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Unequally spaced lateral bracings on compression flanges of steel girders
Structures 3 (2015) 236–243
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Unequally spaced lateral bracings on compression flanges of steel girders Hassan Mehri a,⁎, Roberto Crocetti a, Per Johan Gustafsson b a b
Div. of Structural Engineering, Lund Univ., Box 118, Lund 22100, Sweden Div. of Structural Mechanics, Lund Univ., Box 118, Lund 22100, Sweden
a r t i c l e
i n f o
Article history: Received 26 February 2015 Received in revised form 6 May 2015 Accepted 17 May 2015 Available online 2 July 2015 Keywords: Lateral bracing Stiffness requirement Steel girder
a b s t r a c t 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 from destabilizing forces of compression flanges e.g. near to the twisting supports. A model is presented in this paper, which relates the lateral buckling length of compression flange of steel girders to their lateral torsional buckling moment and can be used to investigate stiffness requirement of lateral bracings applied on the compression flanges between the twisting restraints. Analytical solutions were derived for the effects of bracing locations and bracing stiffness values on buckling length of compression flanges. Moreover, an exact and a simplified solution for the effect of rotational restraint of shorter-spans on critical load value of the compression members with unequally spanned lateral bracings were derived. The model can be suitable for design engineers to preliminary 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 capacity of slender steel columns and beams by limiting their out-of-plane deformations [15]. However, improper restraint against lateral torsional buckling can be detrimental. A number of bridge failures have occurred due to improper lateral bracings. Two examples are: the collapse of The Marcy Bridge [7] in New York, and Y1504 Bridge [2] in Sweden. These bridges were designed with trapezoidal cross-sections and both collapsed due to global lateral torsional buckling during concreting of the deck. No lateral bracings were used in the first case, while stay-inplace corrugated metal sheets were designated to act as lateral stabilizing system in the latter bridge. However, for steel bridge applications, Egilmez et al. [4] showed, both analytically and experimentally, that corrugated metal decks if properly designed and connected to the girders, can significantly reduce lateral deformation of steel girders. Yura et al. [16] derived a simplified expression for global buckling of steel bridge girders which corresponded to the failure mode for the two mentioned bridges. Lateral bracing at partial span near to the abutments can enhance the load carrying capacity of the girders which are prone to global buckling by creating a semi-clamp condition at the supports [16]. Mehri and Crocetti [7] showed that providing relatively “soft” truss bracings along a partial length of bridge span near to the abutments (e.g. α ¼ 0:1 in Fig. 1), the global buckling of The Marcy Bridge could have been avoided; where α is the partial bridge span from the twisting supports at both ends which are laterally braced by means of e.g. either truss ⁎ Corresponding author. Tel.: +46 46 222 7397. E-mail addresses: [email protected] (H. Mehri), [email protected] (R. Crocetti), [email protected] (P.J. Gustafsson).
bracings or corrugated metal decks. An example for the application of the present study can be to estimate critical bending moment value of such bridges by studying required lateral bracing stiffness near to the abutments, see Fig. 1, in order to create semi-clamped boundary conditions. The exact solutions for bracing requirements and load carrying capacity of compression members are only possible for simple cases, with certain boundary and loading conditions. There have been numerous previous studies on bracing requirements of simple beams and columns [17]. Among these studies, there are investigations on critical moment value for a variety of loading and bracing conditions, and different cross-sectional properties. In a number of studies, Timoshenko's energy approach was used to find the optimal locations for bracings of simple beam structures, see e.g. [11,13,14]. Some of the studies have led to predicting conservative values for critical loads and stiffness of bracings that are already included in some code specifications. Finian et al. [6] studied the stability of imperfect steel beams which were restrained by means of a number of discrete elastic bracings, and gave expressions to estimate the magnitudes of bracing forces. Recommendations for critical moment values basically consist of applying a number of coefficients that statistically give lower bound results. For beams for instance, the coefficients are applied to critical moment, obtained from e.g. Timoshenko's approach [10] to consider the effects of different conditions such as moment gradient, load height, cross-sectional symmetries and boundary conditions. Generally, those recommendations have been presented for a limited number of simple cases and can also lead to high discrepancies in the results, especially when a combination of the effects is considered. Winter [15] presented a model with rigid bars, which are hinged at the locations of equally spaced transitional springs, to study the lateral
http://dx.doi.org/10.1016/j.istruc.2015.05.003 2352-0124/© 2015 The Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
H. Mehri et al. / Structures 3 (2015) 236–243
Fig. 1. Global buckling of “narrow” steel girders and the performance of partial-span lateral bracings.
bracing requirements of columns. Winter's model predicts a minimum required (or “ideal”) stiffness value for those particular cases, which is considered to serve equivalent to immovable lateral support. Beyond this threshold stiffness, any increase in brace stiffness will not enhance the critical load of the columns. Galambos [5] discussed column cases with unequal spans. Plaut [8] studied the bracing requirements of columns with single lateral brace at an internal arbitrary point between the pinned or elastic supports. For the mentioned cases, he also showed that the “ideal” brace stiffness only exists when the bracing is at the center of a uniform column at mid-span. Plaut and Yang [9] also studied the behavior of pinned-end columns with two intermediate lateral bracings for two cases: equally spaced bracings, and a case with unequally spaced bracings at a specific location. For a column with three-span lateral bracings, he stated that “The optimal locations of the internal braces are not obvious unless full bracing is possible”. The present paper discusses the indications concerning the optimal locations for unequally spanned lateral bracings which gives the largest critical load for a given brace stiffness value. The brace in this study were placed symmetrically with respect to the midspan of the girders which is normally the case in practice. However, the main purpose of this paper is to analytically investigate the applicability of a proposed simplified model to predict critical moment of laterally braced girders varying the stiffness and location of bracings. This approach can greatly benefit design engineers either before or after performing the final design in order to size the beams and bracings or to check the results obtained from using commercial software. For this purpose, exact and approximate expressions were derived to examine the bending restraint effects of shorter spans on critical load values of beams with unequally spanned lateral bracings. Typical design curves were also given from which critical load values can be determined for different stiffness values and location of symmetrically placed lateral bracings. A curve was also given for the cases in which providing lower stiffness than “full bracing” i.e. bracing stiffness value that serves approximately equal to immovable support is thought to be adequate in design. Finally, a comprehensive example is presented to show the applicability of the proposed approach.
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beams, and the plate of the two dimensional compression flanges. S4R is a common shell element type with four nodes, when modeling of steel plates of columns, beams, and stiffeners is desired. Efforts were undertaken to keep the meshing dimensional ratio equal to unity. At least six elements were used across the width of the flanges. No local buckling was observed in any of the FE analyses. In the linear buckling analyses of the compression flanges, the members were fixed at one end, free to slide in the longitudinal direction at the other end, while free to warp at both ends. The deformations perpendicular to the plane of compression flanges were not permitted in the twodimensional analyses. The three dimensional beams were fixed at one end, free to slide in the longitudinal direction at the other end, while free to warp and restrained against twist at both ends. Bracings were modeled with linearly elastic transitional springs connecting one node of the shell elements to the “ground”. The beam components in the performed FE linear buckling analyses had elastic material properties. Generally speaking, the developed forces in the bracings are relatively little while a significant increase in the resistance can be normally achieved with the relatively soft bracings. As a rule of thumb the bracing forces are normally within the range of 1–2% of the applied force in compression members. Based on the conditions mentioned above, any local effects at the junction of the springs with the shell elements − that might occur in nonlinear analyses with large deformations−were considered negligible. The critical bending moment of a simply supported beam with doubly symmetric cross-section subjected to uniform bending moment about the strongest axis (y–y) can be calculated using the following equation [10]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mcr ¼ ðπ=Lb Þ EIz GJ þ π 2 E2 Iz C w =L2b ;
ð1Þ
where Lb Beam span between the torsional restraints; E and G Young's and shear moduli of beam material, respectively; J, C w , and I z Cross-sectional properties including torsional constant, warping constant, and moment of inertia about weakest axis of beam, respectively, (see Fig. 2). More accurate results can be obtained for I-beams with monosymmetric cross-section considering Iz ¼ Izc þ ðt=cÞIzt [19]; where t, and c are distances from the neutral bending axis of the monosymmetric cross-section to the centroids of tension and compression flanges, respectively; Iz , Izc , and Izt are lateral moments of inertia of the cross-section, the compression flange, and the tension flange, respectively, about the weak axis, i.e. “z” axis in Fig. 2. However, for the purpose of this paper, Iz ≈2I zc can be also used to obtain a reasonably conservative prediction of critical moment values for the common 2
2. Results and discussions
I-shaped steel girder dimensions. Substituting C w with Iz h =4 for doubly-symmetric I-beams (where h is the distance between the centroids of the flanges), GJ ¼ C, and EC w ¼ C 1 , Eq. (1) can be rewritten as:
2.1. Theory and model development
Mcr ¼
In this section of the study, Timoshenko's basic approach [10] for critical moment of simply supported beams under uniform bending moment was used to develop a model which can be used as a general solution for beams that are laterally braced between the twisting supports at the level of their compression flanges. The model relates Timoshenko's beam subjected to equal and opposite bending moments at each end to an equivalent couple forces arising in the flanges of the girders. The model and the derived analytical solutions were verified by means of Finite Element (FE) analyses using ABAQUS commercial software [1]. Quadrilateral, reduced integration, four-node shell element S4R with sufficient meshing was used in the FE analyses to model the flanges and the web of the studied three dimensional
Fig. 2. A model that relates critical moment of beams to critical load of their compression flange.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 π2 EIz h=2L2b 1 þ CL2b =π2 C 1 :
ð2Þ
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Setting γ LT ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ CL2b =π2 C 1 ¼ 1 þ 2C=Ncr h , Eq. (2) can be
written as: Mcr ¼ ðNcr hÞ γ LT :
ð3Þ
Eq. (3) claims that critical moment of I-beams can be approximately modeled by two opposing horizontal forces, γ LT Ncr , applied at the centroids of the flanges, as shown in Fig. 2; where, the compression flange of the beams is regarded as a column with critical load of Ncr ¼ π 2 EIzc =L2b . Table 1 shows the variation of the γLT coefficient with different values of torsional properties (i.e. CL2b =π2 C 1 ); where “γLT ¼ 1” represents beams with “zero” torsional stiffness, i.e. the web and the tension flange give no contribution to controlling lateral torsional deformation of the compression flange, and “γ LT ¼ ∞” represents beams with “infinite” torsional stiffness. Theoretically, knowing the buckling length of the compression flanges, Eq. (3) can also be used to predict the critical moment value for the cases of laterally restrained beams at the level of compression flanges. To generalize Eq. (3), the buckling load of compression flange and γ LT factor can be calculated from Eqs. (4)–(5), respectively. 2
Ncr ¼ π 2 EIzc =le ; γ LT ¼
ð4Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 þ Cle =π2 C 1 ¼ 1 þ 2C=Ncr h
ð5Þ
wherele is the buckling length of the compression flange of the beams, and in this paper is indicated as “le;∞”, “le;k”, or “Lb” which corresponded to cases in which the compression flange of the beams are respectively laterally restrained by means of immovable supports, compression flange of beam is laterally restrained by means of elastic bracings, or the beam is not laterally braced between the twisting supports. Lateral torsional buckling of steel girders involves lateral movement of their compression flange and the twist of the girder cross-section. In the proposed approach presented in Eq. (3), the lateral translational degree-of-freedom of the compression flanges is considered by studying the buckling of the compression flanges (see Eq. (4)). While, the torsional degree-of-freedom of the cross-section along the buckling length of the compression flanges−i.e. “le ” obtained from Eq. (4)−is considered by the γLT coefficient (see Eq. (5)). It must be noted that in the proposed approach the Vlasov condition [12] was assumed for the shape of the cross-section of beams that remains unchanged during buckling analyses. Therefore, no local buckling of cross-section was permitted. 2.1.1. Effects of different loading and boundary conditions Eq. (3) was derived as a general solution, assuming the same boundary and loading conditions as of Eq. (1), i.e. simply supported beam subjected to uniform bending moment. There have been numerous studies on critical moment of simple beams to statistically determine conservative effects of different conditions such as bending moment gradients, loading height and cross-sectional asymmetries on critical moment values obtained by Timoshenko's approach (i.e. Eq. (1)). For instance, for a simply supported doubly-symmetric unbraced beam under a uniformly distributed load on the level of the top flange, the modification 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-
For the purpose of further studies on the proposed approach, the authors suggest an application of buckling length modification factor to account for the effects of distributed load at the level of top flanges, which is a typical case in bridge application e.g. during the construction phase. 2.1.2. Effects of imperfections and nonlinearities Fig. 3 schematically shows a typical buckling curve recommended by Eurocode 3 (2005), which is based on results from numerous laboratory tests. Such curves relate critical compression load,Ncr, or critical bending moment, M cr , values of columns or beams, respectively, to a reduction factor, χ , which can be applied to e.g. plastic capacity of the crosssection in order to calculate the corresponding design values. The effect of different shapes and magnitudes of initial imperfections, residual stresses, material nonlinearities, eccentricities, etc. are conservatively considered in the buckling curves. Thus, using such buckling curves, the corresponding design values can be calculated for given critical load values of beams, or columns obtained from Eq. (3), or Eq. (4), respectively. Winter's rigid bar model [15] gives the minimum required stiffness values for columns with equally spaced lateral bracings. For the cases of unequally spaced bracings, it is proposed that the buckling length can be conservatively considered equal to the length of the largest span [18]. However for the unequally braced columns (or compression flanges), the shorter portions of the beams create some bending restraint to the larger spans due to their different bending stiffness values. In the present study, it is shown that this effect can be significant depending on the bending stiffness ratio of the adjacent spans with unequal lengths. Fig. 3 illustrates the possible overdesign problem which can occur ignoring the mentioned effects; where point “A” represents the critical value obtained approximating buckling length equal to the distance between the bracing points, and point “B” represents the critical value obtained considering the bending restraint effects of the shorter spans; χ A ; and χ B are design reduction factors corresponding to the critical load values marked as “A” and “B”, respectively; N P , or MP are plastic capacities of the column or the beam cross-section; and λrel is lateral or lateral torsional slenderness ratios for the corresponding column 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 that served equal to immovable supports (presented in Section 2.3.), and iii) 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 engineers to preliminary size the cross-section of bridge girders, which are dependent 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
ple beams and can be often difficult to be justified for some specific cases in practice when a combination of effects is being considered. Table 1 Variation of the γ LT coefficient for beams with different torsional properties. CL2b =C 1 γ LT
0.0
0.1
10
100
1000
∞
1.0
1.0
1.4
3.3
10.1
∞
Fig. 3. Schematic illustration of typical design curves for flexural or lateral torsional buckling.
H. Mehri et al. / Structures 3 (2015) 236–243
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Table 2 Variations of γLT values for different IPE profiles, and common built-up steel girders. “Lb ” and “d” are length and depth of beams, respectively.
IPE profiles Built-up girders
Lb =d ¼ 16
Lb =d ¼ 20
Lb =d ¼ 24
1.45–1.60 1.40–2.30
1.60–1.85 1.50–2.60
1.80–2.15 1.60–2.80
bridge girders in steel–concrete composite bridges, and also for some European Standard I-Beam profiles (IPE 100 to IPE 750). In the builtup girders, the dimensions of cross-sections, i.e. the size of the flanges and the webs, were calculated utilizing the commonly used empirical methods for 108 beam cases (varying girder depth from 600 mm to 3000 mm, in 300 mm increments; deck width from 3000 mm to 12000 mm, in 3000 mm increments, and length from 9600 mm to 72000 mm). As can be seen in Table 2, typical γLT values vary within the range of approximately 1.4 to 2.8. In addition, Eq. (3) was used to calculate γ LT values for beam cases (1)–(5) as shown in Fig. 4 varying span lengths between 6 and 36 m; the critical bending moments were obtained using FE buckling analyses. In Fig. 5 γ LT values obtained are plotted against CL2b =C 1 values. The results were also compared to the curve obtained from Eq. (5). The comparison verified that Eq. (5) gives a unique curve for both doublyand 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, much higher stiffness values than the “ideal” stiffness are normally provided to resist lateral deformations of compression members at the brace points. The effect of unequal spanning of lateral bracings on critical load values of compression flange of beams are investigated in this section, assuming that lateral stiffness values of the bracings are large enough to function fairly similar to immovable supports. Fig. 6 illustrates the central-span of the compression flange of a beam which is laterally restrained with immovable supports at a αl distance from either end, as an equivalent column with rotational spring stiffness of βα EI=l at both ends; where EI is the bending stiffness of the cross-section about the strong axis of compression flange, l is the length of the centralspan, 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.
Fig. 5. Values of γ LT for the beam cases: (1)–(5).
Applying boundary conditions EIw″x2¼l
ðwx2¼0 ¼ wx2¼l ¼ 0; EIw″x2¼0 ¼
¼ ðβα EI=lÞθÞ for beam–column “BC” gives:
cl ¼ −βα sinðclÞ=½1 þ cosðclÞ;
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 relationship between α and βα was of interest. For the beam–column “AB” in Fig. 6, applying boundary conditions wx1¼0 ¼ wx1¼αl ¼ 0; EIw″x1¼0 ¼ 0; EIw″x1¼αl ¼ ðβα EI=lÞθ , and dividing moment forceð βα EI=lÞθ by the slope at point B, w0x1¼αl , gives the rotational stiffness at point B: 2
βα ¼ ð3=α ÞðcαlÞ sinðcαlÞ=3½ð sinðcαlÞ−ðcαlÞ cosðcαlÞ;
ð8Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi where cαl ¼ π N=Ncr , and N cr is the buckling load of the beam– column “AB”. Eq. (8) gives an exact solution for the magnitude of bending restraint of the side-spans to the central-span. Substituting the resulted value of βα from Eq. (8) into Eq. (7) gives the buckling length of the compression member, i.e. le;∞ , shown in Fig. 6. However, a simplified solution for the relationship between α and βα , rather than Eq. (8) can be also derived. 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 ¼ 1=ð1−N=N cr Þ to consider the second-order effects of the compression force on the deformations; where N and Ncr are approximately π 2 EI=l
2
Fig. 6. Bending restraint of side-spans with shorter length to the central-span of compression flange restrained by means of immovable lateral supports.
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H. Mehri et al. / Structures 3 (2015) 236–243 2
andπ2 EI=α 2 l , respectively. Thus, the relationship between βα andα can be approximated by the following equation to consider the secondorder effects:
2
βα ≈3=α 1−α :
ð10Þ
Comparisons between the results of Eqs. (8)–(10) showed that the approximate solution obtained from Eq. (10) was in excellent agreement with the exact solution obtained from Eq. (8) (see Fig. 7). Therefore, by substituting the βα values obtained from Eq. (10) (for a given value of α) into Eq. (7), the buckling load of the laterally restrained compression flange can be calculated when lateral stiffness of the braces is sufficiently large. Table 3 shows the results of Eq. (7) for a number of βα values between “zero stiffness”, which represents nil rotational stiffness provided by the side-spans to the central-span, and “infinite stiffness” which means that the 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 significant decreases in the buckling length ratios, i.e. le;∞ =l, of the compression flange in the central-span, especially for the small values of α. In bridge design practice, it is often assumed that the buckling length of the braced girder is conservatively equal to the distance between the bracing points, if adequate stiffness larger than the recommended value is provided for the bracings [3]. The results in Table 3 show that this assumption can lead to a considerable over-design problem (it can be as 2 2 large as a factor of 4, see l =le;∞ for small values of α).
Table 3 Buckling length of compression flange with immovable restraints placed at partial length (αl). α βα cl le;∞ =l le;∞ =Lb
0.2 13.6 5.51 0.57 0.41
0.3 8.22 5.16 0.61 0.38
0.4 5.37 4.82 0.65 0.36
0.5 3.58 4.49 0.70 0.35
0.6 2.36 4.17 0.75 0.34
0.7 1.48 3.88 0.81 0.34
0.8 0.85 3.61 0.87 0.34
0.9 0.37 3.36 0.93 0.33
1.0 0.0 π 1.00 0.33
a This value represented lateral restraints at a distance which was “very” close to the twisting supports but not equal to “zero”.
Setting the determinant of the stiffness matrix to zero gives an exact solution for the normalized relationship between bracing stiffness and corresponding critical load values as in the following equation: 3
3
kl =EI ¼ −ξðclÞ ½ cosðcαlÞ cotð0:5clÞ− sinðcαlÞ=½ξcαl sinðcαlÞ þ cotð0:5clÞ:
ð12Þ Similarly, for the asymmetric mode shape, applying boundary conditions wx1¼0 ¼ EIw″x1¼0 ¼ 0; wx1¼αl ¼ δ; w0x1¼αl ¼ θ for sidespan beam, i.e. beam–column “AB”, and
ðwx2¼0 ¼ δ; w0x2¼0 ¼
EIw″x2¼0:5l
¼ 0Þ for the internal-span beam, i.e. beam– θ; wx2¼0:5l ¼ column “BC”, Eq. (13) gives a normalized solution between the bracing stiffness and the corresponding critical load. For a given lateral stiffness value, critical load is the minimum value obtained from symmetric and asymmetric calculations using Eqs. (12)–(13). 3
3
ð13Þ
For practical uses, the method should be expanded for different stiffness values of lateral bracings. Bracings were represented as linearly elastic translational springs at symmetric locations with respect to the mid-span of the beams, which is mostly the cases in practice (see Fig. 8). Depending on the magnitude of the lateral bracing stiffness, compression flanges of beams can deform in either symmetric or asymmetric shape with respect to the center-span (see Fig. 8). For the symmetric deformation shape, applying boundary conditions wx1¼0 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; w0x2¼0:5l ¼ EIw‴x2¼0:5l ¼ 0 for the internal-span beam, i.e. beam “BC”, Eq. (6) gives: −ξc3 αl cosðcαlÞ−c2 ξc3 cosðcαlÞ þ k=EI ξc2 sinðcαlÞ −ξc2 lα sinðcαlÞ−c cotð0:5clÞ
0.1 28.2 5.88 0.53 0.45
kl =EI ¼ −ξϑðclÞ ½ðα þ 0:5Þ sinðcαl þ 0:5clÞ=½αξ sinðcαlÞ þ 0:5ϑ sinð0:5clÞ
2.4. Beams with elastic lateral restraints
≈0.0a ∞ ≈2π ≈0.50 ≈0.50
δ 0 ¼ θ 0
ð11Þ where ξ ¼ 1=½ sinðcαlÞ−cαl cosðcαlÞ.
Fig. 7. Bending restraint of side-spans with shorter length to buckling length of centralspan, obtained from Eqs. (8)–(10).
where ϑ ¼ 1=½ sinð0:5clÞ−0:5cl cosð0:5clÞ. A typical design curve for the relationship between lateral bracings stiffness and buckling length of the central-span for different locations of lateral bracings is shown in Fig. 9, as the results of linear FE buckling analyses. Comparison between the FE results and the results of Eq. (12) are made in Fig. 10; where the solid lines represented the FE results (for kLb =N0 values between 0.0 and 200.0), and the dashed lines represented the results of Eq. (12), (for kLb =N0 values between 0.0 and 800.0); and N0 ¼ π2 EI=L2b . The results obtained from Eq. (12) were expanded for more values of stiffness than the FE data to show the “excellent” agreement between the two methods. The results showed that the relatively small values of stiffness for symmetrically placed lateral bracings, can significantly decrease the buckling length of the compression flange, especially for the cases where α ≥0:25 (see Fig. 9). For the case of α ¼ 1:0; the points indicated with (i) and (ii) in Fig. 9 represented minimum bracing stiffness values (kLb =N0 being 13.2 and 81, respectively) to change the buckling shapes from a single half-sine wave to
Fig. 8. Compression flange of beams with symmetrically placed lateral elastic bracings.
H. Mehri et al. / Structures 3 (2015) 236–243
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α ¼ 1:0, can be calculated by means of Winter's rigid bars model that are hinged at the locations of braces. Critical load values of the beam shown in Fig. 8 was normalized to critical load values when − ignoring bending restraint of the side spans−the buckling length is “conservatively” assumed to be equal to 2
Fig. 9. Relationship between buckling length of the compression member shown in Fig. 8 and stiffness of lateral bracings.
2
the distance between the bracing points, i.e. l =le;k in Fig. 10. The results shown in Fig. 10 verified that this assumption gives unsafe results for relatively small values of brace stiffness for all locations of bracing. On the other hand, providing relatively soft bracings, e.g. as little as kLb =N0 ¼ 100 , significantly enhanced the critical load values. This capacity is currently ignored in practice when the buckling length is assumed to be equal to “l” [3]. Fig. 11 graphs the results obtained from Eqs. (12)–(13) for different brace stiffness values and brace locations of the compression flange shown in Fig. 8. Comparisons between the two curves showed that shifting from a symmetric to asymmetric buckling mode shape will not occur for small values of α (e.g. values between 0.0 and 0.6) for any stiffness values of the lateral bracings prior to buckling of the internal-span. Fig. 12 shows the results of Eqs. (12)–(13) for α ¼ 0:7 in which the FE results were also included. Obviously, the 2
2
minimum values for the critical load ratios (i.e. l =le;k ) of the internalspan obtained from Eqs. (12)–(13) were in excellent agreement with the FE results. Fig. 13 also depicts required bracing stiffness values for the cases if slightly larger buckling length than the corresponding to equivalent to immovable supports, le;∞ , was desired in design. As it can be concluded in Fig. 13, to approach fairly close to the largest possible critical load of compression flange, a relatively large bracing stiffness was required for the cases with small values of α which might not be economical for particular cases in practice. For this reason, Fig. 13 enables design engineers to examine the use of lower stiffness values to achieve slightly lower load carrying capacity than corresponding to “full bracing”. Fig. 10. The curves showing ratios between critical loads of compression member shown in Fig. 8 and critical load of the same member assuming the buckling length to be equal to the distance between the bracing points.
3. Conclusions
two and three half-sine waves, respectively, where three half-sine wave corresponds to the “ideal” bracing stiffness of the column [15]. These thresholds values for columns with equally spaced lateral bracings i.e.,
Advanced commercial software packages are widely relied on to model complex structures considering nonlinearities, imperfections, etc. in bridge design practice. The results of such numerical analyses should be checked by other approaches. Simplified models can greatly
Fig. 11. Investigations on shifting from a symmetric buckling mode shape, obtained from Eq. (12), to an asymmetric buckling mode shape, obtained from Eq. (13), for the compression flanges shown in Fig. 8.
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H. Mehri et al. / Structures 3 (2015) 236–243
Acknowledgments The financial support from “The Lars Erik Lundbergs Stipendiestiftelse (Dnr 7/2013 and Dnr 2014/05)” to this study is gratefully acknowledged.
Appendix A. Solved example. Determine critical moment, M cr , for the beams shown in Fig. 14. The length of beams is 60.0 m between the twisting supports for both crosssections (S1: doubly-symmetric, and S2: mono-symmetric). The value of α varies between 0.0 (unbraced beam), 0.1, 0.25, and 1.0 (equally spaced lateral bracings); and stiffness of lateral bracings ðkLb =N0 Þ varies between 25, 100, and ∞ (immovable lateral support). Fig. 12. Comparisons between the results of Eqs. (12)–(13) and FE analyses for α ¼ 0:7.
help design engineers to either preliminary size the bridge girders and their bracings, or check the FE results. A model was discussed in this paper which can be used to calculate critical moment values of laterally restrained beams at the level of their compression flanges for given values of bracing stiffness. The model related buckling length of compression flange of steel girders to their lateral torsional critical moment. For this purpose, exact analytical solutions were derived to consider the effects of bending restraints on critical load values, created due to unequal spanning of bracings. This effect has been neglected in practice when the buckling length of compression members is assumed to be equal to the largest distance between the bracing points. The presented paper showed that this assumption can give unsafe results for small bracing stiffness values and can also lead to significant overdesign problem even when relatively soft bracings are used. Typical design curves were also given as the results of analytical solutions for unequally spanned lateral bracings which give required lateral bracing stiffness to achieve the largest possible critical load values. One solved example examined the applicability of the approach for 26 cases of mono-symmetric and doubly-symmetric beams, with different lateral bracing stiffness values and at different locations. The results obtained using the model compared very favorably with those from FE analyses. Besides, the time to perform the analyses using the approach was comparatively short. In addition, the model demonstrated high potential to help in developing and understanding the theory of beam and column bracings.
Fig. 14. Bridge girders and their cross-sectional properties for the solved example.
Solution: N0 ¼ π2 2 105 50 6003 =12 =600002 10−6 ¼ 0:49 ½MN i) For laterally braced cases with kLb =N0 ¼ 25:
α
le;k =Lb Fig. 6
N cr ½MN Eq. (4)
N. B.a 0.1 0.25 0.5 1.0
1.00 0.79 0.57 0.43 0.42
0.49 0.80 1.53 2.63 2.87
γ LT , Eq. (5)
M cr , Eq. (3) ½MNm
FEA (%)
S1
S2
S1
S2
S1
S2
1.95 1.66 1.38 1.24 1.22
2.11 1.77 1.45 1.28 1.26
2.41 3.31 5.30 8.13 8.73
2.60 3.53 5.57 8.44 9.04
+0.8 +4.2 +4.0 +1.7 +2.5⁎
−1.1 +3.1 +1.6 −0.2 +1.4⁎
a N.B.: No lateral bracing. ⁎ refers to a two-half-sine wave shape lateral torsional bucking mode of beam, as observed in FEA.
ii) For laterally braced cases with kLb =N0 ¼ 100:
α
0.1 0.25 0.5 1.0 Fig. 13. Lateral bracing stiffness requirements versus bracing location, when less than full bracing le ¼ le;∞ is desired in design.
le;k =Lb Fig. 6
N cr ½MN Eq. (4)
0.59 0.43 0.36 0.33
1.41 2.67 3.85 4.45
γ LT Eq. (5)
M cr , Eq. (3) ½MNm
FEA (%)
S1
S2
S1
S2
S1
S2
1.41 1.23 1.17 1.15
1.48 1.28 1.20 1.18
4.98 8.23 11.23 12.75
5.24 8.53 11.56 13.07
+5.4 +2.9 +2.8 +3.8⁎⁎
+0.6 +0.2 +0.6 +1.4⁎⁎
⁎⁎ refers to a three-half-sine wave shape of lateral torsional buckling mode of beam, as observed in FEA.
H. Mehri et al. / Structures 3 (2015) 236–243
iii) For laterally braced beam cases with kLb =N0 ¼ ∞:
α
le;∞ =Lb Eqs. (7)&(12)
Ncr ½MN Eq. (4)
0.1 0.25 0.5 1.0
0.44 0.39 0.35 0.33
2.55 3.24 4.03 4.53
γ LT Eq. (5)
M cr , Eq. (3) ½MNm
FEA (%)
S1
S2
S1
S2
S1
S2
1.24 1.19 1.16 1.14
1.29 1.23 1.9 1.17
7.92 9.69 11.68 12.95
8.23 10.01 12.00 13.28
+5.1 +4.4 +0.6** +5.4**
+2.6 +1.3 −0.1** +2.9**
* and ** refer to 2nd and 3rd (i.e. two and three half-sine wave shapes, respectively) modes of lateral torsional buckling of beam, as observed in FEA.
It is evident that there are only slight discrepancies between the results obtained from the proposed approach and the results from FE analyses, whereas FEA demands significantly more time than the proposed approach (which can be done using e.g. an “Excel” sheet for all the 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.
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[4] Egilmez O, Helwig T, et al. Buckling behavior of steel bridge I-girders braced by permanent metal deck forms. J Bridge Eng 2012;17(4):624–33. [5] Galambos TV. Guide to stability design criteria for metal structures. 5th ed. John Wiley & Sons; 1998. [6] McCann F, Wadee MA, et al. Lateral stability of imperfect discretely braced steel beams. J Eng Mech 2013;139(10):1341–9. [7] Mehri H, Crocetti R. Bracing of steel-concrete composite bridge during casting of the deck. Nordic Steel Construction Conf. 2012. Oslo, Norway; 2012. [8] Plaut R. Requirements for lateral bracing of columns with two spans. J Struct Eng 1993;119(10):2913–31. [9] Plaut RH, Yang Y-W. Behavior of three-span braced columns with equal or unequal spans. 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.
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Structures journal homepage: http://www.elsevier.com/locate/structures
Unequally spaced lateral bracings on compression flanges of steel girders Hassan Mehri a,⁎, Roberto Crocetti a, Per Johan Gustafsson b a b
Div. of Structural Engineering, Lund Univ., Box 118, Lund 22100, Sweden Div. of Structural Mechanics, Lund Univ., Box 118, Lund 22100, Sweden
a r t i c l e
i n f o
Article history: Received 26 February 2015 Received in revised form 6 May 2015 Accepted 17 May 2015 Available online 2 July 2015 Keywords: Lateral bracing Stiffness requirement Steel girder
a b s t r a c t 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 from destabilizing forces of compression flanges e.g. near to the twisting supports. A model is presented in this paper, which relates the lateral buckling length of compression flange of steel girders to their lateral torsional buckling moment and can be used to investigate stiffness requirement of lateral bracings applied on the compression flanges between the twisting restraints. Analytical solutions were derived for the effects of bracing locations and bracing stiffness values on buckling length of compression flanges. Moreover, an exact and a simplified solution for the effect of rotational restraint of shorter-spans on critical load value of the compression members with unequally spanned lateral bracings were derived. The model can be suitable for design engineers to preliminary 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 capacity of slender steel columns and beams by limiting their out-of-plane deformations [15]. However, improper restraint against lateral torsional buckling can be detrimental. A number of bridge failures have occurred due to improper lateral bracings. Two examples are: the collapse of The Marcy Bridge [7] in New York, and Y1504 Bridge [2] in Sweden. These bridges were designed with trapezoidal cross-sections and both collapsed due to global lateral torsional buckling during concreting of the deck. No lateral bracings were used in the first case, while stay-inplace corrugated metal sheets were designated to act as lateral stabilizing system in the latter bridge. However, for steel bridge applications, Egilmez et al. [4] showed, both analytically and experimentally, that corrugated metal decks if properly designed and connected to the girders, can significantly reduce lateral deformation of steel girders. Yura et al. [16] derived a simplified expression for global buckling of steel bridge girders which corresponded to the failure mode for the two mentioned bridges. Lateral bracing at partial span near to the abutments can enhance the load carrying capacity of the girders which are prone to global buckling by creating a semi-clamp condition at the supports [16]. Mehri and Crocetti [7] showed that providing relatively “soft” truss bracings along a partial length of bridge span near to the abutments (e.g. α ¼ 0:1 in Fig. 1), the global buckling of The Marcy Bridge could have been avoided; where α is the partial bridge span from the twisting supports at both ends which are laterally braced by means of e.g. either truss ⁎ Corresponding author. Tel.: +46 46 222 7397. E-mail addresses: [email protected] (H. Mehri), [email protected] (R. Crocetti), [email protected] (P.J. Gustafsson).
bracings or corrugated metal decks. An example for the application of the present study can be to estimate critical bending moment value of such bridges by studying required lateral bracing stiffness near to the abutments, see Fig. 1, in order to create semi-clamped boundary conditions. The exact solutions for bracing requirements and load carrying capacity of compression members are only possible for simple cases, with certain boundary and loading conditions. There have been numerous previous studies on bracing requirements of simple beams and columns [17]. Among these studies, there are investigations on critical moment value for a variety of loading and bracing conditions, and different cross-sectional properties. In a number of studies, Timoshenko's energy approach was used to find the optimal locations for bracings of simple beam structures, see e.g. [11,13,14]. Some of the studies have led to predicting conservative values for critical loads and stiffness of bracings that are already included in some code specifications. Finian et al. [6] studied the stability of imperfect steel beams which were restrained by means of a number of discrete elastic bracings, and gave expressions to estimate the magnitudes of bracing forces. Recommendations for critical moment values basically consist of applying a number of coefficients that statistically give lower bound results. For beams for instance, the coefficients are applied to critical moment, obtained from e.g. Timoshenko's approach [10] to consider the effects of different conditions such as moment gradient, load height, cross-sectional symmetries and boundary conditions. Generally, those recommendations have been presented for a limited number of simple cases and can also lead to high discrepancies in the results, especially when a combination of the effects is considered. Winter [15] presented a model with rigid bars, which are hinged at the locations of equally spaced transitional springs, to study the lateral
http://dx.doi.org/10.1016/j.istruc.2015.05.003 2352-0124/© 2015 The Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
H. Mehri et al. / Structures 3 (2015) 236–243
Fig. 1. Global buckling of “narrow” steel girders and the performance of partial-span lateral bracings.
bracing requirements of columns. Winter's model predicts a minimum required (or “ideal”) stiffness value for those particular cases, which is considered to serve equivalent to immovable lateral support. Beyond this threshold stiffness, any increase in brace stiffness will not enhance the critical load of the columns. Galambos [5] discussed column cases with unequal spans. Plaut [8] studied the bracing requirements of columns with single lateral brace at an internal arbitrary point between the pinned or elastic supports. For the mentioned cases, he also showed that the “ideal” brace stiffness only exists when the bracing is at the center of a uniform column at mid-span. Plaut and Yang [9] also studied the behavior of pinned-end columns with two intermediate lateral bracings for two cases: equally spaced bracings, and a case with unequally spaced bracings at a specific location. For a column with three-span lateral bracings, he stated that “The optimal locations of the internal braces are not obvious unless full bracing is possible”. The present paper discusses the indications concerning the optimal locations for unequally spanned lateral bracings which gives the largest critical load for a given brace stiffness value. The brace in this study were placed symmetrically with respect to the midspan of the girders which is normally the case in practice. However, the main purpose of this paper is to analytically investigate the applicability of a proposed simplified model to predict critical moment of laterally braced girders varying the stiffness and location of bracings. This approach can greatly benefit design engineers either before or after performing the final design in order to size the beams and bracings or to check the results obtained from using commercial software. For this purpose, exact and approximate expressions were derived to examine the bending restraint effects of shorter spans on critical load values of beams with unequally spanned lateral bracings. Typical design curves were also given from which critical load values can be determined for different stiffness values and location of symmetrically placed lateral bracings. A curve was also given for the cases in which providing lower stiffness than “full bracing” i.e. bracing stiffness value that serves approximately equal to immovable support is thought to be adequate in design. Finally, a comprehensive example is presented to show the applicability of the proposed approach.
237
beams, and the plate of the two dimensional compression flanges. S4R is a common shell element type with four nodes, when modeling of steel plates of columns, beams, and stiffeners is desired. Efforts were undertaken to keep the meshing dimensional ratio equal to unity. At least six elements were used across the width of the flanges. No local buckling was observed in any of the FE analyses. In the linear buckling analyses of the compression flanges, the members were fixed at one end, free to slide in the longitudinal direction at the other end, while free to warp at both ends. The deformations perpendicular to the plane of compression flanges were not permitted in the twodimensional analyses. The three dimensional beams were fixed at one end, free to slide in the longitudinal direction at the other end, while free to warp and restrained against twist at both ends. Bracings were modeled with linearly elastic transitional springs connecting one node of the shell elements to the “ground”. The beam components in the performed FE linear buckling analyses had elastic material properties. Generally speaking, the developed forces in the bracings are relatively little while a significant increase in the resistance can be normally achieved with the relatively soft bracings. As a rule of thumb the bracing forces are normally within the range of 1–2% of the applied force in compression members. Based on the conditions mentioned above, any local effects at the junction of the springs with the shell elements − that might occur in nonlinear analyses with large deformations−were considered negligible. The critical bending moment of a simply supported beam with doubly symmetric cross-section subjected to uniform bending moment about the strongest axis (y–y) can be calculated using the following equation [10]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mcr ¼ ðπ=Lb Þ EIz GJ þ π 2 E2 Iz C w =L2b ;
ð1Þ
where Lb Beam span between the torsional restraints; E and G Young's and shear moduli of beam material, respectively; J, C w , and I z Cross-sectional properties including torsional constant, warping constant, and moment of inertia about weakest axis of beam, respectively, (see Fig. 2). More accurate results can be obtained for I-beams with monosymmetric cross-section considering Iz ¼ Izc þ ðt=cÞIzt [19]; where t, and c are distances from the neutral bending axis of the monosymmetric cross-section to the centroids of tension and compression flanges, respectively; Iz , Izc , and Izt are lateral moments of inertia of the cross-section, the compression flange, and the tension flange, respectively, about the weak axis, i.e. “z” axis in Fig. 2. However, for the purpose of this paper, Iz ≈2I zc can be also used to obtain a reasonably conservative prediction of critical moment values for the common 2
2. Results and discussions
I-shaped steel girder dimensions. Substituting C w with Iz h =4 for doubly-symmetric I-beams (where h is the distance between the centroids of the flanges), GJ ¼ C, and EC w ¼ C 1 , Eq. (1) can be rewritten as:
2.1. Theory and model development
Mcr ¼
In this section of the study, Timoshenko's basic approach [10] for critical moment of simply supported beams under uniform bending moment was used to develop a model which can be used as a general solution for beams that are laterally braced between the twisting supports at the level of their compression flanges. The model relates Timoshenko's beam subjected to equal and opposite bending moments at each end to an equivalent couple forces arising in the flanges of the girders. The model and the derived analytical solutions were verified by means of Finite Element (FE) analyses using ABAQUS commercial software [1]. Quadrilateral, reduced integration, four-node shell element S4R with sufficient meshing was used in the FE analyses to model the flanges and the web of the studied three dimensional
Fig. 2. A model that relates critical moment of beams to critical load of their compression flange.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 π2 EIz h=2L2b 1 þ CL2b =π2 C 1 :
ð2Þ
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H. Mehri et al. / Structures 3 (2015) 236–243
Setting γ LT ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ CL2b =π2 C 1 ¼ 1 þ 2C=Ncr h , Eq. (2) can be
written as: Mcr ¼ ðNcr hÞ γ LT :
ð3Þ
Eq. (3) claims that critical moment of I-beams can be approximately modeled by two opposing horizontal forces, γ LT Ncr , applied at the centroids of the flanges, as shown in Fig. 2; where, the compression flange of the beams is regarded as a column with critical load of Ncr ¼ π 2 EIzc =L2b . Table 1 shows the variation of the γLT coefficient with different values of torsional properties (i.e. CL2b =π2 C 1 ); where “γLT ¼ 1” represents beams with “zero” torsional stiffness, i.e. the web and the tension flange give no contribution to controlling lateral torsional deformation of the compression flange, and “γ LT ¼ ∞” represents beams with “infinite” torsional stiffness. Theoretically, knowing the buckling length of the compression flanges, Eq. (3) can also be used to predict the critical moment value for the cases of laterally restrained beams at the level of compression flanges. To generalize Eq. (3), the buckling load of compression flange and γ LT factor can be calculated from Eqs. (4)–(5), respectively. 2
Ncr ¼ π 2 EIzc =le ; γ LT ¼
ð4Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 þ Cle =π2 C 1 ¼ 1 þ 2C=Ncr h
ð5Þ
wherele is the buckling length of the compression flange of the beams, and in this paper is indicated as “le;∞”, “le;k”, or “Lb” which corresponded to cases in which the compression flange of the beams are respectively laterally restrained by means of immovable supports, compression flange of beam is laterally restrained by means of elastic bracings, or the beam is not laterally braced between the twisting supports. Lateral torsional buckling of steel girders involves lateral movement of their compression flange and the twist of the girder cross-section. In the proposed approach presented in Eq. (3), the lateral translational degree-of-freedom of the compression flanges is considered by studying the buckling of the compression flanges (see Eq. (4)). While, the torsional degree-of-freedom of the cross-section along the buckling length of the compression flanges−i.e. “le ” obtained from Eq. (4)−is considered by the γLT coefficient (see Eq. (5)). It must be noted that in the proposed approach the Vlasov condition [12] was assumed for the shape of the cross-section of beams that remains unchanged during buckling analyses. Therefore, no local buckling of cross-section was permitted. 2.1.1. Effects of different loading and boundary conditions Eq. (3) was derived as a general solution, assuming the same boundary and loading conditions as of Eq. (1), i.e. simply supported beam subjected to uniform bending moment. There have been numerous studies on critical moment of simple beams to statistically determine conservative effects of different conditions such as bending moment gradients, loading height and cross-sectional asymmetries on critical moment values obtained by Timoshenko's approach (i.e. Eq. (1)). For instance, for a simply supported doubly-symmetric unbraced beam under a uniformly distributed load on the level of the top flange, the modification 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-
For the purpose of further studies on the proposed approach, the authors suggest an application of buckling length modification factor to account for the effects of distributed load at the level of top flanges, which is a typical case in bridge application e.g. during the construction phase. 2.1.2. Effects of imperfections and nonlinearities Fig. 3 schematically shows a typical buckling curve recommended by Eurocode 3 (2005), which is based on results from numerous laboratory tests. Such curves relate critical compression load,Ncr, or critical bending moment, M cr , values of columns or beams, respectively, to a reduction factor, χ , which can be applied to e.g. plastic capacity of the crosssection in order to calculate the corresponding design values. The effect of different shapes and magnitudes of initial imperfections, residual stresses, material nonlinearities, eccentricities, etc. are conservatively considered in the buckling curves. Thus, using such buckling curves, the corresponding design values can be calculated for given critical load values of beams, or columns obtained from Eq. (3), or Eq. (4), respectively. Winter's rigid bar model [15] gives the minimum required stiffness values for columns with equally spaced lateral bracings. For the cases of unequally spaced bracings, it is proposed that the buckling length can be conservatively considered equal to the length of the largest span [18]. However for the unequally braced columns (or compression flanges), the shorter portions of the beams create some bending restraint to the larger spans due to their different bending stiffness values. In the present study, it is shown that this effect can be significant depending on the bending stiffness ratio of the adjacent spans with unequal lengths. Fig. 3 illustrates the possible overdesign problem which can occur ignoring the mentioned effects; where point “A” represents the critical value obtained approximating buckling length equal to the distance between the bracing points, and point “B” represents the critical value obtained considering the bending restraint effects of the shorter spans; χ A ; and χ B are design reduction factors corresponding to the critical load values marked as “A” and “B”, respectively; N P , or MP are plastic capacities of the column or the beam cross-section; and λrel is lateral or lateral torsional slenderness ratios for the corresponding column 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 that served equal to immovable supports (presented in Section 2.3.), and iii) 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 engineers to preliminary size the cross-section of bridge girders, which are dependent 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
ple beams and can be often difficult to be justified for some specific cases in practice when a combination of effects is being considered. Table 1 Variation of the γ LT coefficient for beams with different torsional properties. CL2b =C 1 γ LT
0.0
0.1
10
100
1000
∞
1.0
1.0
1.4
3.3
10.1
∞
Fig. 3. Schematic illustration of typical design curves for flexural or lateral torsional buckling.
H. Mehri et al. / Structures 3 (2015) 236–243
239
Table 2 Variations of γLT values for different IPE profiles, and common built-up steel girders. “Lb ” and “d” are length and depth of beams, respectively.
IPE profiles Built-up girders
Lb =d ¼ 16
Lb =d ¼ 20
Lb =d ¼ 24
1.45–1.60 1.40–2.30
1.60–1.85 1.50–2.60
1.80–2.15 1.60–2.80
bridge girders in steel–concrete composite bridges, and also for some European Standard I-Beam profiles (IPE 100 to IPE 750). In the builtup girders, the dimensions of cross-sections, i.e. the size of the flanges and the webs, were calculated utilizing the commonly used empirical methods for 108 beam cases (varying girder depth from 600 mm to 3000 mm, in 300 mm increments; deck width from 3000 mm to 12000 mm, in 3000 mm increments, and length from 9600 mm to 72000 mm). As can be seen in Table 2, typical γLT values vary within the range of approximately 1.4 to 2.8. In addition, Eq. (3) was used to calculate γ LT values for beam cases (1)–(5) as shown in Fig. 4 varying span lengths between 6 and 36 m; the critical bending moments were obtained using FE buckling analyses. In Fig. 5 γ LT values obtained are plotted against CL2b =C 1 values. The results were also compared to the curve obtained from Eq. (5). The comparison verified that Eq. (5) gives a unique curve for both doublyand 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, much higher stiffness values than the “ideal” stiffness are normally provided to resist lateral deformations of compression members at the brace points. The effect of unequal spanning of lateral bracings on critical load values of compression flange of beams are investigated in this section, assuming that lateral stiffness values of the bracings are large enough to function fairly similar to immovable supports. Fig. 6 illustrates the central-span of the compression flange of a beam which is laterally restrained with immovable supports at a αl distance from either end, as an equivalent column with rotational spring stiffness of βα EI=l at both ends; where EI is the bending stiffness of the cross-section about the strong axis of compression flange, l is the length of the centralspan, 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.
Fig. 5. Values of γ LT for the beam cases: (1)–(5).
Applying boundary conditions EIw″x2¼l
ðwx2¼0 ¼ wx2¼l ¼ 0; EIw″x2¼0 ¼
¼ ðβα EI=lÞθÞ for beam–column “BC” gives:
cl ¼ −βα sinðclÞ=½1 þ cosðclÞ;
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 relationship between α and βα was of interest. For the beam–column “AB” in Fig. 6, applying boundary conditions wx1¼0 ¼ wx1¼αl ¼ 0; EIw″x1¼0 ¼ 0; EIw″x1¼αl ¼ ðβα EI=lÞθ , and dividing moment forceð βα EI=lÞθ by the slope at point B, w0x1¼αl , gives the rotational stiffness at point B: 2
βα ¼ ð3=α ÞðcαlÞ sinðcαlÞ=3½ð sinðcαlÞ−ðcαlÞ cosðcαlÞ;
ð8Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffi where cαl ¼ π N=Ncr , and N cr is the buckling load of the beam– column “AB”. Eq. (8) gives an exact solution for the magnitude of bending restraint of the side-spans to the central-span. Substituting the resulted value of βα from Eq. (8) into Eq. (7) gives the buckling length of the compression member, i.e. le;∞ , shown in Fig. 6. However, a simplified solution for the relationship between α and βα , rather than Eq. (8) can be also derived. 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 ¼ 1=ð1−N=N cr Þ to consider the second-order effects of the compression force on the deformations; where N and Ncr are approximately π 2 EI=l
2
Fig. 6. Bending restraint of side-spans with shorter length to the central-span of compression flange restrained by means of immovable lateral supports.
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andπ2 EI=α 2 l , respectively. Thus, the relationship between βα andα can be approximated by the following equation to consider the secondorder effects:
2
βα ≈3=α 1−α :
ð10Þ
Comparisons between the results of Eqs. (8)–(10) showed that the approximate solution obtained from Eq. (10) was in excellent agreement with the exact solution obtained from Eq. (8) (see Fig. 7). Therefore, by substituting the βα values obtained from Eq. (10) (for a given value of α) into Eq. (7), the buckling load of the laterally restrained compression flange can be calculated when lateral stiffness of the braces is sufficiently large. Table 3 shows the results of Eq. (7) for a number of βα values between “zero stiffness”, which represents nil rotational stiffness provided by the side-spans to the central-span, and “infinite stiffness” which means that the 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 significant decreases in the buckling length ratios, i.e. le;∞ =l, of the compression flange in the central-span, especially for the small values of α. In bridge design practice, it is often assumed that the buckling length of the braced girder is conservatively equal to the distance between the bracing points, if adequate stiffness larger than the recommended value is provided for the bracings [3]. The results in Table 3 show that this assumption can lead to a considerable over-design problem (it can be as 2 2 large as a factor of 4, see l =le;∞ for small values of α).
Table 3 Buckling length of compression flange with immovable restraints placed at partial length (αl). α βα cl le;∞ =l le;∞ =Lb
0.2 13.6 5.51 0.57 0.41
0.3 8.22 5.16 0.61 0.38
0.4 5.37 4.82 0.65 0.36
0.5 3.58 4.49 0.70 0.35
0.6 2.36 4.17 0.75 0.34
0.7 1.48 3.88 0.81 0.34
0.8 0.85 3.61 0.87 0.34
0.9 0.37 3.36 0.93 0.33
1.0 0.0 π 1.00 0.33
a This value represented lateral restraints at a distance which was “very” close to the twisting supports but not equal to “zero”.
Setting the determinant of the stiffness matrix to zero gives an exact solution for the normalized relationship between bracing stiffness and corresponding critical load values as in the following equation: 3
3
kl =EI ¼ −ξðclÞ ½ cosðcαlÞ cotð0:5clÞ− sinðcαlÞ=½ξcαl sinðcαlÞ þ cotð0:5clÞ:
ð12Þ Similarly, for the asymmetric mode shape, applying boundary conditions wx1¼0 ¼ EIw″x1¼0 ¼ 0; wx1¼αl ¼ δ; w0x1¼αl ¼ θ for sidespan beam, i.e. beam–column “AB”, and
ðwx2¼0 ¼ δ; w0x2¼0 ¼
EIw″x2¼0:5l
¼ 0Þ for the internal-span beam, i.e. beam– θ; wx2¼0:5l ¼ column “BC”, Eq. (13) gives a normalized solution between the bracing stiffness and the corresponding critical load. For a given lateral stiffness value, critical load is the minimum value obtained from symmetric and asymmetric calculations using Eqs. (12)–(13). 3
3
ð13Þ
For practical uses, the method should be expanded for different stiffness values of lateral bracings. Bracings were represented as linearly elastic translational springs at symmetric locations with respect to the mid-span of the beams, which is mostly the cases in practice (see Fig. 8). Depending on the magnitude of the lateral bracing stiffness, compression flanges of beams can deform in either symmetric or asymmetric shape with respect to the center-span (see Fig. 8). For the symmetric deformation shape, applying boundary conditions wx1¼0 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; w0x2¼0:5l ¼ EIw‴x2¼0:5l ¼ 0 for the internal-span beam, i.e. beam “BC”, Eq. (6) gives: −ξc3 αl cosðcαlÞ−c2 ξc3 cosðcαlÞ þ k=EI ξc2 sinðcαlÞ −ξc2 lα sinðcαlÞ−c cotð0:5clÞ
0.1 28.2 5.88 0.53 0.45
kl =EI ¼ −ξϑðclÞ ½ðα þ 0:5Þ sinðcαl þ 0:5clÞ=½αξ sinðcαlÞ þ 0:5ϑ sinð0:5clÞ
2.4. Beams with elastic lateral restraints
≈0.0a ∞ ≈2π ≈0.50 ≈0.50
δ 0 ¼ θ 0
ð11Þ where ξ ¼ 1=½ sinðcαlÞ−cαl cosðcαlÞ.
Fig. 7. Bending restraint of side-spans with shorter length to buckling length of centralspan, obtained from Eqs. (8)–(10).
where ϑ ¼ 1=½ sinð0:5clÞ−0:5cl cosð0:5clÞ. A typical design curve for the relationship between lateral bracings stiffness and buckling length of the central-span for different locations of lateral bracings is shown in Fig. 9, as the results of linear FE buckling analyses. Comparison between the FE results and the results of Eq. (12) are made in Fig. 10; where the solid lines represented the FE results (for kLb =N0 values between 0.0 and 200.0), and the dashed lines represented the results of Eq. (12), (for kLb =N0 values between 0.0 and 800.0); and N0 ¼ π2 EI=L2b . The results obtained from Eq. (12) were expanded for more values of stiffness than the FE data to show the “excellent” agreement between the two methods. The results showed that the relatively small values of stiffness for symmetrically placed lateral bracings, can significantly decrease the buckling length of the compression flange, especially for the cases where α ≥0:25 (see Fig. 9). For the case of α ¼ 1:0; the points indicated with (i) and (ii) in Fig. 9 represented minimum bracing stiffness values (kLb =N0 being 13.2 and 81, respectively) to change the buckling shapes from a single half-sine wave to
Fig. 8. Compression flange of beams with symmetrically placed lateral elastic bracings.
H. Mehri et al. / Structures 3 (2015) 236–243
241
α ¼ 1:0, can be calculated by means of Winter's rigid bars model that are hinged at the locations of braces. Critical load values of the beam shown in Fig. 8 was normalized to critical load values when − ignoring bending restraint of the side spans−the buckling length is “conservatively” assumed to be equal to 2
Fig. 9. Relationship between buckling length of the compression member shown in Fig. 8 and stiffness of lateral bracings.
2
the distance between the bracing points, i.e. l =le;k in Fig. 10. The results shown in Fig. 10 verified that this assumption gives unsafe results for relatively small values of brace stiffness for all locations of bracing. On the other hand, providing relatively soft bracings, e.g. as little as kLb =N0 ¼ 100 , significantly enhanced the critical load values. This capacity is currently ignored in practice when the buckling length is assumed to be equal to “l” [3]. Fig. 11 graphs the results obtained from Eqs. (12)–(13) for different brace stiffness values and brace locations of the compression flange shown in Fig. 8. Comparisons between the two curves showed that shifting from a symmetric to asymmetric buckling mode shape will not occur for small values of α (e.g. values between 0.0 and 0.6) for any stiffness values of the lateral bracings prior to buckling of the internal-span. Fig. 12 shows the results of Eqs. (12)–(13) for α ¼ 0:7 in which the FE results were also included. Obviously, the 2
2
minimum values for the critical load ratios (i.e. l =le;k ) of the internalspan obtained from Eqs. (12)–(13) were in excellent agreement with the FE results. Fig. 13 also depicts required bracing stiffness values for the cases if slightly larger buckling length than the corresponding to equivalent to immovable supports, le;∞ , was desired in design. As it can be concluded in Fig. 13, to approach fairly close to the largest possible critical load of compression flange, a relatively large bracing stiffness was required for the cases with small values of α which might not be economical for particular cases in practice. For this reason, Fig. 13 enables design engineers to examine the use of lower stiffness values to achieve slightly lower load carrying capacity than corresponding to “full bracing”. Fig. 10. The curves showing ratios between critical loads of compression member shown in Fig. 8 and critical load of the same member assuming the buckling length to be equal to the distance between the bracing points.
3. Conclusions
two and three half-sine waves, respectively, where three half-sine wave corresponds to the “ideal” bracing stiffness of the column [15]. These thresholds values for columns with equally spaced lateral bracings i.e.,
Advanced commercial software packages are widely relied on to model complex structures considering nonlinearities, imperfections, etc. in bridge design practice. The results of such numerical analyses should be checked by other approaches. Simplified models can greatly
Fig. 11. Investigations on shifting from a symmetric buckling mode shape, obtained from Eq. (12), to an asymmetric buckling mode shape, obtained from Eq. (13), for the compression flanges shown in Fig. 8.
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Acknowledgments The financial support from “The Lars Erik Lundbergs Stipendiestiftelse (Dnr 7/2013 and Dnr 2014/05)” to this study is gratefully acknowledged.
Appendix A. Solved example. Determine critical moment, M cr , for the beams shown in Fig. 14. The length of beams is 60.0 m between the twisting supports for both crosssections (S1: doubly-symmetric, and S2: mono-symmetric). The value of α varies between 0.0 (unbraced beam), 0.1, 0.25, and 1.0 (equally spaced lateral bracings); and stiffness of lateral bracings ðkLb =N0 Þ varies between 25, 100, and ∞ (immovable lateral support). Fig. 12. Comparisons between the results of Eqs. (12)–(13) and FE analyses for α ¼ 0:7.
help design engineers to either preliminary size the bridge girders and their bracings, or check the FE results. A model was discussed in this paper which can be used to calculate critical moment values of laterally restrained beams at the level of their compression flanges for given values of bracing stiffness. The model related buckling length of compression flange of steel girders to their lateral torsional critical moment. For this purpose, exact analytical solutions were derived to consider the effects of bending restraints on critical load values, created due to unequal spanning of bracings. This effect has been neglected in practice when the buckling length of compression members is assumed to be equal to the largest distance between the bracing points. The presented paper showed that this assumption can give unsafe results for small bracing stiffness values and can also lead to significant overdesign problem even when relatively soft bracings are used. Typical design curves were also given as the results of analytical solutions for unequally spanned lateral bracings which give required lateral bracing stiffness to achieve the largest possible critical load values. One solved example examined the applicability of the approach for 26 cases of mono-symmetric and doubly-symmetric beams, with different lateral bracing stiffness values and at different locations. The results obtained using the model compared very favorably with those from FE analyses. Besides, the time to perform the analyses using the approach was comparatively short. In addition, the model demonstrated high potential to help in developing and understanding the theory of beam and column bracings.
Fig. 14. Bridge girders and their cross-sectional properties for the solved example.
Solution: N0 ¼ π2 2 105 50 6003 =12 =600002 10−6 ¼ 0:49 ½MN i) For laterally braced cases with kLb =N0 ¼ 25:
α
le;k =Lb Fig. 6
N cr ½MN Eq. (4)
N. B.a 0.1 0.25 0.5 1.0
1.00 0.79 0.57 0.43 0.42
0.49 0.80 1.53 2.63 2.87
γ LT , Eq. (5)
M cr , Eq. (3) ½MNm
FEA (%)
S1
S2
S1
S2
S1
S2
1.95 1.66 1.38 1.24 1.22
2.11 1.77 1.45 1.28 1.26
2.41 3.31 5.30 8.13 8.73
2.60 3.53 5.57 8.44 9.04
+0.8 +4.2 +4.0 +1.7 +2.5⁎
−1.1 +3.1 +1.6 −0.2 +1.4⁎
a N.B.: No lateral bracing. ⁎ refers to a two-half-sine wave shape lateral torsional bucking mode of beam, as observed in FEA.
ii) For laterally braced cases with kLb =N0 ¼ 100:
α
0.1 0.25 0.5 1.0 Fig. 13. Lateral bracing stiffness requirements versus bracing location, when less than full bracing le ¼ le;∞ is desired in design.
le;k =Lb Fig. 6
N cr ½MN Eq. (4)
0.59 0.43 0.36 0.33
1.41 2.67 3.85 4.45
γ LT Eq. (5)
M cr , Eq. (3) ½MNm
FEA (%)
S1
S2
S1
S2
S1
S2
1.41 1.23 1.17 1.15
1.48 1.28 1.20 1.18
4.98 8.23 11.23 12.75
5.24 8.53 11.56 13.07
+5.4 +2.9 +2.8 +3.8⁎⁎
+0.6 +0.2 +0.6 +1.4⁎⁎
⁎⁎ refers to a three-half-sine wave shape of lateral torsional buckling mode of beam, as observed in FEA.
H. Mehri et al. / Structures 3 (2015) 236–243
iii) For laterally braced beam cases with kLb =N0 ¼ ∞:
α
le;∞ =Lb Eqs. (7)&(12)
Ncr ½MN Eq. (4)
0.1 0.25 0.5 1.0
0.44 0.39 0.35 0.33
2.55 3.24 4.03 4.53
γ LT Eq. (5)
M cr , Eq. (3) ½MNm
FEA (%)
S1
S2
S1
S2
S1
S2
1.24 1.19 1.16 1.14
1.29 1.23 1.9 1.17
7.92 9.69 11.68 12.95
8.23 10.01 12.00 13.28
+5.1 +4.4 +0.6** +5.4**
+2.6 +1.3 −0.1** +2.9**
* and ** refer to 2nd and 3rd (i.e. two and three half-sine wave shapes, respectively) modes of lateral torsional buckling of beam, as observed in FEA.
It is evident that there are only slight discrepancies between the results obtained from the proposed approach and the results from FE analyses, whereas FEA demands significantly more time than the proposed approach (which can be done using e.g. an “Excel” sheet for all the 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.
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