DSpace at VNU: Natural frequency for torsional vibration of simply supported steel I-girders with intermediate bracings

Natural frequency for torsional vibration of simply supported steel I-girders with intermediate bracings Canh Tuan Nguyena, Jiho Moonb, Van Nam Lec, Hak-Eun Leea,n a Civil, Environmental

Natural frequency for torsional vibration of simply supported steel I-girders with intermediate bracings

Canh Tuan Nguyena, Jiho Moonb, Van Nam Lec, Hak-Eun Leea,n

a

Civil, Environmental & Architectural Engineering, Korea University, 5-1 Anam-dong, Sungbuk-gu, Seoul 136-701, South Korea

b 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

a r t i c l e i n f o

Article history:

Received 31 May 2010

Received in revised form

19 November 2010

Accepted 1 December 2010

Available online 8 January 2011

Keywords:

Natural frequency

Torsional vibration

Stiffness requirement

Steel I-girder

Torsional bracing

a b s t r a c t s

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

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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 inFig 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]

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 thin-walled 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, neglect-ing the effect of warpneglect-ing 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

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n

Corresponding author Tel.: + 82 2 3290 3315; fax: + 82 2 928 5217.

E-mail address: helee@korea.ac.kr (H.-E Lee).

that prevent lateral displacement This differs from the case where

torsional bracings are described as a cross beam or X-bracing

system as shown inFig 1

The mode shape of an I-girder with intermediate torsional

bracings differs depending on the stiffness and number of bracings

Fig 2shows the mode shape of the I-girder with central torsional

bracings The total required stiffness RTis 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 RTas shown inFig 2

Thus, stiffness and number of bracings are the major parameters

that affect the natural frequency of torsional vibration

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 The natural frequency for torsional vibration of an I-girder with intermediate torsional bracings is derived using Lagrange’s equa-tion[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

Fig 1 Types of intermediate torsional bracings.

Nomenclature

E Young’s modulus

G shear modulus of elasticity

Ig mass moment of inertia about centroidal axis

Ip polar moment about centroidal axis

Iw warping constant

J pure torsional constant

k order of bracing points

L span length of I-girders

n number of torsional bracings

q time-dependent generalized function

R* summation of required stiffness Rm

Rm required stiffness that changes the mode shape from

mth to (m+ 1)th

Rn required stiffness that changes the mode shape from

nth to (n+ 1)th

RT total torsional stiffness requirement from theory

RFEM

T total torsional stiffness requirement from FEM

T kinetic energy

V potential energy

W torsional slenderness

x,y,z principle coordinate axes

f admissible shape function

r material density

o fundamental natural frequency of braced I-girder

from theory

oFEM fundamental natural frequency braced I-girder

from FEM

oo fundamental natural frequency of unbraced I-girder

from theory

oFEM

o fundamental natural frequency unbraced I-girder from FEM

om natural frequency corresponding to the mth

mode shape

om intermediate natural frequency between the mth and

(m+ 1)th mode shapes

on natural frequency corresponding to the nth

mode shape

on intermediate natural frequency between the nth and

(n +1)th mode shapes

mode shape

c twisting angle of the cross section of main girder

ck twisting angle of the cross section of main girder at

bracing points

longitudinal directions, respectively The rotation of the cross

section is represented by the twisting anglec, and the torsional

bracings are considered to be elastic rotational restraints

repre-sented by torsional stiffness R as shown inFig 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

T ¼1

2

Z L

0

wherecis the twisting angle about z-axis; Igis the mass moment of

inertia about centroidal axis defined as Ig¼rIp, whereris the mass

density of the material (M/L3); and Ipis the polar moment about the

centroidal axis

The potential energy including the warping effect can be given

as

V ¼1

2

Z L

0

ðEIwc00 2þGJcu2Þdzþ1

2

Xn

k ¼ 1

where E is Young’s modulus, G is the shear modulus of elasticity, Iw

is the warping constant, J is the pure torsional constant, andckis

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)¼f(L) ¼0 and

(b) f00(0) ¼f00(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, functionccan be defined as

cðz,tÞ ¼n þ 2X

i ¼ 1

where

fiðzÞ ¼ sin ipz

L

 

for i ¼ 1, 2, , n þ2: ð4Þ Substituting Eq (3) in Eqs (1) and (2) with Lagrange’s equation,

an equation of motion can be obtained as

X

n þ 2

j ¼ 1

mijq€jþn þ 2X

j ¼ 1

kijqj¼0 for i ¼ 1, 2, :::, n þ2: ð5Þ

where

mij¼

Z L

0

and

kij¼

Z L

0

ðEIwf00

if00

jþGJfiufjuÞdz þ RXn

i ¼ 1

Xn

j ¼ 1

Assuming that the time dependence of qjis harmonic, a system

of homogeneous equations, which represents the eigenvalue problem, can be given as

X

n þ 2

j ¼ 1

ðkijo2mijÞqj¼0 for i ¼ 1, 2, :::, n þ2 ð8Þ Then, Eq (8) can be expressed in a matrix form as

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

om¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p

L=m

 2

GJ

Ig

1 þ p

L=m

 2

EIw

GJ

!

þ ðn þ 1Þ R

IgL

v u for m ¼ 1, 2, 3, , n1 when RrR ð10aÞ

on¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p

L

 2GJ

Ig

AnþBnW22Bn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ 2AnW2Þ2þ LR

2p2GJ

 2

s

þCnRL

p2GJ

0

@

1 A

v u t

on þ 1¼ p

L=ðn þ 1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GJ

Ig

L=ðn þ 1Þ

EIw

GJ

! v

u

when R 4RT

ð10cÞ where

An¼ ðn þ1Þ2þ1; Bn¼ ðn þ 1Þ4þ6ðn þ1Þ2þ1; Cn¼n þ 1 ð11Þ

In Eq (10),omandonrepresent intermediate natural frequen-cies with contributions of torsional stiffness, (omromrom þ 1

andonronron þ 1), whereom,on, andon + 1represent natural frequencies corresponding to the mth, nth, and (n + 1)th mode shapes, respectively Torsional slenderness W is defined as

ðp=LÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

EIw=GJ p

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, Rmand RTare 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 (referFig 4) It is also noted that Eq (10a) is available when RrR, 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 RoRrRT Eq (10c) can be used to calculate the natural frequency when R4RT

Fig 3 I-girder with intermediate torsional bracings.

2.2 Stiffness requirement

The relationship between natural frequencies and stiffness is

shown inFig 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  1)th mode shape is defined as R When R4RT, full bracing is

provided, and the natural frequency is equal toon + 1

A torsional natural frequency corresponding to an arbitrary mth

mode shape is given as

L=m

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

GJ

Ig

1 þ p

L=m

 2

EIw

GJ

! v

u

ð12Þ

The required stiffness Rm can be obtained in increments of

natural frequencies fromomtoom + 1and are expressed as

Rm¼ IgL

n þ1

ðo2

m þ 1o2

Thus, the total amount of stiffness Rto obtain the nth mode is

simplified as

R¼SRm¼ IgL

n þ 1

  Xn1

m ¼ 1

ðo2

m þ 1o2

and Rcan be expressed in terms ofo1andonas follows by the

summation of right term in Eq (14):

R¼ IgL

n þ 1

Substituting Eq (12) when m ¼1 and m¼n in Eq (15) yields

R

¼p2GJ

L ðn1Þ 1 þðn

2þ1ÞW2

ð16Þ where Ris 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

fromontoon + 1, as shown inFig 4 Thus, Rnis obtained as

Rn¼p2GJ

L

ðanW4þbnW2þwnÞ

2ðn þ 1Þð1 þdnW2Þ ð17Þ

where

an¼16ðn þ 1Þ64ðn þ 1Þ4þ4ðn þ1Þ21;

bn¼16ðn þ1Þ4þ4ðn þ 1Þ22; and

wn¼4ðn þ 1Þ21; dn¼6ðn þ1Þ2þ1 ð18Þ

Finally, the total stiffness requirement RTcan be determined

by the summation of Rin Eq (16) and Rnin Eq (17), and it is

given by

RT¼p2GJ

where

Fn¼f2ðnÞW4þf3ðnÞW2þf4ðnÞ

2f1ðnÞð1 þ f5ðnÞW2Þ ð20Þ

and

f1ðnÞ ¼ n þ 1

f2ðnÞ ¼ 28ðn þ 1Þ648ðn þ 1Þ5þ70ðn þ 1Þ456ðn þ 1Þ3þ16ðn þ 1Þ28ðn þ 1Þ1

f3ðnÞ ¼ 30ðn þ 1Þ432ðn þ 1Þ3þ18ðn þ 1Þ212ðn þ 1Þ2

f4ðnÞ ¼ 6ðn þ 1Þ24ðn þ 1Þ1

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 Rand

RTwith Eqs (16)–(21); (b) comparing R with Rand 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 inFig 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 cen-troidal 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 inTable 1 Convergence studies are conducted to obtain the refined analysis model and the results are shown inFig 6 It can be found that the ratio oFEM

o =oo converges to 1.0 with increase in the number of elements of the flange, where oo and oFEM

o are the 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

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 ratioo/oooroFEM

o =ooand the non-dimensional torsional stiffness ratio R/RT, respectively, whereois

the natural frequency proposed in this study,oFEMis the natural

frequency obtained from finite element analyses, andoois 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

FromFig 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 RTis also examined The verification results for RTare

shown inFig 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

non-dimensional total stiffness requirement RFEM

T =RT and torsional slenderness W, respectively, where RTis the stiffness requirement

calculated using Eq (19), while RFEM is the stiffness requirement

obtained from frequency analyses It can be seen that the proposed

RTshow a good agreement with those of the finite element analysis regardless of the number of bracing points n and amount of

Table 1

Profiles of analysis models.

Fig 6 Results of convergence study (IG1).

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

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

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

engi-neering example and parametric studies are conducted to

inves-tigate 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 inTable 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

Fig 8 Comparisons of required stiffness for I-girder with intermediate torsional

bracings: (a) n¼ 1, (b) n¼ 2, and (c) n ¼3.

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.

Fig 10 Details of connection between main girder and cross beam.

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 inTable 2 Cross beams are assumed

to be connected to the main girders along the centroidal line One-side 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 inFig 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 inFig 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 inFig 11 InFig 11, Mzis a bending moment at each end of the cross beams, andcis a twisting angle of the cross section of the main girders at mid-span Thus, an elastic total torsional stiffness is obtained as R¼DMz/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 inTable 3

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

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

Fig 11 Deformed shapes of cross beams under bending.

Table 3

Results of total torsional stiffness of bracings.

Torsional stiffness

times (  10 6

N mm)

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

inFig 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

FromFig 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 ReferTable 3)

The results of the finite element analysis match well with those of

the lower bound solution The average error between the

theore-tical 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

Fig 15shows 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 I-girders 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 estima-tions 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 This research was supported by the grant from POSCO Corpora-tion, 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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