Formulas: Design by Simple Methods of Analysis

Equation A-4-9

For nonslender-element columns, the nominal strength for flexural buckling of compression members shall be determined using the provisions of Chapter E with steel properties as stipulated in Section 4.2.3b(a). Equation A-4-9 shall be used in lieu of Equations E3-2 and E3-3 to calculate the nominal compressive strength for flexural buckling:

where

$F_{y}(T)$ is the yield stress at elevated temperature and $F_{e}(T)$ is the critical elastic buckling stress calculated from Equation E3-4 with the elastic modulus, $E(T)$, at elevated temperature. $F_{y}(T)$ and $E(T)$ are obtained using coefficients from Table A-4.2.1.

Equation A-4-10

The strength of gravity-only columns that do not provide resistance to lateral loads is permitted to be increased by the rotational restraints from cooler columns in the stories above and below the story exposed to the fire. This increased strength applies to fires on only one floor and should not be used for multiple story fires. It is permitted to account for the increase in design strength by reducing the column slenderness, $L_{c} / r$, used to calculate $F_{e}(T)$ in Equation A-4-9 to $L_{c}(T) / r$ as follows:

Equation A-4-10M

where

Lc = effective length of member, in. (mm)

• = KL
• $K=1.0$ for gravity-only columns
• $L=$ laterally unbraced length of the member, in. (mm)
• $T=$ temperature of structural steel, ${ }^{\circ} \mathrm{F}\left({ }^{\circ} \mathrm{C}\right)$

$n=1$ for columns with cooler columns both above and below

• $n=2$ for columns with cooler columns either above or below only
• $r \quad=$ radius of gyration, in. (mm)

Equation A-4-11

For filled composite columns, the nominal strength for compression shall be determined using the provisions of Section I2.2 with steel and concrete properties as stipulated in Section 4.2.3b. Equation A-4-11 shall be used in lieu of Equations I2-2 and I2-3 to calculate the nominal compressive strength for flexural buckling:

where

$P_{n o}(T)$ is calculated at elevated temperature using Equations I2-9, I2-10, and I2-11. $P_{e}(T)$ is calculated at elevated temperature using Equation I2-4. $E I_{\text {eff }}(T)$ is calculated at elevated temperature using Equations I2-12 and I2-13. $F_{y}(T), f_{c}^{\prime}(T), E_{s}(T)$, and $E_{c}(T)$ are obtained using coefficients from Tables A-4.2.1 and A-4.2.2.

Equation A-4-12

For filled composite plate shear walls, the nominal strength for compression shall be determined using the provisions of Section I2.3 with steel and concrete properties as stipulated in Section 4.2.3b and Equation A-4-12 used in lieu of Equations I2-2 and I2-3 to calculate the nominal compressive strength for flexural buckling:

where

$P_{n o}(T)$ is calculated at elevated temperature using Equation I2-15. $P_{e}(T)$ is calculated at elevated temperature using Equation I2-4. $E I_{\text {eff }}(T)$ is calculated at elevated temperatures using Equation II-1. $F_{y}(T), f_{c}^{\prime}(T), E_{s}(T)$, and $E_{c}(T)$ are obtained using coefficients from Tables A-4.2.1 and A-4.2.2.

Formula 6

(1) When $L_{b} \leq L_{r}(T)$

Equation A-4-13

Formula 8

(2) When $L_{b}>L_{r}(T)$

Equation A-4-14

Equation A-4-15

Equation A-4-16

Equation A-4-17

Equation A-4-18

Equation A-4-20

Alternatively, the nominal flexural strength of a composite beam, $M_{n}(T)$, is permitted to be calculated using the bottom flange temperature, $T$, as follows:

Found in

• STRUCTURAL DESIGN FOR FIRE CONDITIONS BY ANALYSIS