Module 8: Steel Beams (Flexural Members) - Theory & Concepts

Module 8: Steel Beams (Flexural Members)

Steel Flexural Members

Beams in structural steel framing carry gravity loads to columns and provide lateral stability to the floor system. The primary design limit state for most steel beams is flexural yielding or lateral-torsional buckling. Because steel is ductile and strong, beams are often designed to reach their full plastic capacity.

Bending Stresses and Plastic Moment Capacity

The behavior of a steel beam under increasing load differs vastly from timber. Timber remains largely elastic until failure. Steel, due to its immense ductility, can yield extensively.

Checklist

Elastic Range ( $M_y$ ): The moment that causes the extreme outer fibers (flanges) to reach the yield stress ( $F_y$ ). The core of the beam remains elastic. $M_y = F_y \times S_x$ (where $S_x$ is the elastic section modulus).
Plastic Range ( $M_p$ ): As the moment increases beyond $M_y$ , the yielding progresses inward toward the neutral axis. Eventually, the entire cross-section (flanges and web) yields. This is the Plastic Moment, the absolute maximum bending capacity of the section.
Calculated using the plastic section modulus ( $Z_x$ ), where:

$$
M_p = F_y Z_x
$$

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The ratio $Z_x / S_x$ is called the shape factor. For a standard W-shape, it is approximately $1.10$ to $1.15$ , indicating a 10-15% reserve strength beyond first yield.

Key Takeaways

Steel beams are designed based on their Plastic Moment ( $M_p$ ), not just the initial yield moment ( $M_y$ ).
The plastic section modulus ( $Z_x$ ) is used instead of the elastic section modulus ( $S_x$ ).
W-shapes possess a reserve strength of 10-15% beyond initial yielding due to their shape factor.

Compactness and Width-to-Thickness Ratios

Before determining a beam's moment capacity, you must confirm that its cross-section is "compact." A compact section can fully yield and develop its plastic moment (

M_p

) without the flanges or web buckling locally first.

Local Buckling Parameters

To ensure local buckling does not govern, we check the width-to-thickness ratio (

\\lambda

) of the beam's individual plate elements (the flanges and the web) against specific code limits.

Compact Limit ( $\\lambda_p$ ): If the ratio of every element is less than or equal to its respective $\\lambda_p$ , the section is compact. It can reach $M_p$ without local buckling. Most standard W-shapes are rolled specifically to be compact for $F_y = 50 \\text{ ksi}$ steel.
Noncompact Limit ( $\\lambda_r$ ): If the ratio exceeds $\\lambda_p$ but is less than $\\lambda_r$ , the section is noncompact. The extreme fibers will yield, but local buckling will prevent the section from reaching the full plastic moment ( $M_p$ ).
Slender Limit (exceeds $\\lambda_r$ ): The element is so thin it will buckle elastically before any yielding occurs.

Laterally Supported Beams (Zone 1)

To reach the plastic moment (

M_p

), the compression flange of the beam must be prevented from buckling sideways. The length between lateral bracing points is the unbraced length (

L_b

Yielding Limit State

If the unbraced length (

L_b

) is very short or the beam is continuously braced (e.g., a steel beam supporting a concrete slab connected by shear studs), the beam will never buckle laterally. The only limit state is yielding (reaching the plastic moment).

$$
M_n = M_p = F_y Z_x
$$

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This defines Zone 1. The limiting unbraced length for full plastic yielding is defined as $L_p$ . If $L_b \le L_p$ , the nominal capacity is $M_p$ .

$$
L_p = 1.76 r_y \\sqrt{\\frac{E}{F_y}}
$$

Key Takeaways

Zone 1 applies when the unbraced length ( $L_b$ ) is less than or equal to $L_p$ .
In Zone 1, lateral-torsional buckling is prevented, and the beam can reach its full plastic moment ( $M_p$ ).

Laterally Unsupported Beams (Zones 2 & 3)

If the unbraced length (

L_b

) exceeds

L_p

, the nominal moment capacity (

M_n

) begins to decrease due to the increasing probability of Lateral-Torsional Buckling (LTB).

Lateral-Torsional Buckling (LTB)

L_b

increases, the compression flange becomes unstable, twists laterally, and causes the entire beam to buckle out-of-plane before reaching

M_p

. The NSCP defines two zones for LTB based on an upper unbraced length limit,

L_r

Zone 2: Inelastic LTB ( $L_p \lt L_b \le L_r$ ): Intermediate unbraced length; beam fails by inelastic lateral-torsional buckling. Nominal strength ( $M_n$ ) linearly decreases from $M_p$ to a lower residual stress level ( $0.7 F_y S_x$ ). Requires calculating the bending coefficient ( $C_b$ ) based on the moment gradient.

$$
M_n = C_b \\left[ M_p - (M_p - 0.7 F_y S_x) \\left( \\frac{L_b - L_p}{L_r - L_p} \\right) \\right] \\le M_p
$$

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Zone 3: Elastic LTB ( $L_b \gt L_r$ ): Long unbraced length; beam fails purely by elastic lateral-torsional buckling before any yielding occurs. The nominal strength ( $M_n$ ) drops off sharply following a theoretical elastic buckling curve dependent on the section's torsional constant ( $J$ ) and warping constant ( $C_w$ ).

$$
M_n = F_{cr} S_x \le M_p
$$

Key Takeaways

Lateral-torsional buckling drastically reduces the moment capacity as the unbraced length ( $L_b$ ) increases beyond critical limits ( $L_p$ , $L_r$ ).
Zone 2 ( $L_p \lt L_b \le L_r$ ) is governed by an inelastic linear transition formula.
Zone 3 ( $L_b \gt L_r$ ) is governed by an elastic buckling curve.
The modification factor $C_b$ accounts for non-uniform moment distributions along the unbraced length, often increasing the calculated $M_n$ .

Holes in the Tension Flange

Net Flange Area Limit

In many cases, beams must have holes drilled through their flanges to attach to columns or support other members. If these holes are in the tension flange, they reduce the area available to resist the tension component of the bending moment.

If the holes are relatively small, the immense ductility of steel allows it to yield and redistribute stresses around the hole without losing its overall plastic moment capacity ( $M_p$ ).
However, if the holes are too large (specifically, if $F_u A_{fn} < Y_t F_y A_{fg}$ ), the tension flange will rupture at the net section before the rest of the cross-section can yield. In this case, the nominal moment capacity ( $M_n$ ) must be reduced based on the net area of the tension flange.

Shear Strength of Steel Beams

Shear in a W-shape is carried almost entirely by the web.

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Unlike concrete or wood, shear stress is assumed to be uniformly distributed across the entire area of the web ( $A_w = d \times t_w$ ).
The nominal shear strength ( $V_n$ ) relates the web's area ( $d t_w$ ) to the steel's shear yield stress ( $0.6 F_y$ ).
Depending on the web's height-to-thickness ratio ( $h/t_w$ ), failure is governed by yielding ( $C_v = 1.0$ , common for W-shapes) or inelastic/elastic web buckling ( $C_v \lt 1.0$ ).

$$
V_n = 0.6 F_y A_w C_v
$$

Key Takeaways

Shear capacity is derived from the web's area ( $A_w$ ) and the steel's shear yield stress ( $0.6 F_y$ ).
For most standard W-shapes, web buckling does not govern, and $C_v = 1.0$ .

Deflection Limits

Checklist

Evaluating beam stiffness to ensure serviceability under live and dead loads.
Calculated deflections ( $\Delta_{actual}$ ) must remain within code-specified limits (e.g., $L/360$ for live load, $L/240$ for total load) to prevent damage to non-structural elements (e.g., plaster, glass). Deflection is checked using service (unfactored) loads even in LRFD design.

Key Takeaways

Deflection is checked under service (unfactored) loads.
Live load deflection and total load deflection must both fall under specified limits to ensure serviceability.

Biaxial Bending and the Shear Center

When a beam is loaded simultaneously in both the X and Y axes, or when a load is applied such that it induces twisting, additional design considerations are required.

Biaxial Bending

Similar to timber, steel beams loaded about both principal axes (e.g., roof purlins) must be designed for biaxial bending. The interaction of these moments is typically evaluated using a linear interaction equation where the sum of the ratios of required moment to available moment for each axis must be less than or equal to 1.0.

The Shear Center

For a beam to bend without twisting, the applied transverse load must pass through a specific point on the cross-section known as the Shear Center.

Doubly Symmetric Shapes: For W-shapes, the shear center perfectly aligns with the geometric centroid. A vertical load applied to the top flange passes through the shear center and causes only bending.
Singly Symmetric and Asymmetric Shapes: For shapes like Channels (C-shapes) or Angles, the shear center is located completely outside the physical web. If a load is applied through the centroid (or directly to the web) of a channel, it will induce a massive torsional moment, causing the beam to twist dramatically.
To avoid torsion in these shapes, loads must be applied specifically at the shear center, or the beam must be heavily braced against twisting.

Combined Axial and Bending Stresses

Steel members often experience simultaneous axial loads (tension or compression) and bending moments. This occurs in beam-columns, portal frames, and truss chords subjected to transverse loads. The NSCP/AISC prescribes interaction equations to ensure these combined forces do not exceed the member's capacity.

Interaction Equations (H1-1)

The interaction between axial force and bending moment is evaluated using a ratio-based approach. The sum of the utilization ratios must be less than or equal to

1.0

$$
\\text{If } \\frac{P_r}{P_c} \\ge 0.2: \\quad \\frac{P_r}{P_c} + \\frac{8}{9} \\left( \\frac{M_{rx}}{M_{cx}} + \\frac{M_{ry}}{M_{cy}} \\right) \\le 1.0
$$

$$
\\text{If } \\frac{P_r}{P_c} \\lt 0.2: \\quad \\frac{P_r}{2P_c} + \\left( \\frac{M_{rx}}{M_{cx}} + \\frac{M_{ry}}{M_{cy}} \\right) \\le 1.0
$$

Checklist

$P_r$ = Required axial strength (factored load)
$P_c$ = Available axial strength ( $\phi P_n$ )
$M_r$ = Required flexural strength (factored moment)
$M_c$ = Available flexural strength ( $\phi M_n$ )

Lateral-Torsional Buckling (LTB) Zones

Steel beam flexural capacity depends entirely on the unbraced length (

L_b

) compared to the limiting lengths

L_p

(plastic limit) and

L_r

(elastic limit).

The Three Zones of LTB

Zone 1: Yielding ( $L_b \le L_p$ ): The beam is adequately braced and will reach its full plastic moment ( $M_p$ ) before any buckling occurs.
Zone 2: Inelastic LTB ( $L_p \lt L_b \le L_r$ ): The beam buckles laterally after some portions of the cross-section have yielded. The nominal moment capacity ( $M_n$ ) linearly decreases from $M_p$ down to the residual yield moment ( $0.7 F_y S_x$ ). A moment gradient modifier ( $C_b$ ) is applied to account for non-uniform bending moments along the span.
Zone 3: Elastic LTB ( $L_b \gt L_r$ ): The beam buckles laterally while the entire cross-section is still in the elastic range. Failure is sudden and governed solely by the member's lateral and torsional stiffness, not its yield strength.

Key Takeaways

The plastic moment ( $M_p$ ) is the absolute upper bound of a flexural member's bending capacity.
Lateral-torsional buckling drastically reduces the moment capacity as the unbraced length ( $L_b$ ) increases beyond critical limits ( $L_p$ , $L_r$ ).
Web elements must provide adequate shear resistance ( $0.6 F_y A_w$ ) to prevent shear yielding or web buckling.
Members subjected to combined axial load and bending must satisfy specific interaction equations (H1-1) to ensure their structural integrity is not compromised by the coupled effects.

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