Formulas: Calculation of Required Strengths

Formula 1

Reduced stiffnesses, $E I^{*}=0.8 \tau_{b} E I$ and $E A^{*}=0.8 E A$ , are used in the method provided in this section, just as it is for the direct analysis method of Chapter C.

E I^{*}=0.8 \tau_{b} E I

Formula 2

E A^{*}=0.8 E A

Formula 3

The $\tau_{b}$ factor is similar to the inelastic stiffness reduction factor implied in the column curve to account for loss of stiffness under high compression loads when $\alpha P_{r}>0.5 P_{y}$ , and the 0.8 factor accounts for additional softening under combined axial compression and bending.

\alpha P_{r}>0.5 P_{y}

Formula 4

Hence, the combination of the applied loads ( $P$ and $w_{y}$ ), initial out-of-plane imperfection ( $\delta_{o x}=L / 1,000$ ), and the resulting deflections and twist ( $\delta_{x}, \delta_{y}$ , and $\theta$ ) produce both major-axis and minor-axis bending moments, which at midspan are

\delta_{o x}=L / 1,000

Equation C-A-1-1

M_{u x}^{\prime}=\left(\frac{w_{y} L}{8}+P \delta_{y}\right) \cos \theta-P\left(\delta_{o x}+\delta_{x}\right) \sin \theta

Equation C-A-1-2

M_{u y}^{\prime}=\left(\frac{w_{y} L}{8}+P \delta_{y}\right) \sin \theta+P\left(\delta_{o x}+\delta_{x}\right) \cos \theta

Found in

DESIGN BY ELASTIC ANALYSIS