## A quick recap of strong & weak axis bending?

A beam's flexural capacities are broken into two main categories: strong axis bending and weak axis bending. Weak axis bending is about the axis that has the lower moment of inertial (normally considered the y-y axis).

Weak axis bending is allowed a lower safety factor than strong axis bending since Lateral Torsional Buckling is not possible (since Ix is stronger than Iy LTB can't occur)

Note: These calculations work for I-shaped members and channels bent about their minor axis.

## How to solve for Weak Axis Bending

The nominal strength of the beam (Mn) shall be the lower value of:

1. Yielding (around the plastic moment)
2. Flange Local Buckling)

### 1) Yielding

 Yielding around the plastic moment is found by: M_n = M_p = F_yZ_y \le 1.6F_yS_y

where:

Mn = The nominal flexural strength (kip-in, lb-in, kip-ft, etc.)
Mp = Plastic Moment
Fy = The Yield Strength of the Steel (e.g. 36 ksi, 46 ksi, 50 ksi).
Zy = The plastic section modulus along the y or weak axis (can be found in the AISC 13th edition in Table 1-1) (in3)
Sy = The Section Modulus along the y or weak axis (similar to the plastic section modulus) (in3)

Important: Don't forget that Mn is the nominal moment which still needs to be divided by Ωb (for ASD = 1.67) or multiplied by ϕb (for LRFD = 0.9) to find the design flexural strength.

### 2a) Flange Local Buckling (compact flanges)

For sections with compact flanges the yielding state will apply (as found in Section 1 above).

### 2b) Flange Local Buckling (non-compact flanges)

 For sections with non-compact flanges: M_n = \left[M-p - M_p - 0.7F_yS_y)\left(\frac{\Lambda-\Lambda_{pf}}{\Lambda_{rf} - \Lambda_{pf}}\right)\right]

where:

Mn = The nominal flexural strength (kip-in, lb-in, kip-ft, etc.)
Mp = Plastic Moment (solved in Section 1 above)
Fy = The Yield Strength of the Steel (e.g. 36 ksi, 46 ksi, 50 ksi).
Sy = The Section Modulus along the y or weak axis (similar to the plastic section modulus) (in3)
λ = \frac{b}{t} , where b = base of the beam & t = thickness of the flange
λpf = λp = the limiting slenderness ratio for a compact flange (From the AISC 360, Section B4)
λrf = λr = the limiting slenderness ratio for a non-compact flange (From the AISC 360, Section B4)

### 2c) Flange Local Buckling (slender flanges)

 For sections with slender flanges: M_n = F_{cr}S_y

where:

Mn = The nominal flexural strength (kip-in, lb-in, kip-ft, etc.)
Fcr = \frac{0.69E}{\left(\frac{b_f}{2t_f}\right)^2} , where bf = the base of the flange, & tf = the thickness of the flange.
Sy = The Section Modulus along the y or weak axis (similar to the plastic section modulus) (in3)

## References

1. American Institute of Steel Construction (AISC 360), "Steel Construction Manual 13th edition", 2005
• Weak axis bending is discussed in Section F6 of the specs.
• Table B4.1 in the Specifications (AISC 360) has also been referenced to find the compact and non-compact slenderness limits.
• Most of the required variables to solve the above complex equations (for standard sized beams) can be found in Table 1 of the manual.

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