Beams: Flexure

Beams: Flexure

Beams are structural members primarily subjected to transverse loads that induce bending moments and shear forces. The flexural (bending) strength of a steel beam depends on its cross-sectional properties, its material yield strength, and critically, the unbraced length of its compression flange.

Flexural Strength Concepts

The behavior of a beam under bending is characterized by the elastic stress distribution, yielding of the extreme fibers, and eventually, full plastification of the cross-section.

Key Properties

Section Modulus ( $S$ ): A geometric property representing the elastic bending resistance. $S = I/c$ , where $c$ is the distance from the neutral axis to the extreme fiber. Stress is calculated as $f_b = M/S$ .
Yield Moment ( $M_y$ ): The bending moment that causes the extreme fibers to just reach the yield stress ( $F_y$ ). $M_y = F_y \times S_x$ . Beyond this point, the outer fibers yield, but the inner core remains elastic.
Plastic Modulus ( $Z$ ): A geometric property representing the fully plastic bending resistance. It is the sum of the first moments of the areas above and below the equal-area axis. $Z_x$ is roughly $1.10$ to $1.15$ times $S_x$ for standard W-shapes.
Plastic Moment ( $M_p$ ): The theoretical maximum moment a section can carry before it acts as a "plastic hinge." The entire cross-section (both tension and compression zones) has reached the yield stress. $M_p = F_y \times Z_x$ .

Compactness Classification and Local Buckling

Before a beam can reach its full plastic moment capacity (

M_p

), its individual plate elements (flanges and web) must not buckle locally. Sections are classified based on the width-to-thickness ratios (

\lambda

) of these elements.

The nominal moment capacity of a beam may be limited by Flange Local Buckling (FLB) or Web Local Buckling (WLB) if the elements are too thin relative to their width. AISC Table B4.1b provides the limiting ratios

\lambda_p

(plastic limit) and

\lambda_r

(yield limit).

Compact Section: $\lambda \le \lambda_p$ for both the flange and the web. The elements are thick enough to develop the full plastic capacity ( $M_p$ ) and sustain significant inelastic rotation before local buckling occurs. The vast majority of standard rolled W-shapes are compact for flexure ( $F_y = 50$ ksi).
Non-Compact Section: $\lambda_p < \lambda \le \lambda_r$ for either the flange or the web. The extreme fibers yield before local buckling occurs, but the element will buckle locally before the entire section can fully plastify. The nominal strength ( $M_n$ ) is interpolated linearly between $M_p$ and $0.7 F_y S_x$ .
Slender Section: $\lambda > \lambda_r$ . The element will buckle elastically before the extreme fibers reach the yield stress. The nominal strength ( $M_n$ ) is governed by elastic buckling equations and is significantly lower than $M_y$ . This is common for built-up plate girders but rare for rolled W-shapes.

Lateral-Torsional Buckling (LTB)

If a beam's compression flange is not continuously braced against lateral movement, the beam can fail globally due to Lateral-Torsional Buckling (LTB). The compression flange acts like a slender column attempting to buckle sideways. Because the tension flange tries to remain straight, this lateral movement is accompanied by twisting (torsion) of the entire cross-section.

The critical moment that causes LTB depends fundamentally on the unbraced length ( $L_b$ ), which is the distance between points of lateral support for the compression flange.

Zones of Behavior

AISC categorizes the flexural capacity of a compact beam into three distinct zones depending on its unbraced length (

L_b

) relative to two limits:

L_p

(the plastic limit length) and

L_r

(the inelastic limit length).

Beam Flexural Strength (M_n) vs. Unbraced Length (L_b)

Unbraced Length, L_b: 10 ft

Moment Gradient Factor, C_b: 1

Loading chart...

Beam Parameters (W18x35 approx):

L_p (Plastic Limit): 4.31 ft
L_r (Inelastic Limit): 13.5 ft
M_p (Plastic Capacity): 277.1 k-ft

Current State:

Unbraced Length: 10 ft
Zone: Zone 2 (Inelastic LTB)
Nominal Strength (M_n): 209.5 k-ft

LTB Capacity Equations (AISC Chapter F2)

Zone 1: Plastic Yielding ( $L_b \le L_p$ ) The beam is considered fully braced or is very short. It can develop its full plastic moment. LTB does not occur before the plastic hinge forms.

M_n = M_p = F_y Z_x

Zone 2: Inelastic LTB ( $L_p < L_b \le L_r$ ) The beam fails by inelastic lateral-torsional buckling. Residual stresses in the flanges initiate partial yielding before the beam buckles globally. The nominal strength is interpolated linearly between

M_p

and the moment causing onset of yielding (

0.7 F_y S_x

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

Zone 3: Elastic LTB ( $L_b > L_r$ ) The beam is long and unbraced. It fails purely by elastic lateral-torsional buckling before any part of the cross-section yields. The strength is governed by the critical buckling stress (

F_{cr}

M_n = F_{cr} S_x \le M_p

F_{cr} = \frac{C_b \pi^2 E}{(L_b/r_{ts})^2} \sqrt{1 + 0.078 \frac{J c}{S_x h_o} \left(\frac{L_b}{r_{ts}}\right)^2}

Where:

r_{ts}

is the effective radius of gyration,

J

is the torsional constant,

c=1.0

for doubly symmetric I-shapes, and

h_o

is the distance between flange centroids.

C_b

The basic LTB equations assume the worst-case scenario: a uniform bending moment along the entire unbraced length. In reality, the moment usually varies (e.g., higher at midspan, lower near supports). A non-uniform moment is less severe because only a small portion of the unbraced segment is subjected to the maximum compressive stress.

The $C_b$ factor accounts for this beneficial non-uniform moment distribution, amplifying the nominal LTB capacity (

M_n

) calculated in Zones 2 and 3.

Moment Gradient Factor

Calculates the beneficial effect of non-uniform bending moments.

$$
C_b = \\frac{12.5 M_{max}}{2.5 M_{max} + 3 M_A + 4 M_B + 3 M_C}
$$

Using $C_b = 1.0$ is always conservative (it assumes uniform moment).
The amplified capacity $C_b \times M_{n(LTB)}$ can never exceed the full plastic moment $M_p$ .

Biaxial Bending

Design of members subjected to bending about both principal axes simultaneously.

Purlins on sloped roofs, crane girders, and corner columns are often subjected to loads that do not align with either the strong (x-x) or weak (y-y) axis, causing bending about both axes simultaneously (biaxial bending).

The design must satisfy an interaction equation to ensure the combined stresses do not exceed the capacity. AISC uses a linear interaction formula for simple cases:

Biaxial Bending Interaction Equation

Checks that the combined moment ratio about both principal axes is less than or equal to 1.0.

$$
\\frac{M_{ux}}{\\phi M_{nx}} + \\frac{M_{uy}}{\\phi M_{ny}} \\le 1.0 \\text{ (LRFD)}
$$

Because LTB does not occur during weak-axis bending (the member is bending about its least stiff axis already),

M_{ny}

is simply the minimum of the plastic moment (

M_{py} = F_y Z_y

) and the flange local buckling capacity (

M_{n(FLB)} = 1.6 F_y S_y

Hole Reductions in Tension Flanges

Accounting for the loss of area due to bolted connections.

When holes are drilled or punched in the tension flange for bolted connections (e.g., at a splice or support), the net area of the tension flange is reduced. While yielding of the gross area governs overall flexural capacity, the rupture of the tension flange at the net section must be checked.

AISC Section F13 specifies that the flexural strength does not need to be reduced if the nominal tensile rupture strength of the net section is greater than or equal to the nominal yield strength of the gross section:

F_u A_{fn} \ge Y_t F_y A_{fg}

, no reduction is required. (Where

Y_t = 1.0

for

F_y/F_u \le 0.8

, else

1.1

If the inequality is not met, a reduced nominal moment capacity based on the net area must be used.

Continuous Beams and Plastic Design

Advanced analysis method utilizing the ductility of compact steel sections.

In traditional elastic design, a statically indeterminate continuous beam is considered to have reached its capacity when the single most highly stressed location reaches the yield moment (

M_y

However, compact steel sections possess immense ductility. When the full plastic moment (

M_p

) is reached at the highest moment location (usually over a support), the cross-section cannot resist any additional moment. However, it does not fail. Instead, it yields extensively, behaving like a plastic hinge.

Moment Redistribution and Mechanism Formation

Once a plastic hinge forms over a support, it continues to hold the moment

M_p

while rotating freely. Any additional load applied to the beam is then distributed to the less stressed regions (like the midspan), effectively transferring bending moments away from the critical section. This is called moment redistribution.

The beam will not collapse until enough plastic hinges form to turn the statically indeterminate structure into an unstable mechanism. Plastic Design explicitly relies on this reserved strength, allowing for lighter, more efficient structures compared to traditional elastic analysis. AISC Appendix 1 provides the rigorous requirements for inelastic (plastic) analysis, including stricter compactness limits (

\lambda_{pd}

) and unbraced length limits (

L_{pd}

) to ensure sufficient rotational ductility at the hinges.

Lateral Bracing Detailing

Methods for effectively bracing the compression flange against LTB.

The unbraced length (

L_b

) is the distance between points of effective lateral support. However, not all framing connections constitute "effective" bracing. To be considered a brace point, the bracing element must possess both sufficient strength (to resist the lateral buckling force) and sufficient stiffness (to prevent significant lateral movement).

Nodal vs. Relative Bracing (AISC Appendix 6)

AISC defines two main categories of bracing systems:

Nodal Bracing: Bracing that prevents lateral movement at a specific, discrete point along the beam (a "node"), without relying on other bracing points. Example: A cross-beam framing directly into the side of the compression flange.
Relative Bracing: Bracing that controls the lateral movement of one point relative to another adjacent point. Example: Diagonal cross-bracing (X-bracing) placed between two parallel main girders.

Lateral vs. Torsional Bracing

Lateral Bracing: Physically prevents the lateral translation of the compression flange. To be effective, the brace must be attached directly to, or very close to, the compression flange. Attaching a brace only to the tension flange is highly ineffective at stopping LTB.
Torsional Bracing: Prevents the cross-section from twisting. Because LTB involves both lateral movement and twisting, preventing twist is often sufficient to prevent LTB entirely. Example: Full-depth web stiffeners connected to a floor deck, or a rigid moment connection to a cross-beam.

Design Strength Summary

For LRFD:

LRFD Flexural Design Requirement

Checks that the design flexural strength exceeds the required moment.

$$
\\phi_b M_n \\ge M_u
$$

For ASD:

ASD Flexural Design Requirement

Checks that the allowable flexural strength exceeds the required moment.

$$
M_n / \\Omega_b \\ge M_a
$$

Shear Center and Torsion

The point through which applied loads must pass to prevent twisting of the cross-section.

When a transverse load is applied to a beam, it induces shear stresses. If the resultant of these shear stresses does not pass through the centroid of the cross-section, it creates a twisting moment (torsion). The Shear Center is the specific point on a cross-section where a transverse load can be applied without causing any torsion.

Properties of the Shear Center

Doubly Symmetric Shapes: For W-shapes and HSS, the shear center perfectly coincides with the geometric centroid. Loading through the centroid causes only bending, no twisting.
Singly Symmetric Shapes: For channels (C-shapes) and structural tees (WT-shapes), the shear center lies on the axis of symmetry but does not coincide with the centroid. For a channel, it lies entirely outside the web, away from the flanges.
Unsymmetric Shapes: The shear center must be calculated using the shear flow distribution.

If a beam is loaded off its shear center, it undergoes combined bending and torsion, significantly complicating the stress analysis and reducing its capacity.

Key Takeaways

The full plastic moment ( $M_p = F_y Z_x$ ) is the absolute theoretical maximum flexural capacity of a compact steel beam.
The nominal capacity of a beam can be governed by three limit states: Flange Local Buckling (FLB), Web Local Buckling (WLB), or Lateral-Torsional Buckling (LTB). Most W-shapes are compact and exempt from FLB/WLB.
Lateral-Torsional Buckling (LTB) limits the moment capacity depending on the unbraced length ( $L_b$ ) of the compression flange.
AISC categorizes beam LTB capacity into three zones based on $L_p$ and $L_r$ : Plastic (Zone 1, no LTB), Inelastic LTB (Zone 2, straight line interpolation), and Elastic LTB (Zone 3, Euler curve).
The $C_b$ factor accounts for non-uniform moment gradients within an unbraced segment, significantly increasing the calculated LTB capacity.
Biaxial bending requires an interaction equation to check the combined effects of strong-axis and weak-axis moments.
Compact, adequately braced continuous beams can utilize moment redistribution and plastic hinge formation to safely carry loads far beyond initial yielding.

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Flexural Strength Concepts

Key Properties

Compactness Classification and Local Buckling

Lateral-Torsional Buckling (LTB)

Zones of Behavior

Beam Flexural Strength (M_n) vs. Unbraced Length (L_b)

LTB Capacity Equations (AISC Chapter F2)

Moment Gradient Factor (CbC_bCb​)

Moment Gradient Factor

Biaxial Bending

Biaxial Bending Interaction Equation

Hole Reductions in Tension Flanges

Continuous Beams and Plastic Design

Moment Redistribution and Mechanism Formation

Lateral Bracing Detailing

Nodal vs. Relative Bracing (AISC Appendix 6)

Lateral vs. Torsional Bracing

Design Strength Summary

LRFD Flexural Design Requirement

ASD Flexural Design Requirement

Shear Center and Torsion

Properties of the Shear Center

Moment Gradient Factor ( $C_b$ )