Fundamentals of beam bracing

ENGINEERING JOURNAL / FIRST QUARTER / 2001 / 11 INTRODUCTION The purpose of this paper is to provide a fairly compre-hensive view of the subject of beam stability bracing. Factors that affect bracing requirements will be discussed and design methods proposed which are illustrated by design examples. The design examples emphasize simplic- ity. Before going into specific topics related to beam brac- ing, some important concepts developed for column bracing by Winter (1960) will be presented because these concepts will be extended to beams later. For a perfectly straight column with a discrete midheight brace stiffness βL, the relationship between Pcr and βL is shown in Figure 1 (Timoshenko and Gere, 1961). The col- umn buckles between brace points at full or ideal bracing; in this case the ideal brace stiffness βi = 2Pe/ Lb where Pe = π2EI/Lb2. Any brace with stiffness up to the ideal value will increase the column buckling load. Winter (1960) showed that effective braces require not only adequate stiffness but also sufficient strength. The strength requirement is directly related to the magnitude of the initial out-of- straightness of the member to be braced. The heavy solid line in Figure 2(a) shows the relationship between ∆T, the total displacement at midheight, and P for a column with a hinge assumed at the midheight brace point (Winter’s model), an initial out-of-straightness ∆o at mid- height and a midheight brace stiffness equal to the ideal value. For P = 0, ∆T = ∆o. When P increases and approaches the buckling load, π2EI/Lb2, the total deflection ∆T becomes very large. For example, when the applied load is within five percent of the buckling load, ∆T = 20∆o. If a brace stiffness twice the value of the ideal stiffness is used, much smaller deflections occur. When the load just reaches the buckling load, ∆T = 2∆o. For βL = 3βi and P = Pe, ∆T = 1.5∆o. The brace force, Fbr, is equal to (∆T - ∆o )βL and is directly related to the magnitude of the initial imperfection. If a member is fairly straight, the brace force will be small. Conversely, members with large initial out-of-straightness will require larger braces. If the brace stiffness is equal to the ideal value, then the brace force gets very large as the buckling load is approached because ∆T gets very large as shown in Figure 2(a). For example, at P = 0.95Pcr and ∆o = Lb/500, the brace force is 7.6 percent of Pe which is off the scale of the graph. Theoretically the brace force will be infinity when the buckling load is reached if the ideal brace stiffness is used. Thus, a brace system will not be satisfac- tory if the theoretical ideal stiffness is provided because the brace forces get too large. If the brace stiffness is overde- signed, as represented by βL = 2βi and 3βi curves in Figure 2(b), then the brace forces will be more reasonable. For a brace stiffness twice the ideal value and a ∆o = Lb/500, the brace force is only 0.8%Pe at P = Pe, not infinity as in the ideal brace stiffness case. For a brace stiffness ten times the ideal value, the brace force will reduce even further to 0.44 percent. At Pcr the brace force cannot be less than 0.4%P corresponding to ∆T = ∆o (an infinitely stiff brace) for ∆o = Lb/500. For design Fbr = 1%P is recommended based on a brace stiffness of twice the ideal value and an initial out-of- straightness of Lb/500 because the Winter model gives slightly unconservative results for the midspan brace prob- lem (Plaut, 1993). Published bracing requirements for beams usually only consider the effect of brace stiffness because perfectly straight beams are considered. Such solutions should not be used directly in design. Similarly, design rules based on strength considerations only, such as a 2 percent rule, can result in inadequate bracing systems. Both strength and stiffness of the brace system must be checked. BEAM BRACING SYSTEMS Beam bracing is a much more complicated topic than col- umn bracing. This is due mainly to the fact that most col- umn buckling involves primarily bending whereas beam buckling involves both flexure and torsion. An effective beam brace resists twist of the cross section. In general, Fundamentals of Beam Bracing JOSEPH A. YURA Joseph A.Yura is Cockrell Family Regents Chair in civil engi- neering, University of Texas at Austin. Fig. 1. Effect of brace stiffness. 12 / ENGINEERING JOURNAL / FIRST QUARTER / 2001 bracing may be divided into two main categories; lateral and torsional bracing as illustrated in Figure 3. Lateral bracing restrains lateral displacement as its name implies. The effectiveness of a lateral brace is related to the degree that twist of the cross section is restrained. For a simply supported I-beam subjected to uniform moment, the center of twist is located at a point outside the tension flange; the top flange moves laterally much more than the bottom flange. Therefore, a lateral brace restricts twist best when it is located at the top flange. Lateral bracing attached at the bottom flange of a simply supported beam is almost totally ineffective. A torsional brace can be differentiated from a lateral brace in that twist of the cross section is restrained directly, as in the case of twin beams with a cross frame or diaphragm between the members. The cross frame loca- tion, while able to displace laterally, is still considered a brace point because twist is prevented. Some systems such as concrete slabs can act both as lateral and torsional braces. Bracing that controls both lateral movement and twist is more effective than lateral or torsional braces acting alone (Tong and Chen, 1988; Yura and Phillips, 1992). However, since bracing requirements are so minimal, it is more prac- tical to develop separate design recommendations for these two types of systems. Lateral bracing can be divided into four categories: rela- tive, discrete (nodal), continuous and lean-on. A relative brace system controls the relative lateral movement between two points along the span of the girder. The top flange horizontal truss system shown in Figure 4 is an example of a relative brace system. The system relies on the fact that if the individual girders buckle laterally, points a and b would move different amounts. Since the diagonal brace prevents points a and b from moving different amounts, lateral buckling cannot occur except between the brace points. Typically, if a perpendicular cut anywhere along the span length passes through one of the bracing members, the brace system is a relative type. Discrete sys- tems can be represented by individual lateral springs along the span length. Temporary guy cables attached to the top flange of a girder during erection would be a discrete brac- ing system. A lean-on system relies on the lateral buckling strength of lightly loaded adjacent girders to laterally sup- port a more heavily loaded girder when all the girders are horizontally tied together. In a lean-on system all girders must buckle simultaneously. In continuous bracing sys- tems, there is no “unbraced” length. In this paper only rel- ative and discrete systems that provide full bracing will be considered. Design recommendations for lean-on systems and continuous lateral bracing are given elsewhere (Yura, Phillips, Raju, and Webb, 1992). Torsional brace systems can be discrete or continuous (decking) as shown in Figure 3. Both types are considered herein. Some of the factors that affect brace design are shown in Figure 5. A lateral brace should be attached where it best offsets the twist. For a cantilever beam in (a), the best loca- tion is the top tension flange, not the compression flange. Top flange loading reduces the effectiveness of a top flange brace because such loading causes the center of twist to shift toward the top flange as shown in (b), from its position below the flange when the load is at the midheight of the beam. Larger lateral braces are required for top flange load- ing. If cross members provide bracing above the top flange, case (c), the compression flange can still deflect laterally if stiffeners do not prevent cross-section distortion. In the fol- lowing sections the effect of loading conditions, load loca- tion, brace location and cross-section distortion on brace requirements will be presented. All the cases considered were solved using an elastic finite element program identi- fied as BASP in the figures (Akay, Johnson, and Will, 1977; Choo, 1987). The solutions and the design recommenda- Fig. 2. Braced Winter column with initial out-of-straightness. ENGINEERING JOURNAL / FIRST QUARTER / 2001 / 13 tions presented are consistent with the work of others: Kirby and Nethercot (1979), Lindner and Schmidt (1982), Medland (1980), Milner (1977), Nakamura (1988), Naka- mura and Wakabayashi (1981), Nethercot (1989), Taylor and Ojalvo (1966), Tong and Chen (1988), Trahair and Nethercot (1982), Wakabayashi and Nakamura (1983), and Wang and Nethercot (1989). LATERAL BRACING OF BEAMS Behavior The uniform moment condition is the basic case for lateral buckling of beams. If a lateral brace is placed at the midspan of such a beam, the effect of different brace sizes (stiffness) is illustrated by the finite element solutions for a W16×26 section 20-ft long in Figure 6. For a brace attached to the top (compression) flange, the beam buckling capacity initially increases almost linearly as the brace stiffness increases. If the brace stiffness is less than 1.6 k/in., the beam buckles in a shape resembling a half sine curve. Even though there is lateral movement at the brace point, the load increase can be more than three times the unbraced case. The ideal brace stiffness required to force the beam to buckle between lateral supports is 1.6 k/in. in Fig. 3. Types of beam bracing. Fig. 4. Relative bracing. Fig. 5. Factors that affect brace stiffness. Fig. 6. Effect of lateral brace location. 14 / ENGINEERING JOURNAL / FIRST QUARTER / 2001 this example. Any brace stiffness greater than this value does not increase the beam buckling capacity and the buck- led shape is a full sine curve. When the brace is attached at the top flange, there is no cross section distortion. No stiff- ener is required at the brace point. A lateral brace placed at the centroid of the cross section requires an ideal stiffness of 11.4 kips/in. if a 4 × 1/4 stiffener is attached at midspan and 53.7 kips/in. (off scale) if no stiffener is used. Substantially more bracing is required for the no stiffener case because of web distortion at the brace point. The centroidal bracing system is less efficient than the top flange brace because the centroidal brace force causes the center of twist to move above the bottom flange and closer to the brace point, which is undesirable for lat- eral bracing. For the case of a beam with a concentrated centroid load at midspan, shown in Figure 7, the moment varies along the length. The ideal centroid brace (110 kips/in.) is 44 times larger than the ideal top flange brace (2.5 kips/in.). For both brace locations, cross-section distortion had a minor effect on Pcr (less than 3 percent). The maximum beam moment at midspan when the beam buckles between the braces is 1.80 times greater than the uniform moment case which is close to the Cb factor of 1.75 given in specifications (AISC, AASHTO). This higher buckling moment is the main rea- son why the ideal top flange brace requirement is 1.56 times greater (2.49 versus 1.6 kips/in.) than the uniform moment case. Figure 8 shows the effects of load and brace position on the buckling strength of laterally braced beams. If the load is at the top flange, the effectiveness of a top flange brace is greatly reduced. For example, for a brace stiffness of 2.5 kips/in., the beam would buckle between the ends and the midspan brace at a centroid load close to 50 kips. If the load is at the top flange, the beam will buckle at a load of 28 kips. For top flange loading, the ideal top flange brace would have to be increased to 6.2 kips/in. to force buckling between the braces. The load position effect must be con- sidered in the brace design requirements. This effect is even more important if the lateral brace is attached at the centroid. The results shown in Figure 8 indicate that a cen- troid brace is almost totally ineffective for top flange load- ing. This is not due to cross section distortion since a stiffener was used at the brace point. The top flange load- ing causes the center of twist at buckling to shift to a posi- tion close to mid-depth for most practical unbraced lengths, as shown in Figure 5. Since there is virtually no lateral dis- placement near the centroid for top flange loading, a lateral brace at the centroid will not brace the beam. Because of cross-section distortion and top flange loading effects, lat- eral braces at the centroid are not recommended. Lateral braces must be placed near the top flange of simply sup- ported and overhanging spans. Design recommendations will be developed only for the top flange lateral bracing sit- uation. Torsional bracing near the centroid or even the bot- tom flange can be effective as discussed later. The load position effect discussed above assumes that the load remains vertical during buckling and passes through the plane of the web. In the laboratory, a top flange loading condition is achieved by loading through a knife-edge at the middle of the flange. In actual structures the load is applied to the beams through secondary members or the slab itself. Loading through the deck can provide a beneficial “restor- ing” effect illustrated in Figure 9. As the beam tries to buckle, the contact point shifts from mid-flange to the flange tip resulting in a restoring torque that increases the buckling capacity. Unfortunately, cross-section distortion severely limits the benefits of tipping. Lindner and Schmidt (1982) developed a solution for the tipping effect, which considers the flange-web distortion. Test data (Lindner and Schmidt, 1982; Raju, Webb, and Yura, 1992) indicate that a cross member merely resting (not positively attached) on the top flange can significantly increase the lateral buckling capacity. The restoring solution is sensitive to the initial Fig. 7. Midspan load at centroid. Fig. 8. Effect of brace and load position. ENGINEERING JOURNAL / FIRST QUARTER / 2001 / 15 shape of the cross section and location of the load point on the flange. Because of these difficulties, it is recommended that the restoring effect not be considered in design. When a beam is bent in double curvature, the compres- sion flange switches from the top flange to the bottom flange at the inflection point. Beams with compression in both the top and bottom flanges along the span have more severe bracing requirements than beams with compression on just one side as illustrated by the comparison of the cases given in Figure 10. The solid lines are finite element solu- tions for a 20-ft long W16×26 beam subjected to equal but opposite end moments and with lateral bracing at the midspan inflection point. For no bracing the buckling moment is 1,350 kip-in. A brace attached to one flange is ineffective for reverse curvature because twist at midspan is not prevented. If lateral bracing is attached to both flanges, the buckling moment increases nonlinearly as the brace stiffness increases to 24 kips/in., the ideal value shown by the black dot. Greater brace stiffness has no effect because buckling occurs between the brace points. The ideal brace stiffness for a beam with a concentrated midspan load is 2.6 kips/in. at Mcr = 2,920 kip-in. as shown by the dashed lines. For the two load cases the moment diagrams between brace points are similar, maximum moment at one end and zero moment at the other end. In design, a Cb value of 1.75 is used for these cases which corresponds to an expected max- imum moment of 2,810 kip-in. The double curvature case reached a maximum moment 25 percent higher because of warping restraint provided at midspan by the adjacent ten- sion flange. In the concentrated load case, no such restraint is available since the compression flanges of both unbraced segments are adjacent to each other. On the other hand, the brace stiffness at each flange must be 9.2 times the ideal value of the concentrated load case to achieve the 25 per- cent increase. Since warping restraint is usually ignored in design Mcr = 2,810 kip-in. is the maximum design moment. At this moment level, the double curvature case requires a brace stiffness of 5.6 kips/in. which is about twice that required for the concentrated load case. The results in Fig- ure 10 show that not only is it incorrect to assume that an inflection point is a brace point but also that bracing requirements for beams with inflection points are greater than cases of single curvature. For other cases of double curvature, such as uniformly loaded beams with end restraint (moments), the observations are similar. Up to this point, only beams with a single midspan lateral brace have been discussed. The bracing effect of a beam with multiple braces is shown in Figure 11. The response of a beam with three equally spaced braces is shown by the solid line. When the lateral brace stiffness, βL, is less than 0.14 kips/in., the beam will buckle in a single wave. In this region a small increase in brace stiffness greatly increases the buckling load. For 0.14 < βL < 1.14, the buckled shape switches to two waves and the relative effectiveness of the lateral brace is reduced. For 1.4 < βL < 2.75, the buckled shape is three waves. The ideal brace stiffness is 2.75 kips/in. at which the unbraced length can be considered 10 ft. For the 20-ft span with a single brace at midspan dis- cussed previously which is shown by the dashed line, a Fig. 9. Tipping effect. Fig. 10. Beams with inflection points. Fig. 11. Multiple lateral bracing. 16 / ENGINEERING JOURNAL / FIRST QUARTER / 2001 brace stiffness of only 1.6 kips/in. was required to reduce the unbraced length to 10 ft. Thus the number of lateral braces along the span affects the brace requirements. Simi- lar behavior has been derived for columns (Timoshenko and Gere, 1961) where changing from one brace to three braces required an increase in ideal column brace stiffness of 1.71, which is the same as that shown in Figure 11 for beams, 2.75/1.6 = 1.72. Yura and Phillips (1992) report the results of a test pro- gram on the lateral and torsional bracing of beams for com- parison with the theoretical studies presented above. Some typical test results show good correlation with the finite ele- ment solutions in Figure 12. Since the theoretical results were reliable, significant variables from the theory were included in the development of the design recommenda- tions given in the following section. In summary, moment gradient, brace location, load location, brace stiffness and number of braces affect the buckling strength of laterally braced beams. The effect of cross-section distortion can be effectively eliminated by placing the lateral brace near the top flange. Lateral Brace Design In the previous section it was shown that the buckling load increases as the brace stiffness increases until full bracing causes the beam to buckle between braces. In many instances the relationship between bracing stiffness and buckling load is nonlinear as evidenced by the response shown in Figure 11 for multiple braces. A general design equation has been developed for braced beams, which gives good correlation with exact solutions for the entire range of zero bracing to full bracing (Yura et al., 1992). That braced beam equation is applicable to both continuous and discrete bracing systems, but it is fairly complicated. In most design situations full bracing is assumed or desired, that is, buck- ling be