# D4.E.6.3 Members Subject to Combined Forces

For each of the following interaction equations, the value of the RATIO parameter is used in lieu of 1.0 when it is specified (Refer to D4.E.7 Design Parameters).

## 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). ω1y and ω1z are calculated as per Cl. 13.8.5 or as specified in the CMY and CMZ design parameters, respectively.

1. For Class 1 and Class 2 sections of I-shaped members (Cl. 13.8.2):
$C f C r + 0.85 U 1 x M f x M r x + β U 1 y M f y M r y ≤ 1.0$
where
 Cf , Mf = the maximum load effects, including stability, as specified in Cl. 8.4. β = 0.6 + 0.4λy ≤ 0.85

The capacity of the member is investigated for the following:

1. Cross sectional strength with β = 0.6, where
1. Cr as specified in Cl. 13.3 with λ = 0
2. Mr as specified in Cl. 13.5
3. U1x and U1y as specified in Cl. 13.8.4 but not less than 1.0. Design parameters SSY and SSZ are used to evaluate these coefficients.
2. Overall member strength, where
1. Cr as specified in Cl. 13.3 with K = 1, except for uniaxial bending, in which case Cr is based on the axis of bending
2. Mr as specified in Cl. 13.5
3. U1x and U1y 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.
3. Lateral torsional buckling strength, when applicable, where
1. Cr as specified in Cl. 13.3
2. Mrx as specified in Cl. 13.6
3. Mry as specified in Cl. 13.5
4. U1x and U1y 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 U1x is not less than 1.0). Design parameters SSY and SSZ are used to evaluate these coefficients.
2. For Class 1 and Class 2 square or circular HSS sections per S16-19 only (Cl. 13.8.3):
• For square sections:
$C f C r + 0.85 U 1 x M f x M r x + 0.50 U 1 y M f y M r y ≤ 1.0$
• For circular sections:
$C f C r + 0.85 U 1 x M f x 2 + U 1 y M f y 2 M r x ≤ 1.0$
3. For all other cases (Cl13.8.3 per S16-09 / 14 or Cl 13.8.4 of S16-19):
$C f C r + U 1 x M f x M r x + U 1 y M f y M r y ≤ 1.0$

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):

$T f T r + M f M r ≤ 1.0$
where
 Mr = 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):

• $M f M r − T f Z M r A ≤ 1.0$ for Class 1 and Class 2 sections
• $M f M r − T f S M r A ≤ 1.0$ for Class 3 and Class 4 sections
where
 Mr = 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):

• $M fx M rx + M fy M ry − T f Z x M rx A ≤ 1.0$ for Class 1 and Class 2 sections
• $M fx M rx + M fy M ry − T f S x M rx A ≤ 1.0$ for Class 3 and Class 4 sections
where
 Mrx = the moment resistance as specified in Cl. 13.6.1.
where
 Mry = 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):
$M f x M r x + M f y M r y ≤ 1.0$

## 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):

 $0.727 M f M r + 0.455 V f V r ≤ 1.0$ (S16-09 / 14)

 $M f M r ≤ 1.0$

 $V f V r ≤ 1.0$

where
 Mr = the value determined in accordance with Cl. 13.5 of Cl 13.6 as applicable Vr = the value determined in accordance with Cl. 13.4

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

$2.20 - 1.60 M f M r$

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