C-6.36.3 beam bracing

Beam bracing must control twist of the section, by either restraining lateral movement of the compression flange (lateral bracing) or by restraining twist of the section (torsional bracing). Similar to columns, lateral bracing can be either point bracing or panel bracing. An example of panel bracing is a shear diaphragm parallel to the plane of the compression flange that is supported by steel joists attached to the compression flange of a simply supported beam. An example of a point brace is the stiffness from guy cables, or a strut anchored to a rigid point, such as the core of the building, dead men, or theoretically rigid anchors. For maximum effectiveness, lateral braces should be connected as close to the compression flange as possible. Lateral bracing systems that are attached near the beam shear center or near the tension flange are generally ineffective in limiting the compression flange lateral displacements. An example of a torsional brace is a cross-frame or vertical diaphragm element spanning between adjacent girders that restrains twist of the girder. For a given torsional brace stiffness, the effectiveness of the brace is not generally sensitive to its location on the cross section, barring any distortion of the cross section; therefore, floor beams in a through-girder system can provide effective torsional bracing despite being connected to the tension flange.

For beams subjected to reverse-curvature bending, an unbraced inflection point cannot be considered a braced point because significant twist can occur at that point (Ziemian, 2010). Bracing provided near an inflection point must be attached at or near both flanges to prevent twist; alternatively, torsional bracing can be provided. A lateral brace on one flange is ineffective near an inflection point.

The beam bracing requirements of this section are based predominantly on the recommendations from Yura (2001).

6.3.1 Lateral Bracing

For beam lateral bracing, the following stiffness requirement is derived following Winter’s approach (Winter, 1958, 1960):

$\beta_{br}=\frac{2N_{i}C_{t}P_{f}C_{d}}{\phi L_{br}}$ (C-A-6-7)

where

• $C_{d}$ = double curvature factor, which accounts for the potential larger demands on the lateral bracing in unbraced lengths containing inflection points, applied only to the point brace closest to the inflection point or to the panel brace corresponding to the unbraced length containing the inflection point, as well as the panel brace in the adjacent unbraced length closest to the inflection point
• point $=1+\left(M_{S} / M_{L}\right)^{2}$, where the $C_{d}$ factor is applicable as defined
• $=1.0$, otherwise
• $C_{t}=1.0$ for centroidal loading
• $1+\left(1.2 / n\right)$ for top-flange loading
• $I_{y c}=$ moment of inertia of the compression flange about its principal axis within the plane of the web, in. ${ }^{4}\left(\mathrm{~mm}^{4}\right)$
• $M_{L}=$ absolute value of the maximum moment causing compression in the braced flange within the overall length, composed of an unbraced length containing an inflection point and the adjacent unbraced length closest to the inflection point, kip-ft (N-mm)
• $M_{S}=$ absolute value of the maximum moment causing tension in the braced flange within the overall length, composed of the unbraced length containing an inflection point and the adjacent unbraced length closest to the inflection point, kip-ft (N-mm)
• $N_{i}=1.0$ for panel bracing
• $=\left(4-2 / n\right)$ for point bracing
• $P_{f}=$ beam compressive flange force, kips (N)
• $n \quad=$ number of intermediate braces

The $C_{d}$ factor varies between 1 and 2 , and is applied only to the point brace closest to the inflection point or to the panel brace corresponding to the unbraced length containing the inflection point and the adjacent panel brace closest to the inflection point. The term $2 N_{i} C_{i}$ can be conservatively approximated as 10 for any number of point braces and 4 for panel bracing, and $P_{f}$ can be approximated by $M_{r} / h_{o}$, which simplifies Equation C-A-6-7 to the stiffness requirements given by Equations A-6-6a and A-6-6b and A-6-8a and A-6-8b. Equation C-A-6-7 can be used in lieu of Equations A-6-6a and A-6-6b and A-6-8a and A-6-8b.

The brace strength requirement for panel bracing is

(C-A-6-8a)

and for point bracing is

(C-A-6-8b)

These requirements are based on an assumed initial lateral displacement of the compression flange of $\Delta_{o}=0.002 L_{b r}$. The brace strength requirements of Equations A-6-5 and A-6-7 are derived from Equations C-A-6-8a and C-A-6-8b by assuming top flange loading ( $C_{t}=2$ ). Equations C-A-6-8a and C-A-6-8b can be used in lieu of Equations A-6-5 and A-6-7, respectively.

6.3.2 Torsional Bracing

Torsional bracing can either be attached continuously along the length of the beam (for example, a metal deck or slab) or located at discrete points along the length of the member (for example, cross-frames or secondary beam members). With respect to the girder response, torsional bracing attached to the tension flange is just as effective as a brace attached at mid-depth or to the compression flange, as long as distortion of the beam cross section is controlled. Although the girder response is generally not sensitive to the brace location, the position of the brace on the cross section influences the stiffness of the brace itself. For example, a torsional brace attached below midheight of the section will tend to bend in single curvature with a flexural stiffness of $2 E I / L$ based on the brace properties, while a brace attached on the top flange will tend to bend in reverse curvature with a flexural stiffness of $6 E I / L$ based on the brace properties. Partially restrained connections of the bracing to the girder being braced can be used if their flexibility is considered in evaluating the torsional brace stiffness (Ziemian, 2010).

The torsional brace requirements are based on the buckling strength of a beam with a continuous torsional brace along its length, as presented in Taylor and Ojalvo (1966) and modified for cross-section distortion in Yura (2001):

(C-A-6-9)

The term $C_{b u} M_{o}$ is the buckling strength of the beam without torsional bracing. $C_{n}=1.2$ when there is top flange loading and $C_{n}=1.0$ for centroidal loading. $\overline{\beta_{T}}=n \beta_{T} / L$ is the continuous torsional brace stiffness per unit length or its equivalent when $n$ point braces, each with a stiffness $\beta_{T}$, are used along the span, $L$, and the factor 2 accounts for initial out-of-straightness (the continuous torsional ideal bracing stiffness is thus taken as $\overline{\beta_{T}} / 2$ ). Neglecting the unbraced beam buckling term gives a conservative estimate of the torsional brace stiffness requirement as expressed in Equation A-6-11. For a doubly symmetric I -shaped cross section, $I_{y e f f}$ is equal to the moment of inertia about the principal axis within the plane of the web of the section, $I_{y}$. For a singly symmetric I -shaped cross section,

(C-A-6-10)

where

$I_{y c}$ and $I_{y t}=$ respective moments of inertia of compression and tension flanges about their principal axes within the plane of the web, in. ${ }^{4}\left(\mathrm{~mm}^{4}\right)$

• $\begin{aligned} c & =\text { distance from the neutral axis to the extreme compressive fibers, in. } \\ & \text { (mm) }\end{aligned}$
• $t$ = distance from the neutral axis to the extreme tensile fibers, in. (mm)

The strength requirements have changed from the 2016 expression, which was found to underestimate the brace moments for large, unbraced lengths where elastic lateral deflections and twisting can become large at moments approaching the elastic critical buckling limit (Liu and Helwig, 2020). The strength requirement for beam torsional bracing is developed based upon the critical initial twist imperfection of $\theta_{o}=0.002 L_{b r} / h_{o}$, where $h_{o}$ is equal to the depth of the beam (Wang and Helwig,

2005). Based on the use of an effective bracing stiffness equal to 3 times the ideal torsional bracing stiffness, the torsional bracing required moment resistance may be estimated as $M_{r b}=\beta_{T} \theta_{o}$. Using the formulation of Equation A-6-11, without $\phi$ or $\Omega$, the torsional bracing strength requirement given by Equation A-6-9 is obtained. It is noted that this strength requirement was previously given in the Commentary to the 2016 Specification but $\beta_{T}$ is now modified to use a factor of 3 (instead of 2) on the ideal bracing stiffness. Reichenbach et al. (2021) found that a $\beta_{T}$ of 3 times the ideal bracing value is necessary to limit the additional elastic twist rotation under load approximately to $\theta_{o}$ at the elastic critical buckling limit, resulting in a total twist rotation of $2 \theta_{o}$ under the load. The lower bound of $0.02 M_{r}$ in Equation A-6-9, which was the strength requirement given in the 2016 AISC Specification (AISC, 2016), tends to govern at smaller unbraced lengths where the stiffness of the beam is reduced due to inelasticity.

The $\beta_{s e c}$ term in Equation A-6-10 and defined in Equations A-6-12 and A-6-13 accounts for cross-section distortion. A web stiffener at the brace point reduces cross-sectional distortion and improves the effectiveness of a torsional brace. When a cross-frame is attached near both flanges or a vertical diaphragm element is approximately the same depth as the girder, then web distortion will be insignificant and $\beta_{s e c}$ may be taken as infinity. The required bracing flexural stiffness, $\beta_{b r}$, given by Equation A-6-10 is obtained by solving the following expression, which represents the brace system stiffness including distortion effects:

$\frac{1}{\beta_{T}}=\frac{1}{\beta_{b r}}+\frac{1}{\beta_{s e c}}$

(C-A-6-11)

Yura (2001) provides additional guidance regarding the handling of cross-section distortional flexibility for cases where the bracing system is attached through only a portion of the depth of the member being braced.

Parallel chord trusses with both chords subjected only to flexural loading and with both chords extended to the end of the span and attached to supports can be treated the same as beams. In Equations A-6-5 through A-6-9, $M_{r}$ may be taken as the maximum compressive chord force times the depth of the truss to determine the torsional brace strength and stiffness requirements. Cross-section distortion effects, $\beta_{s e c}$, need not be considered when full-depth cross-frames are used for bracing. When either chord does not extend to the end of the span, consideration should be given to the control of twisting near the ends of the span by the use of cross-frames or ties.

For torsionally braced systems comprosed of two- and three-girder systems, the girder major-axis bending stiffness can lead to flexibility that lowers the effectiveness of the torsional bracing systems. Solutions are provided for estimating the girder major-axis bending stiffness (Yura, 2001; Helwig et al., 1993). For narrow girder systems (two- and three-girder systems with close girder spacing), the girders can be susceptible to the system mode of buckling, which is not sensitive to the brace spacing (Yura et al., 2008; Han and Helwig, 2020).

Beams—Point Torsional Bracing Combined with Lateral Bracing at the Compression Flange. Prado and White (2015) and Lokhande and White (2015) have suggested that for beams having point torsional bracing combined with panel or point

lateral bracing on the flange subjected to flexural compression, the required torsional and lateral brace stiffnesses can be reduced relative to the base values specified in Sections 6.3.1 and 6.3.2, but should satisfy the following interaction equation:

(C-A-6-12)

where

$\beta_{L b r}=$ actual or provided lateral brace stiffness, kip/in. (N/mm)

$\beta_{L b r o}=$ required lateral brace stiffness given by Equation A-6-6 for panel bracing or Equation A-6-8 for point bracing acting alone, kip/in. (N/mm)

$\beta_{T b r}=$ actual or provided torsional brace stiffness, kip-in./rad (N-mm/rad)

βTbro = required torsional brace stiffness given by Equation A-6-10 acting alone, kip-in./rad (N-mm/rad)

Beams—Point Torsional Bracing Combined with Lateral Bracing at the Tension

Flange. For beams having point torsional bracing combined with panel or point lateral bracing on the flange subjected to flexural tension, Equation C-A-6-12 applies and, in addition, the required torsional brace stiffness should be greater than or equal to the smaller of $\beta_{P b r o} h^{2}$ or $\beta_{T b r o}$,

where

$\begin{aligned} \beta_{P b r o} & =\text { required point lateral brace stiffness given by Equation A-6-8, calculated } \\ & \text { using the unbraced length between the torsional brace points, kip/in. } \\ & \text { (N/mm) }\end{aligned}$

$h_{o} \quad$ = distance between the flange centroids, in. (mm)

The provisions of Sections 6.3.1 and 6.3.2 apply for the lateral and the torsional brace strength requirements.

Reduction in Beam Bracing Requirements with Combined Torsional and Lateral Bracing. Equation C-A-6-12 recognizes the typical reduction in the beam torsional and lateral bracing stiffness requirements when lateral and torsional bracing are used in combination, thus restraining both twisting and lateral movement at the braced points. This linear interaction equation is known to provide a conservative estimate of the bracing stiffness requirements in cases where the lateral bracing is provided at or near the flange subjected to flexural compression (Yura et al., 1992; Prado and White, 2015; Lokhande and White, 2015).

For situations where the lateral bracing is located at or near the flange subjected to flexural tension, the lateral bracing system is ineffective on its own. However, a point torsional brace works effectively as a lateral brace to the compression flange, in the limit that the lateral bracing system stiffness becomes large. Prado and White (2015) and Lokhande and White (2015) show that the point lateral bracing stiffness requirement of Equations A-6-8a and A-6-8b, denoted by $\beta_{p b r o}$, when multiplied by $h_{o}^{2}$, serves as an accurate to conservative minimum limit on the required torsional bracing stiffness obtained from Equation C-A-6-12 for the case of point torsional bracing combined with lateral bracing at the tension flange. Furthermore, where $\beta_{p b r o} h_{o}^{2}$ is greater than the base torsional bracing stiffness requirement from Equation A-6-10, the torsional bracing stiffness need not be greater than the requirement from Equation A-6-10.

The minimum required strength of the separate lateral and torsional bracing components is still governed by the provisions of Sections 6.3.1 and 6.3.2. The strength demands on the separate brace components are not necessarily reduced by the combination.