Thin-Walled Structures 49 (2011) 534–542 Contents lists available at ScienceDirect 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 December 2010 Available online 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 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 3290 3315; fax: + 82 928 5217 E-mail address: helee@korea.ac.kr (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 The mode shape of an I-girder with intermediate torsional bracings differs depending on the stiffness and number of bracings Fig 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 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 + 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 Natural frequency and stiffness requirement for torsional vibration of I-girder with intermediate bracings 2.1 Natural frequency Fig Types of intermediate torsional bracings Fig 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 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 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 ỵ2 X kij o2 mij ịqj ẳ for i ẳ 1, 2, :::, n ỵ2 8ị jẳ1 Then, Eq (8) can be expressed in a matrix form as ẵKo2 ẵMịfqg ẳ f0g: Fig 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 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 L _2 T¼ Ig c dz ð1Þ 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 L X Vẳ EIw c00 ỵGJ cu2 ịdzỵ R c2 2ị 2 kẳ1 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 cz,tị ẳ n ỵ2 X fi zịqi tị 3ị iẳ1 where fi zị ẳ sin ipz L for i ẳ 1, 2, , n ỵ2: 9ị 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 ẳ t 1ỵ ỵn þ 1Þ Ig Ig L L=m L=m GJ for m ¼ 1, 2, 3, , nÀ1 when R r RÃ ð10aÞ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u 2 LR C RL u p GJ @ n on ẳ t An ỵ Bn W 2Bn ỵ 2An W ị ỵ ỵ A L Ig 2p2 GJ p GJ when RÃ o R rRT vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u 2 uGJ p EIw t on ỵ ẳ 1ỵ L=n ỵ 1ị Ig L=n þ 1Þ GJ p ð10bÞ when R 4RT ð10cÞ where An ẳ n ỵ1ị2 ỵ 1; Bn ẳ n ỵ 1ị4 ỵ 6n ỵ1ị2 ỵ 1; Cn ẳ n ỵ In Eq (10), om and on represent intermediate natural frequencies with contributions of torsional stiffness, (om r om r om ỵ and on r on r on ỵ ), where om, on, and on + 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 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 ỵ2 X mij q j ỵ jẳ1 n ỵ2 X kij qj ẳ for i ẳ 1, 2, :::, n ỵ2: 5ị j¼1 where Z mij ¼ L Ig fi fj dz 6ị and kij ẳ Z L EIw f00i f00j ỵ GJfiu fju ịdz ỵ R n X n X iẳ1jẳ1 fi fj 11ị 7ị Fig 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 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 + A torsional natural frequency corresponding to an arbitrary mth mode shape is given as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u 2 p u EIw tGJ ỵ p 12ị om ẳ L=m Ig L=m GJ The required stiffness Rm can be obtained in increments of natural frequencies from om to om + and are expressed as Ig L o2m ỵ o2m ị 13ị Rm ẳ n ỵ1 Thus, the total amount of stiffness RÃ to obtain the nth mode is simplified as X nÀ1 Ig L o2 o2m ị 14ị R ẳ SRm ẳ nỵ1 m ẳ mỵ1 and R can be expressed in terms of o1 and on as follows by the summation of right term in Eq (14): Ig L o2n o21 ị 15ị R ẳ nỵ1 Substituting Eq (12) when m ¼1 and m¼n in Eq (15) yields RÃ ẳ p2 GJ L n1ị ỵn2 ỵ1ịW 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 Thus, Rn is obtained as Rn ẳ p2 GJ an W ỵ bn W ỵ wn ị 2n ỵ 1ị1 ỵ dn W ị L 17ị where an ẳ 16n ỵ 1ị6 4n ỵ 1ị4 ỵ 4n ỵ1ị2 1; bn ẳ 16n ỵ1ị4 ỵ 4n ỵ 1ị2 2; and wn ẳ 4n ỵ 1ị2 1; dn ẳ 6n ỵ1ị2 ỵ 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) 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 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 Convergence studies are conducted to obtain the refined analysis model and the results are shown in Fig 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 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 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 19ị where Fn ẳ f2 nịW ỵ f3 nịW ỵ f4 nị 2f1 nị1 ỵ f5 nịW ị 20ị and f1 nị ẳ n ỵ f2 nị ẳ 28n ỵ 1ị6 48n ỵ 1ị5 ỵ 70n ỵ 1ị4 56n ỵ 1ị3 ỵ 16n ỵ 1ị2 8n ỵ 1ị1 f3 nị ẳ 30n ỵ 1ị4 32n þ 1Þ3 þ 18ðn þ 1Þ2 À12ðn þ 1ÞÀ2 f4 nị ẳ 6n ỵ 1ị2 4n ỵ 1ị1 f5 nị ẳ 6n ỵ 1ị2 ỵ 21ị 537 Fig Boundary conditions of analysis model 538 C.T Nguyen et al / Thin-Walled Structures 49 (2011) 534–542 Table 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 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 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 Comparisons of natural frequency of I-girder with intermediate torsional bracings (IG1): (a) n ¼1, (b) n¼ 2, and (c) n¼ 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 Typical finite element analysis models for twin I-girder bridge: (a) n¼1 and (b) n ¼3 Table 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 2440 844 642 400 300 50,000 6000 6000 6000 6000 Fig 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 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 and cross beams, respectively Detailed dimensions of the main girder and cross beams are given in Table 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 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 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 to 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 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 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 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 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 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 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intermediate torsional bracings is derived