unequally spaced lateral bracings on compression flanges of steel girders
TRANSCRIPT
Structures 3 (2015) 236–243
Contents lists available at ScienceDirect
Structures
j ourna l homepage: ht tp : / /www.e lsev ie r .com/ locate /s t ructures
Unequally spaced lateral bracings on compression flanges of steel girders
Hassan Mehri a,⁎, Roberto Crocetti a, Per Johan Gustafsson b
a Div. of Structural Engineering, Lund Univ., Box 118, Lund 22100, Swedenb Div. of Structural Mechanics, Lund Univ., Box 118, Lund 22100, Sweden
⁎ Corresponding author. Tel.: +46 46 222 7397.E-mail addresses: [email protected] (H. Mehri)
(R. Crocetti), [email protected] (P.
http://dx.doi.org/10.1016/j.istruc.2015.05.0032352-0124/© 2015 The Institution of Structural Engineers
a b s t r a c t
a r t i c l e i n f oArticle history:Received 26 February 2015Received in revised form 6 May 2015Accepted 17 May 2015Available online 2 July 2015
Keywords:Lateral bracingStiffness requirementSteel girder
In the bridge sector, lateral bracings can be provided e.g. in the form of metal decks or horizontal truss bracings.Those bracings are more efficient at the regions of maximum lateral shear deformations generated fromdestabilizing forces of compression flanges e.g. near to the twisting supports. A model is presented in thispaper, which relates the lateral buckling length of compression flange of steel girders to their lateral torsionalbucklingmoment and can be used to investigate stiffness requirement of lateral bracings applied on the compres-sion flanges between the twisting restraints. Analytical solutionswere derived for the effects of bracing locationsand bracing stiffness values on buckling length of compression flanges. Moreover, an exact and a simplifiedsolution for the effect of rotational restraint of shorter-spans on critical load value of the compression memberswith unequally spanned lateral bracings were derived. Themodel can be suitable for design engineers to prelim-inary size the cross-section of beams and lateral bracings.
© 2015 The Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
1. Introduction and background
Relatively little lateral bracings can greatly enhance load carrying ca-pacity of slender steel columns and beams by limiting their out-of-planedeformations [15]. However, improper restraint against lateral torsionalbuckling can be detrimental. A number of bridge failures have occurreddue to improper lateral bracings. Two examples are: the collapse of TheMarcy Bridge [7] in New York, and Y1504 Bridge [2] in Sweden. Thesebridges were designed with trapezoidal cross-sections and both col-lapsed due to global lateral torsional buckling during concreting of thedeck. No lateral bracings were used in the first case, while stay-in-place corrugated metal sheets were designated to act as lateral stabiliz-ing system in the latter bridge. However, for steel bridge applications,Egilmez et al. [4] showed, both analytically and experimentally, thatcorrugated metal decks if properly designed and connected to thegirders, can significantly reduce lateral deformation of steel girders.
Yura et al. [16] derived a simplified expression for global buckling ofsteel bridge girders which corresponded to the failure mode for the twomentioned bridges. Lateral bracing at partial span near to the abutmentscan enhance the load carrying capacity of the girders which are prone toglobal buckling by creating a semi-clamp condition at the supports [16].Mehri and Crocetti [7] showed that providing relatively “soft” truss brac-ings along a partial length of bridge span near to the abutments (e.g.α ¼0:1 in Fig. 1), the global buckling of The Marcy Bridge could have beenavoided; where α is the partial bridge span from the twisting supportsat both ends which are laterally braced by means of e.g. either truss
, [email protected]. Gustafsson).
. Published by Elsevier Ltd. All rights
bracings or corrugated metal decks. An example for the application ofthe present study can be to estimate critical bending moment value ofsuch bridges by studying required lateral bracing stiffness near to theabutments, see Fig. 1, in order to create semi-clamped boundaryconditions.
The exact solutions for bracing requirements and load carryingcapacity of compression members are only possible for simple cases,with certain boundary and loading conditions. There have been numer-ous previous studies on bracing requirements of simple beams andcolumns [17]. Among these studies, there are investigations on criticalmoment value for a variety of loading and bracing conditions, and dif-ferent cross-sectional properties. In a number of studies, Timoshenko'senergy approach was used to find the optimal locations for bracings ofsimple beam structures, see e.g. [11,13,14]. Some of the studies haveled to predicting conservative values for critical loads and stiffness ofbracings that are already included in some code specifications. Finianet al. [6] studied the stability of imperfect steel beams which wererestrained by means of a number of discrete elastic bracings, and gaveexpressions to estimate the magnitudes of bracing forces.
Recommendations for critical moment values basically consist ofapplying a number of coefficients that statistically give lower boundresults. For beams for instance, the coefficients are applied to criticalmoment, obtained from e.g. Timoshenko's approach [10] to considerthe effects of different conditions such asmoment gradient, load height,cross-sectional symmetries and boundary conditions. Generally, thoserecommendations have been presented for a limited number of simplecases and can also lead to high discrepancies in the results, especiallywhen a combination of the effects is considered.
Winter [15] presented a model with rigid bars, which are hinged atthe locations of equally spaced transitional springs, to study the lateral
reserved.
Fig. 1.Global buckling of “narrow” steel girders and theperformanceof partial-span lateralbracings.
237H. Mehri et al. / Structures 3 (2015) 236–243
bracing requirements of columns. Winter's model predicts a minimumrequired (or “ideal”) stiffness value for those particular cases, which isconsidered to serve equivalent to immovable lateral support. Beyondthis threshold stiffness, any increase in brace stiffness will not enhancethe critical load of the columns. Galambos [5] discussed column caseswith unequal spans. Plaut [8] studied the bracing requirements of col-umns with single lateral brace at an internal arbitrary point betweenthe pinned or elastic supports. For thementioned cases, he also showedthat the “ideal” brace stiffness only exists when the bracing is at the cen-ter of a uniform column at mid-span. Plaut and Yang [9] also studied thebehavior of pinned-end columns with two intermediate lateral bracingsfor two cases: equally spaced bracings, and a case with unequally spacedbracings at a specific location. For a column with three-span lateral brac-ings, he stated that “The optimal locations of the internal braces are not ob-vious unless full bracing is possible”. The present paper discusses theindications concerning the optimal locations for unequally spanned later-al bracings which gives the largest critical load for a given brace stiffnessvalue. The brace in this study were placed symmetrically with respect tothe midspan of the girders which is normally the case in practice.
However, the main purpose of this paper is to analytically investi-gate the applicability of a proposed simplified model to predict criticalmoment of laterally braced girders varying the stiffness and locationof bracings. This approach can greatly benefit design engineers eitherbefore or after performing the final design in order to size the beamsand bracings or to check the results obtained from using commercialsoftware. For this purpose, exact and approximate expressionswere de-rived to examine the bending restraint effects of shorter spans on criti-cal load values of beams with unequally spanned lateral bracings.Typical design curves were also given from which critical load valuescan be determined for different stiffness values and location of symmet-rically placed lateral bracings. A curve was also given for the cases inwhich providing lower stiffness than “full bracing” i.e. bracing stiffnessvalue that serves approximately equal to immovable support is thoughtto be adequate in design. Finally, a comprehensive example is presentedto show the applicability of the proposed approach.
Fig. 2. A model that relates critical moment of beams to critical load of their compressionflange.
2. Results and discussions
2.1. Theory and model development
In this section of the study, Timoshenko's basic approach [10] forcritical moment of simply supported beams under uniform bendingmoment was used to develop a model which can be used as a generalsolution for beams that are laterally braced between the twisting sup-ports at the level of their compression flanges. The model relatesTimoshenko's beam subjected to equal and opposite bending momentsat each end to an equivalent couple forces arising in the flanges of thegirders. The model and the derived analytical solutions were verifiedby means of Finite Element (FE) analyses using ABAQUS commercialsoftware [1]. Quadrilateral, reduced integration, four-node shellelement S4R with sufficient meshing was used in the FE analyses tomodel the flanges and the web of the studied three dimensional
beams, and the plate of the two dimensional compression flanges. S4Ris a common shell element type with four nodes, when modeling ofsteel plates of columns, beams, and stiffeners is desired. Efforts wereundertaken to keep the meshing dimensional ratio equal to unity. Atleast six elements were used across the width of the flanges. No localbuckling was observed in any of the FE analyses. In the linear bucklinganalyses of the compression flanges, the members were fixed at oneend, free to slide in the longitudinal direction at the other end, whilefree to warp at both ends. The deformations perpendicular to theplane of compression flanges were not permitted in the two-dimensional analyses. The three dimensional beams were fixed at oneend, free to slide in the longitudinal direction at the other end, whilefree to warp and restrained against twist at both ends. Bracings weremodeled with linearly elastic transitional springs connecting one nodeof the shell elements to the “ground”. The beam components in the per-formed FE linear buckling analyses had elastic material properties. Gen-erally speaking, the developed forces in the bracings are relatively littlewhile a significant increase in the resistance can be normally achievedwith the relatively soft bracings. As a rule of thumb the bracing forcesare normally within the range of 1–2% of the applied force in compres-sion members. Based on the conditions mentioned above, any localeffects at the junction of the springs with the shell elements−thatmight occur in nonlinear analyses with large deformations−were con-sidered negligible.
The critical bending moment of a simply supported beam withdoubly symmetric cross-section subjected to uniform bendingmomentabout the strongest axis (y–y) can be calculated using the followingequation [10]:
Mcr ¼ π=Lbð ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEIzGJ þ π2E2IzCw=L
2b
q; ð1Þ
where
Lb Beam span between the torsional restraints;E and G Young's and shear moduli of beam material, respectively;J, Cw , and Iz Cross-sectional properties including torsional constant,
warping constant, and moment of inertia about weakestaxis of beam, respectively, (see Fig. 2).
More accurate results can be obtained for I-beams withmonosymmetric cross-section considering Iz ¼ Izc þ t=cð ÞIzt [19];where t, and c are distances from the neutral bending axis of themonosymmetric cross-section to the centroids of tension and compres-sion flanges, respectively; Iz, Izc, and Izt are lateral moments of inertia ofthe cross-section, the compression flange, and the tension flange,respectively, about the weak axis, i.e. “z” axis in Fig. 2. However, forthe purpose of this paper, Iz≈2Izc can be also used to obtain a reasonablyconservative prediction of critical moment values for the common
I-shaped steel girder dimensions. Substituting Cw with Izh2=4 for
doubly-symmetric I-beams (where h is the distance between the cen-troids of the flanges),GJ ¼ C, and ECw ¼ C1, Eq. (1) can be rewritten as:
Mcr ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπ2EIzh=2L2b
� �21þ CL2b=π2C1
� �r: ð2Þ
238 H. Mehri et al. / Structures 3 (2015) 236–243
Setting γLT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ CL2b=π2C1
q¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2C=Ncrh
2q
, Eq. (2) can be
written as:
Mcr ¼ Ncrhð Þ � γLT : ð3Þ
Eq. (3) claims that critical moment of I-beams can be approximatelymodeled by two opposing horizontal forces, γLTNcr , applied at the cen-troids of the flanges, as shown in Fig. 2; where, the compression flangeof the beams is regarded as a columnwith critical load ofNcr ¼ π2EIzc=L
2b.
Table 1 shows the variation of the γLT coefficient with different
values of torsional properties (i.e. CL2b=π2C1); where “γLT ¼ 1” repre-
sents beams with “zero” torsional stiffness, i.e. the web and the tensionflange give no contribution to controlling lateral torsional deformationof the compression flange, and “γLT ¼ ∞” represents beams with “infi-nite” torsional stiffness.
Theoretically, knowing the buckling length of the compressionflanges, Eq. (3) can also be used to predict the critical moment valuefor the cases of laterally restrained beams at the level of compressionflanges. To generalize Eq. (3), the buckling load of compression flangeand γLT factor can be calculated from Eqs. (4)–(5), respectively.
Ncr ¼ π2EIzc=l2e ; ð4Þ
γLT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Cl2e=π2C1
q¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2C=Ncrh
2q
ð5Þ
where le is the buckling length of the compressionflange of the beams,and in this paper is indicated as “le;∞”, “le;k”, or “Lb”which corresponded tocases in which the compression flange of the beams are respectively lat-erally restrained bymeans of immovable supports, compression flange ofbeam is laterally restrained by means of elastic bracings, or the beam isnot laterally braced between the twisting supports.
Lateral torsional buckling of steel girders involves lateral movementof their compression flange and the twist of the girder cross-section. Inthe proposed approach presented in Eq. (3), the lateral translationaldegree-of-freedom of the compression flanges is considered by study-ing the buckling of the compression flanges (see Eq. (4)).While, the tor-sional degree-of-freedom of the cross-section along the buckling lengthof the compression flanges− i.e. “le” obtained from Eq. (4)− is consid-ered by theγLTcoefficient (see Eq. (5)). It must be noted that in the pro-posed approach the Vlasov condition [12] was assumed for the shape ofthe cross-section of beams that remains unchanged during bucklinganalyses. Therefore, no local buckling of cross-section was permitted.
2.1.1. Effects of different loading and boundary conditionsEq. (3)was derived as a general solution, assuming the same bound-
ary and loading conditions as of Eq. (1), i.e. simply supported beamsubjected to uniform bending moment. There have been numerousstudies on critical moment of simple beams to statistically determineconservative effects of different conditions such as bending momentgradients, loading height and cross-sectional asymmetries on criticalmoment values obtained by Timoshenko's approach (i.e. Eq. (1)). Forinstance, for a simply supported doubly-symmetric unbraced beamunder a uniformly distributed load on the level of the top flange, themodification coefficient to be applied to Eq. (1) was suggested [17] to
be equal to 0:8 ¼ 1:12� 1:4−1� �
. Recommendations are limited to sim-
ple beams and can be often difficult to be justified for some specificcases in practice when a combination of effects is being considered.
Table 1Variation of the γLT coefficient for beams with different torsional properties.
CL2b=C1 0.0 0.1 10 100 1000 ∞
γLT 1.0 1.0 1.4 3.3 10.1 ∞
For the purpose of further studies on the proposed approach, the au-thors suggest an application of buckling lengthmodification factor to ac-count for the effects of distributed load at the level of top flanges, whichis a typical case in bridge application e.g. during the construction phase.
2.1.2. Effects of imperfections and nonlinearitiesFig. 3 schematically shows a typical buckling curve recommended by
Eurocode 3 (2005), which is based on results fromnumerous laboratorytests. Such curves relate critical compression load,Ncr, or critical bendingmoment, Mcr , values of columns or beams, respectively, to a reductionfactor, χ , which can be applied to e.g. plastic capacity of the cross-section in order to calculate the corresponding design values. The effectof different shapes and magnitudes of initial imperfections, residualstresses, material nonlinearities, eccentricities, etc. are conservativelyconsidered in the buckling curves. Thus, using such buckling curves,the corresponding design values can be calculated for given criticalload values of beams, or columns obtained from Eq. (3), or Eq. (4),respectively.
Winter's rigid bar model [15] gives the minimum required stiffnessvalues for columns with equally spaced lateral bracings. For the casesof unequally spaced bracings, it is proposed that the buckling lengthcan be conservatively considered equal to the length of the largestspan [18]. However for the unequally braced columns (or compressionflanges), the shorter portions of the beams create some bending re-straint to the larger spans due to their different bending stiffness values.In the present study, it is shown that this effect can be significantdepending on the bending stiffness ratio of the adjacent spans withunequal lengths. Fig. 3 illustrates the possible overdesign problemwhich can occur ignoring thementioned effects; where point “A” repre-sents the critical value obtained approximating buckling length equal tothe distance between the bracing points, and point “B” represents thecritical value obtained considering the bending restraint effects of theshorter spans; χA; and χB are design reduction factors correspondingto the critical load values marked as “A” and “B”, respectively; NP , orMP are plastic capacities of the column or the beam cross-section; andλrel is lateral or lateral torsional slenderness ratios for the correspondingcolumn or beam, respectively.
The model explained in Section 2.1 was examined for beams with:i) no lateral restraints (presented in Section 2.2), ii) lateral bracings thatserved equal to immovable supports (presented in Section 2.3.), andiii) lateral bracings that had elastic stiffness (presented in Section 2.4.),all at the location of compression flanges:.
2.2. Laterally unrestrained beams
Empirical recommendations are commonly used by design engi-neers to preliminary size the cross-section of bridge girders, which aredependent on the length of girders, the width of their concrete decks,and etc. Table 2 shows the values of γLT for typical I-shaped built-up
Fig. 3. Schematic illustration of typical design curves for flexural or lateral torsionalbuckling.
Fig. 5. Values of γLT for the beam cases: (1)–(5).
Table 2Variations of γLT values for different IPE profiles, and common built-up steel girders. “Lb”and “d” are length and depth of beams, respectively.
Lb=d ¼ 16 Lb=d ¼ 20 Lb=d ¼ 24
IPE profiles 1.45–1.60 1.60–1.85 1.80–2.15Built-up girders 1.40–2.30 1.50–2.60 1.60–2.80
239H. Mehri et al. / Structures 3 (2015) 236–243
bridge girders in steel–concrete composite bridges, and also for someEuropean Standard I-Beam profiles (IPE 100 to IPE 750). In the built-up girders, the dimensions of cross-sections, i.e. the size of the flangesand the webs, were calculated utilizing the commonly used empiricalmethods for 108 beam cases (varying girder depth from 600 mm to3000 mm, in 300 mm increments; deck width from 3000 mm to12000 mm, in 3000 mm increments, and length from 9600 mm to72000 mm). As can be seen in Table 2, typical γLT values vary withinthe range of approximately 1.4 to 2.8.
In addition, Eq. (3) was used to calculate γLT values for beam cases(1)–(5) as shown in Fig. 4 varying span lengths between 6 and 36 m;the critical bendingmomentswere obtained using FE buckling analyses.
In Fig. 5 γLT values obtained are plotted against CL2b=C1 values. Theresults were also compared to the curve obtained from Eq. (5). Thecomparison verified that Eq. (5) gives a unique curve for both doubly-and mono-symmetric beams with the assumptions made earlier.
2.3. Beams with immovable lateral restraints
In general, for many of the cases encountered in practice, muchhigher stiffness values than the “ideal” stiffness are normally providedto resist lateral deformations of compression members at the bracepoints. The effect of unequal spanning of lateral bracings on criticalload values of compression flange of beams are investigated in thissection, assuming that lateral stiffness values of the bracings are largeenough to function fairly similar to immovable supports. Fig. 6 illus-trates the central-span of the compression flange of a beam which islaterally restrained with immovable supports at a αl distance fromeither end, as an equivalent column with rotational spring stiffness ofβαEI=l at both ends;whereEI is the bending stiffness of the cross-sectionabout the strong axis of compression flange, l is the length of the central-span, and βα varies depending on α values.
For beam–column “BC” in Fig. 6, the general differential equation is:
EIw‴x þ Nw″
x ¼ 0 ð6Þ
where wx is the lateral deflection of the beam–columns “AB” and“BC”.
Fig. 4. The studied beam cases: (1)–(5), each with varying lengths = 6, 12, 24 and 36 m.
Applying boundary conditions ðwx2¼0 ¼ wx2¼l ¼ 0; EIw″x2¼0 ¼
EIw″x2¼l ¼ βαEI=lð ÞθÞ for beam–column “BC” gives:
cl ¼ −βα sin clð Þ= 1þ cos clð Þ½ �; ð7Þ
where c2 ¼ N=EI.Eq. (7) has no close-form solution and should be solved numerically
for given values of βα . An analytical solution for the relationshipbetween α and βα was of interest.
For the beam–column “AB” in Fig. 6, applying boundary conditionswx1¼0 ¼ wx1¼αl ¼ 0; EIw″
x1¼0 ¼ 0; EIw″x1¼αl ¼ βαEI=lð Þθ� �
, and dividingmoment force βαEI=lð Þθ by the slope at point B, w0
x1¼αl , gives the rota-tional stiffness at point B:
βα ¼ 3=αð Þ cαlð Þ2 sin cαlð Þ=3 ð sin cαlð Þ− cαlð Þ cos cαlð Þ½ �; ð8Þ
where cαl ¼ πffiffiffiffiffiffiffiffiffiffiffiffiffiN=Ncr
p, and Ncr is the buckling load of the beam–
column “AB”.Eq. (8) gives an exact solution for themagnitude of bending restraint
of the side-spans to the central-span. Substituting the resulted value ofβα from Eq. (8) into Eq. (7) gives the buckling length of the compres-sion member, i.e. le;∞ , shown in Fig. 6. However, a simplified solutionfor the relationship between α and βα , rather than Eq. (8) can be alsoderived. The following relationship between α and βα can be written,when second-order effects are neglected:
βα ¼ 3=α: ð9Þ
Eq. (9) can be modified by applying the amplification factor η ¼1= 1−N=Ncrð Þ to consider the second-order effects of the compression
force on the deformations; where N and Ncr are approximately π2EI=l2
Fig. 6. Bending restraint of side-spanswith shorter length to the central-span of compres-sion flange restrained by means of immovable lateral supports.
Table 3Buckling length of compression flange with immovable restraints placed at partial length(αl).
α ≈0.0a 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0βα ∞ 28.2 13.6 8.22 5.37 3.58 2.36 1.48 0.85 0.37 0.0cl ≈2π 5.88 5.51 5.16 4.82 4.49 4.17 3.88 3.61 3.36 πle;∞=l ≈0.50 0.53 0.57 0.61 0.65 0.70 0.75 0.81 0.87 0.93 1.00le;∞=Lb ≈0.50 0.45 0.41 0.38 0.36 0.35 0.34 0.34 0.34 0.33 0.33
a This value represented lateral restraints at a distance which was “very” close to thetwisting supports but not equal to “zero”.
240 H. Mehri et al. / Structures 3 (2015) 236–243
andπ2EI=α2l2, respectively. Thus, the relationship between βα andα canbe approximated by the following equation to consider the second-order effects:
βα≈3=α 1−α2� �: ð10Þ
Comparisons between the results of Eqs. (8)–(10) showed that theapproximate solution obtained from Eq. (10) was in excellent agree-ment with the exact solution obtained from Eq. (8) (see Fig. 7). There-fore, by substituting the βα values obtained from Eq. (10) (for a givenvalue of α) into Eq. (7), the buckling load of the laterally restrainedcompression flange can be calculated when lateral stiffness of thebraces is sufficiently large.
Table 3 shows the results of Eq. (7) for a number ofβα values between“zero stiffness”, which represents nil rotational stiffness provided by theside-spans to the central-span, and “infinite stiffness”which means thatthe side-spans act as a fully-clamped support to the central-span.
To summarize, for different positions of the lateral supports (αl),critical moment values of beams can be calculated by substituting le;∞obtained from Table 3 into Eqs. (3)–(5). Table 3 showed significantdecreases in the buckling length ratios, i.e. le;∞=l, of the compressionflange in the central-span, especially for the small values of α. In bridgedesign practice, it is often assumed that the buckling length of thebraced girder is conservatively equal to the distance between the brac-ing points, if adequate stiffness larger than the recommended value isprovided for the bracings [3]. The results in Table 3 show that this as-sumption can lead to a considerable over-design problem (it can be aslarge as a factor of 4, see l2=l2e;∞ for small values of α).
2.4. Beams with elastic lateral restraints
For practical uses, the method should be expanded for differentstiffness values of lateral bracings. Bracingswere represented as linearlyelastic translational springs at symmetric locations with respect to themid-span of the beams, which is mostly the cases in practice (seeFig. 8). Depending on the magnitude of the lateral bracing stiffness,compression flanges of beams can deform in either symmetric or asym-metric shape with respect to the center-span (see Fig. 8).
For the symmetric deformation shape, applying boundary conditionswx1¼0 ¼ EIw″
x1¼0 ¼ 0;wx1¼αl ¼ δ;w0x1¼αl ¼ θ
� �for the side-span beam,
i.e. beam “AB”, and wx2¼0 ¼ δ;w0x2¼0 ¼ θ;w0
x2¼0:5l ¼ EIw‴x2¼0:5l ¼ 0
� �for
the internal-span beam, i.e. beam “BC”, Eq. (6) gives:
ξc3 � cos cαlð Þ þ k=EIξc2 � sin cαlð Þ
�−ξc3αl � cos cαlð Þ−c2
−ξc2lα � sin cαlð Þ−c � cot 0:5clð Þ�
δθ
¼ 0
0
ð11Þwhere ξ ¼ 1= sin cαlð Þ−cαl � cos cαlð Þ½ �.
Fig. 7. Bending restraint of side-spans with shorter length to buckling length of central-span, obtained from Eqs. (8)–(10).
Setting the determinant of the stiffnessmatrix to zero gives an exactsolution for the normalized relationship between bracing stiffness andcorresponding critical load values as in the following equation:
kl3=EI ¼ −ξ clð Þ3 cos cαlð Þ � cot 0:5clð Þ− sin cαlð Þ½ �= ξcαl � sin cαlð Þ þ cot 0:5clð Þ½ �:ð12Þ
Similarly, for the asymmetric mode shape, applying boundaryconditions wx1¼0 ¼ EIw″
x1¼0 ¼ 0;wx1¼αl ¼ δ;w0x1¼αl ¼ θ
� �for side-
span beam, i.e. beam–column “AB”, and ðwx2¼0 ¼ δ;w0x2¼0 ¼
θ;wx2¼0:5l ¼ EIw″x2¼0:5l ¼ 0Þ for the internal-span beam, i.e. beam–
column “BC”, Eq. (13) gives a normalized solution between the bracingstiffness and the corresponding critical load. For a given lateral stiffnessvalue, critical load is the minimum value obtained from symmetric andasymmetric calculations using Eqs. (12)–(13).
kl3=EI ¼ −ξϑ clð Þ3 α þ 0:5ð Þ � sin cαlþ 0:5clð Þ½ �= αξ � sin cαlð Þ þ 0:5ϑ � sin 0:5clð Þ½ �ð13Þ
where ϑ ¼ 1= sin 0:5clð Þ−0:5cl � cos 0:5clð Þ½ �.A typical design curve for the relationship between lateral bracings
stiffness and buckling length of the central-span for different locationsof lateral bracings is shown in Fig. 9, as the results of linear FE bucklinganalyses. Comparison between the FE results and the results of Eq. (12)aremade in Fig. 10; where the solid lines represented the FE results (forkLb=N0 values between 0.0 and 200.0), and the dashed lines represent-ed the results of Eq. (12), (for kLb=N0 values between 0.0 and 800.0);
and N0 ¼ π2EI=L2b . The results obtained from Eq. (12) were expandedfor more values of stiffness than the FE data to show the “excellent”agreement between the two methods. The results showed that the rel-atively small values of stiffness for symmetrically placed lateral brac-ings, can significantly decrease the buckling length of the compressionflange, especially for the cases where α≥0:25 (see Fig. 9). For the caseof α ¼ 1:0; the points indicated with (i) and (ii) in Fig. 9 representedminimum bracing stiffness values (kLb=N0 being 13.2 and 81, respec-tively) to change the buckling shapes from a single half-sine wave to
Fig. 8. Compression flange of beams with symmetrically placed lateral elastic bracings.
Fig. 10. The curves showing ratios between critical loads of compression member shownin Fig. 8 and critical load of the samemember assuming the buckling length to be equal tothe distance between the bracing points.
Fig. 9. Relationship between buckling length of the compression member shown in Fig. 8and stiffness of lateral bracings.
241H. Mehri et al. / Structures 3 (2015) 236–243
two and three half-sinewaves, respectively, where three half-sinewavecorresponds to the “ideal” bracing stiffness of the column [15]. Thesethresholds values for columns with equally spaced lateral bracings i.e.,
Fig. 11. Investigations on shifting from a symmetric buckling mode shape, obtained from Eq. (flanges shown in Fig. 8.
α ¼ 1:0, can be calculated by means of Winter's rigid bars model thatare hinged at the locations of braces.
Critical load values of the beam shown in Fig. 8 was normalized tocritical load values when − ignoring bending restraint of the sidespans−the buckling length is “conservatively” assumed to be equal to
the distance between the bracing points, i.e. l2=l2e;k in Fig. 10. The resultsshown in Fig. 10 verified that this assumption gives unsafe results forrelatively small values of brace stiffness for all locations of bracing. Onthe other hand, providing relatively soft bracings, e.g. as little askLb=N0 ¼ 100 , significantly enhanced the critical load values. Thiscapacity is currently ignored in practice when the buckling length isassumed to be equal to “l” [3].
Fig. 11 graphs the results obtained from Eqs. (12)–(13) for differ-ent brace stiffness values and brace locations of the compressionflange shown in Fig. 8. Comparisons between the two curves showedthat shifting from a symmetric to asymmetric buckling mode shapewill not occur for small values of α (e.g. values between 0.0 and0.6) for any stiffness values of the lateral bracings prior to bucklingof the internal-span. Fig. 12 shows the results of Eqs. (12)–(13) forα ¼ 0:7 in which the FE results were also included. Obviously, the
minimum values for the critical load ratios (i.e. l2=l2e;k) of the internal-span obtained from Eqs. (12)–(13) were in excellent agreement withthe FE results.
Fig. 13 also depicts required bracing stiffness values for the casesif slightly larger buckling length than the corresponding to equiva-lent to immovable supports, le;∞, was desired in design. As it can beconcluded in Fig. 13, to approach fairly close to the largest possiblecritical load of compression flange, a relatively large bracing stiffnesswas required for the cases with small values of αwhich might not beeconomical for particular cases in practice. For this reason, Fig. 13 en-ables design engineers to examine the use of lower stiffness values toachieve slightly lower load carrying capacity than corresponding to“full bracing”.
3. Conclusions
Advanced commercial software packages are widely relied on tomodel complex structures considering nonlinearities, imperfections,etc. in bridge design practice. The results of such numerical analysesshould be checked by other approaches. Simplified models can greatly
12), to an asymmetric buckling mode shape, obtained from Eq. (13), for the compression
Fig. 14. Bridge girders and their cross-sectional properties for the solved example.
Fig. 12. Comparisons between the results of Eqs. (12)–(13) and FE analyses for α ¼ 0:7.
αle;k=Lb Ncr MN½ � γLT , Eq. (5)
Mcr , Eq. (3)MNm½ � FEA (%)
242 H. Mehri et al. / Structures 3 (2015) 236–243
help design engineers to either preliminary size the bridge girders andtheir bracings, or check the FE results. A model was discussed in thispaper which can be used to calculate critical moment values of laterallyrestrained beams at the level of their compression flanges for givenvalues of bracing stiffness. The model related buckling length of com-pression flange of steel girders to their lateral torsional critical moment.For this purpose, exact analytical solutionswere derived to consider theeffects of bending restraints on critical load values, created due to un-equal spanning of bracings. This effect has been neglected in practicewhen the buckling length of compression members is assumed to beequal to the largest distance between the bracing points. The presentedpaper showed that this assumption can give unsafe results for smallbracing stiffness values and can also lead to significant overdesign prob-lem even when relatively soft bracings are used.
Typical design curves were also given as the results of analytical so-lutions for unequally spanned lateral bracings which give required lat-eral bracing stiffness to achieve the largest possible critical loadvalues. One solved example examined the applicability of the approachfor 26 cases of mono-symmetric and doubly-symmetric beams, withdifferent lateral bracing stiffness values and at different locations. Theresults obtained using the model compared very favorably with thosefromFE analyses. Besides, the time to perform the analyses using the ap-proach was comparatively short. In addition, the model demonstratedhigh potential to help in developing and understanding the theory ofbeam and column bracings.
Fig. 13. Lateral bracing stiffness requirements versus bracing location, when less than fullbracing le ¼ le;∞
� �is desired in design.
Acknowledgments
The financial support from “The Lars Erik LundbergsStipendiestiftelse (Dnr 7/2013 and Dnr 2014/05)” to this study isgratefully acknowledged.
Appendix A. Solved example.
Determine critical moment,Mcr, for the beams shown in Fig. 14. Thelength of beams is 60.0 m between the twisting supports for both cross-sections (S1: doubly-symmetric, and S2: mono-symmetric). The valueof α varies between 0.0 (unbraced beam), 0.1, 0.25, and 1.0 (equallyspaced lateral bracings); and stiffness of lateral bracings kLb=N0ð Þ variesbetween 25, 100, and ∞ (immovable lateral support).
Solution:
N0 ¼ π2 2� 105� �
50� 6003=12� �
=600002 � 10−6 ¼ 0:49 MN½ �
i) For laterally braced cases with kLb=N0 ¼ 25:
ii) For laterally braced cases with kLb=N0 ¼ 100:
Fig. 6 Eq. (4)S1 S2 S1 S2 S1 S2
N. B.a 1.00 0.49 1.95 2.11 2.41 2.60 +0.8 −1.10.1 0.79 0.80 1.66 1.77 3.31 3.53 +4.2 +3.10.25 0.57 1.53 1.38 1.45 5.30 5.57 +4.0 +1.60.5 0.43 2.63 1.24 1.28 8.13 8.44 +1.7 −0.21.0 0.42 2.87 1.22 1.26 8.73 9.04 +2.5⁎ +1.4⁎
a N.B.: No lateral bracing.⁎ refers to a two-half-sine wave shape lateral torsional bucking mode of beam, as ob-
served in FEA.
αle;k=LbFig. 6
Ncr MN½ �Eq. (4)
γLT
Eq. (5)Mcr , Eq. (3)MNm½ � FEA (%)
S1 S2 S1 S2 S1 S2
0.1 0.59 1.41 1.41 1.48 4.98 5.24 +5.4 +0.60.25 0.43 2.67 1.23 1.28 8.23 8.53 +2.9 +0.20.5 0.36 3.85 1.17 1.20 11.23 11.56 +2.8 +0.61.0 0.33 4.45 1.15 1.18 12.75 13.07 +3.8⁎⁎ +1.4⁎⁎
⁎⁎ refers to a three-half-sine wave shape of lateral torsional buckling mode of beam, asobserved in FEA.
243H. Mehri et al. / Structures 3 (2015) 236–243
iii) For laterally braced beam cases with kLb=N0 ¼ ∞:
αle;∞=LbEqs. (7)&(12)
Ncr MN½ �Eq. (4)
γLT
Eq. (5)Mcr , Eq. (3)MNm½ � FEA (%)
S1 S2 S1 S2 S1 S2
0.1 0.44 2.55 1.24 1.29 7.92 8.23 +5.1 +2.60.25 0.39 3.24 1.19 1.23 9.69 10.01 +4.4 +1.30.5 0.35 4.03 1.16 1.9 11.68 12.00 +0.6** −0.1**1.0 0.33 4.53 1.14 1.17 12.95 13.28 +5.4** +2.9**
* and ** refer to 2nd and3rd (i.e. two and three half-sinewave shapes, respectively)modesof lateral torsional buckling of beam, as observed in FEA.
It is evident that there are only slight discrepancies between theresults obtained from the proposed approach and the results from FEanalyses, whereas FEA demands significantly more time than the pro-posed approach (which can be done using e.g. an “Excel” sheet for allthe calculations).
References
[1] ABAQUS/CAE. V 6.13 [computer software]. RI: Simulia; 2013.[2] Ålenius M. Finite element modelling of composite bridge stability; 2003.[3] EC3. Design of steel structures — Part 1-1: general rules and rules for buildings. EN
1993–1–1:2005. Stockholm, Sweden: Swedish Standards Institute; 2005.
[4] Egilmez O, Helwig T, et al. Buckling behavior of steel bridge I-girders braced bypermanent metal deck forms. J Bridge Eng 2012;17(4):624–33.
[5] Galambos TV. Guide to stability design criteria for metal structures. 5th ed. JohnWiley & Sons; 1998.
[6] McCann F, Wadee MA, et al. Lateral stability of imperfect discretely braced steelbeams. J Eng Mech 2013;139(10):1341–9.
[7] Mehri H, Crocetti R. Bracing of steel-concrete composite bridge during casting of thedeck. Nordic Steel Construction Conf. 2012. Oslo, Norway; 2012.
[8] Plaut R. Requirements for lateral bracing of columns with two spans. J Struct Eng1993;119(10):2913–31.
[9] Plaut RH, Yang Y-W. Behavior of three-span braced columns with equal or unequalspans. J Struct Eng 1995;121(6):986–94.
[10] Timoshenko S, Gere JM. Theory of elastic stability. New York: McGraw-Hill; 1961.[11] Trahair N, Nethercot D. Bracing requirements in thin-walled structures. Developments
in Thin-Walled StructuresElsevier Applied Science Publishers; 1984. p. 93–130.[12] Vlasov VZ. Thin-walled elastic beams. Jerusalem: Israel program for scientific
translations; 1961.[13] Wang CM, Goh CJ, et al. An energy approach to elastic stability analysis of multiply
braced monosymmetric I-beams. Mech Struct Mach 1989;17(4):415–29.[14] Wang CM, Kitipornchai S, et al. Buckling of braced monosymmetric cantilevers. Int J
Mech Sci 1987;29(5):321–37.[15] Winter G. Lateral bracing of columns and beams. ASCE 1960;125(1):807–26.[16] Yura J, Helwig T, et al. Global lateral buckling of I-shaped girder systems. J Struct Eng
2008;134(9):1487–94.[17] Yura JA. Bracing for stability, state-of-the-art. Proceedings of the 13th Structures
Congress. Part 1 (of 2), April 3, 1995–April 5, 1995, Boston, MA, USAASCE; 1995.[18] Yura JA. Winter's bracing approach revisited. Eng Struct 1996;18(10):821–5.[19] Yura JA. Fundamentals of beam bracing. Eng J 2001:11–26.