D4.E.6.3 Members Subject to Combined Forces

Axial compression and bending

The member strength and stability for sections subjected to axial compression and uniaxial or biaxial bending is obtained through the use of interaction equations. In these equations, the additional bending caused by the action of the axial load is accounted for by using amplification factors (Cl. 13.8). ω_1y and ω_1z are calculated as per Cl. 13.8.5 or as specified in the CMY and CMZ design parameters, respectively.

For Class 1 and Class 2 sections of I-shaped members (Cl. 13.8.2):

\frac{C_{f}}{C_{r}} + \frac{0.85 U_{1 x} M_{f x}}{M_{r x}} + \frac{β U_{1 y} M_{f y}}{M_{r y}} \leq 1.0

where

C_f , M_f	=	the maximum load effects, including stability, as specified in Cl. 8.4.
β	=	0.6 + 0.4λ_y ≤ 0.85

The capacity of the member is investigated for the following:

Cross sectional strength with β = 0.6, where
1. C_r as specified in Cl. 13.3 with λ = 0
2. M_r as specified in Cl. 13.5
3. U_1x and U_1y as specified in Cl. 13.8.4 but not less than 1.0. Design parameters SSY and SSZ are used to evaluate these coefficients.
Overall member strength, where
1. C_r as specified in Cl. 13.3 with K = 1, except for uniaxial bending, in which case C_r is based on the axis of bending
2. M_r as specified in Cl. 13.5
3. U_1x and U_1y are taken as 1.0 for members in an unbraced frame, and as specified in Cl. 13.8.4 for members in a braced frame. Design parameters SSY and SSZ are used to evaluate these coefficients.
Lateral torsional buckling strength, when applicable, where
1. C_r as specified in Cl. 13.3
2. M_rx as specified in Cl. 13.6
3. M_ry as specified in Cl. 13.5
4. U_1x and U_1y are taken as 1.0 for members in an unbraced frame, and as specified in Cl. 13.8.4 for members in a braced frame (where U_1x is not less than 1.0). Design parameters SSY and SSZ are used to evaluate these coefficients.

For Class 1 and Class 2 square or circular HSS sections per S16-19 only (Cl. 13.8.3):
- For square sections:
  $\frac{C_{f}}{C_{r}} + \frac{0.85 U_{1 x} M_{f x}}{M_{r x}} + \frac{0.50 U_{1 y} M_{f y}}{M_{r y}} \leq 1.0$
- For circular sections:
  $\frac{C_{f}}{C_{r}} + \frac{0.85 \sqrt{{(U_{1 x} M_{f x})}^{2} + {(U_{1 y} M_{f y})}^{2}}}{M_{r x}} \leq 1.0$
For all other cases (Cl13.8.3 per S16-09 / 14 or Cl 13.8.4 of S16-19):
$\frac{C_{f}}{C_{r}} + \frac{U_{1 x} M_{f x}}{M_{r x}} + \frac{U_{1 y} M_{f y}}{M_{r y}} \leq 1.0$

The capacity of the member is investigated for the following per Cl.13.8.2:
1. Cross sectional strength
2. Overall member strength
3. Lateral torsional buckling strength,

Axial tension and bending

Members subjected to axial tension and bending must satisfy the following equation (Cl. 13.9.1):

\frac{T_{f}}{T_{r}} + \frac{M_{f}}{M_{r}} \leq 1.0

where

M_r

=

the moment resistance as specified in Cl. 13.5.

Note: For I shapes with equal flanges designed to S16-19, Class 1 and 2 sections are proportioned as per Cl. 13.9.2.

Additionally, the following equations must be satisfied for laterally unsupported members (Cl. 13.9.2 of S16-09 /14):

$\frac{M_{f}}{M_{r}} - \frac{T_{f} Z}{M_{r} A} \leq 1.0$ for Class 1 and Class 2 sections
$\frac{M_{f}}{M_{r}} - \frac{T_{f} S}{M_{r} A} \leq 1.0$ for Class 3 and Class 4 sections

where

M_r

=

the moment resistance as specified in Cl. 13.6.

The following equations must be satisfied for laterally unsupported members designed per S16-19 (Cl. 13.9.3 of S16-19):

$\frac{M_{fx}}{M_{rx}} + \frac{M_{fy}}{M_{ry}} - \frac{T_{f} Z_{x}}{M_{rx} A} \leq 1.0$ for Class 1 and Class 2 sections
$\frac{M_{fx}}{M_{rx}} + \frac{M_{fy}}{M_{ry}} - \frac{T_{f} S_{x}}{M_{rx} A} \leq 1.0$ for Class 3 and Class 4 sections

where

M_rx

=

the moment resistance as specified in Cl. 13.6.1.

where

M_ry

=

the moment resistance as specified in Cl. 13.5.

Biaxial Bending

For bending about both axis, the following equation must be satisfied (Cl. 13.8):

\frac{M_{f x}}{M_{r x}} + \frac{M_{f y}}{M_{r y}} \leq 1.0

Shear and Bending

For S16-09 / 14, to resist the combined effects of shear and bending, all of the following equations must be satisfied (Cl. 14.6):

0.727 \frac{M_{f}}{M_{r}} + 0.455 \frac{V_{f}}{V_{r}} \leq 1.0

(S16-09 / 14)

\frac{M_{f}}{M_{r}} \leq 1.0

\frac{V_{f}}{V_{r}} \leq 1.0

where

M_r	=	the value determined in accordance with Cl. 13.5 of Cl 13.6 as applicable
V_r	=	the value determined in accordance with Cl. 13.4

For S16-19: for beams with webs with F_s > 0.60 F_y to resist the combined effects of shear and bending, the shear resistance, V_r, is multiplied by the following reduction factor (Cl. 14.6):

[2.20 - 1.60 \frac{M_{f}}{M_{r}}]

The shear resistance is not reduced below 0.60 ϕA_wF_y. Also, the shear resistance is not increased due to this calculation.