Formulas: general provisions

Formula 1

The bracing requirements in Sections 6.2 and 6.3 generally are not sufficient to permit the development of member strengths based on $L_{c}$ or $L_{b}$ smaller than $L_{b r}$; that is, the development of column or beam strengths based on a corresponding effective length factor of $K<1$.

Formula 2

The ideal bracing stiffness for this column associated with $L_{c}=L_{b r}=L$; that is, the bracing stiffness necessary to develop a column critical buckling load of $P_{c r}=P_{e}=\pi^{2} E I / L_{b r}^{2}$, is $P_{b} / L_{b r}$.

Formula 3

Formula 4

A brace having 5 times this stiffness is required for the column to reach a critical load of $95 \%$ of $P_{c r}=\pi^{2} E I /\left(0.7 L_{b r}\right)^{2}$ based on $L_{c}=0.7 L_{b r}$.

Formula 5

Formula 6

$\beta = \beta_i$ — color: black; symbol: Solid line

Formula 7

$\Delta_o = 0.002L_{br}$ (text_label - position: x = 1.6, y = 0.3)

Formula 8

For columns, the value that is often used for the initial out-of-alignment, $\Delta_{o}=0.002 L_{b r}$, equals the initial out-of-plumbness provided in the AISC Code of Standard Practice for Steel Buildings and Bridges (AISC, 2022a).

Formula 9

In Figure C-A-6.3, for a brace stiffness of $\beta_{b r}=2 \beta_{i}$ and the initial imperfection of $\Delta_{o}=L_{b r} / 500$, the resulting brace force, $P_{b r}$, for this point-bracing system is equal to $0.4 \%$ of $P_{e}$ at $P / P_{e}=1.0$.

Formula 10

Formula 11

Similarly, for torsional bracing of beams, the important imperfection is an initial rotation, $\theta_{o}=L_{b r} /\left(500 h_{o}\right)$ (Wang and Helwig, 2005), where $h_{o}$ is the distance between flange centroids.

Formula 12

For such cases, Chen and Tong (1994) recommend the use of an average initial displacement due to erection tolerances of $\Delta_{o}=L_{b r} /\left(500 \sqrt{n_{o}}\right)$, where $n_{o}$ is the number of columns, each with a random $\Delta_{o}$, stabilized by the bracing system.

Equation C-A-6-1

where

• $\begin{aligned} L_{b r} & =\text { unbraced length within the panel under consideration, in. (mm) }\end{aligned}$
• $\begin{aligned} P_{r} & =\text { sum of the required axial forces in the columns being stabilized, kips (N) }\end{aligned}$
• $\begin{aligned} V_{1 s t} & =\text { first-order shear force in the bracing system due to gravity and/or lateral } \\ & \text { loading on the structure, temperature effects, etc., kips (N) }\end{aligned}$

$\Delta_{o \text {,total }}=$ total relative displacement between the ends of the unbraced length under consideration due to erection tolerances, first-order effects of gravity and/or lateral loads on the structure, and first-order effects (i.e., the effects prior to amplification from member axial compression) from any other sources such as temperature movement, connection slip, etc., in. (mm)

Equation C-A-6-3

Equation C-A-6-4

Found in

• GENERAL PROVISIONS