Steel Beam Flexural Design

The design of steel beams for flexure (bending) involves ensuring that the moment capacity of the selected steel shape is greater than or equal to the maximum applied bending moment caused by the loads. In standard codes like AISC, flexural strength depends heavily on the beam's unbraced length and its susceptibility to local and global buckling.

1. Yielding and the Plastic Moment

For a compact steel section that is fully braced against lateral movement, the nominal flexural strength ($M_n$) is governed by the yielding of the entire cross-section, forming a plastic hinge.

The plastic moment capacity ($M_p$) is calculated as:

$M_p = F_y Z$

Where:

• $F_y$ is the specified minimum yield stress of the steel.
• $Z$ is the plastic section modulus about the axis of bending.

The allowable design strength is $\phi_b M_n$ in LRFD (Load and Resistance Factor Design), where $\phi_b = 0.90$ for flexure.

2. Lateral-Torsional Buckling (LTB)

If the compression flange of a beam is not adequately braced against lateral displacement, the beam may fail prematurely due to lateral-torsional buckling before reaching its full plastic moment capacity.

The unbraced length ($L_b$) is the distance between points braced against lateral displacement of the compression flange. The AISC specification defines two critical lengths, $L_p$ and $L_r$:

• $L_b \le L_p$: The beam is fully braced. It can reach $M_p$.
• $L_p < L_b \le L_r$: Inelastic lateral-torsional buckling occurs. The strength linearly transitions between $M_p$ and the buckling moment.
• $L_b > L_r$: Elastic lateral-torsional buckling occurs. The strength is heavily reduced and depends on the elastic buckling moment.

3. Design Procedure

1. Determine Factored Loads: Calculate the maximum factored bending moment ($M_u$) using appropriate load combinations.
2. Determine Bracing: Identify the unbraced length $L_b$.
3. Select Trial Shape: Select a trial section from steel tables based on required $Z$ (assuming fully braced initially).
4. Check Capacity: Calculate $\phi_b M_n$ considering LTB effects based on $L_b$. Ensure $\phi_b M_n \ge M_u$.
5. Check Shear and Deflection: Flexural design must always be paired with checks for shear strength and serviceability (deflection limits).