shows an example of distortion of the section obtained from finite
element analysis.

541


Fig. 15 shows the variation in natural frequency for torsional
vibration with the number of bracings and total torsional stiffness.
The natural frequency is considerably increased with increase in
the number of bracing points for the CB1 type cross beam. However,
for a cross beam having a relatively low total torsional stiffness
such as CB4, the increment of natural frequency is not large. Thus, it
can be concluded that larger cross beams are more effective in
increasing the torsional natural frequency when the numbers of
bracing points increase.

5. Conclusions
This paper presents a simple analytical solution for the natural
frequency and stiffness requirement for the torsional vibration of Igirders with intermediate torsional bracings. Firstly, the natural
frequencies for torsional vibration are derived using Lagrange’s
equation for an arbitrary number of bracing points as given in Eq.
(10). The total required stiffness, which provides the full support, is
also derived as shown in Eq. (19). The proposed equations are then
successfully verified by comparing them with the results of finite
element analysis.
The proposed equations are applied to twin I-girder systems,
which are commonly used in civil engineering practices as an
application of the proposed equations. Also, a parametric study is
performed to investigate the effects of the total torsional stiffness
of the bracings and the number of cross beams on natural
frequencies for the torsional vibration of twin I-girder systems.
From the results, the lower bound solution provides good estimations of natural frequency for the torsional vibration of the twin
I-girder systems. Finally, it is found that the natural frequency
for torsional vibration is considerably increased with increase in
the total torsional stiffness of the bracings and the number of

bracing points. The increment of natural frequency is significantly
affected by the total torsional stiffness of the bracing and larger
cross beams are more effective when the number of bracing points
increases.

Acknowledgments

Fig. 14. Web distortions at connections between main girders and cross beams.

This research was supported by the grant from POSCO Corporation, the Ministry of Land, Transport and Maritime of Korean
Government through the Core Research Institute at Seoul National
University for Core Engineering Technology Development of Super
Long Span Bridge R&D Center.
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Fig. 15. Variations of natural frequency with number of bracings and total torsional
stiffness cross beam.

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