AASHTO Compactness Classification

Welcome to the final lesson of our first module. In our previous sessions, we have established a firm grasp of the AASHTO LRFD philosophy, from load factors and combinations on the "demand" side to material properties and resistance factors on the "capacity" side of the design equation. We know the fundamental equation we must satisfy is $\sum η_{i} γ_{i} Q_{i} \leq ϕ R_{n}$ .

Having defined the steel's yield strength ( $F_{y}$ ) and the appropriate resistance factor ( $ϕ$ ), our next task is to determine the nominal resistance, $R_{n}$ . For members subjected to compression, either from axial loads or flexure, the resistance is not solely a function of the material's yield strength. It is critically dependent on the stability of the cross-section's components. A thin plate element under compression can buckle locally before the member as a whole can develop its full strength.

This lesson focuses on how AASHTO LRFD accounts for this phenomenon. Our objective is to classify cross-section elements as compact, noncompact, or slender using AASHTO's specified width-to-thickness limits. Understanding this classification is the prerequisite for calculating the nominal flexural or compressive resistance of any steel member, forming the final piece of our foundational knowledge before we dive into the specifics of tub girders.

1. Local Buckling and Cross-Section Classification

The concept of local buckling governs the strength of steel compression elements. Imagine compressing a thin steel plate along its edges; it will eventually wrinkle or "buckle" out-of-plane. The stress at which this occurs depends on its width-to-thickness ratio ( $b / t$ ), its support conditions, and its material properties.

AASHTO categorizes the behavior of plate elements under compression into three distinct classes based on their slenderness:

Compact: The element is sufficiently "stocky" (has a low width-to-thickness ratio). It can withstand compressive strains well into the plastic range without local buckling. This allows the cross-section to develop its full plastic moment capacity, $M_{p}$ , and exhibit significant rotational ductility.
Noncompact: The element has an intermediate slenderness. It can reach its yield stress, $F_{y}$ , but local buckling will occur in the inelastic range shortly after. The section cannot develop the full plastic moment, and its capacity is limited by inelastic local buckling.
Slender: The element is so thin (has a high width-to-thickness ratio) that local buckling will occur in the elastic range, before the stress in the element reaches yield, $F_{y}$ . Its capacity is limited by elastic buckling.

AASHTO defines the boundaries between these classifications with limiting slenderness ratios, denoted $λ_{p}$ (for the compact/noncompact boundary) and $λ_{r}$ (for the noncompact/slender boundary). If $λ$ is the element's actual width-to-thickness ratio:

If $λ \leq λ_{p}$ , the element is Compact.
If $λ_{p} < λ \leq λ_{r}$ , the element is Noncompact.
If $λ > λ_{r}$ , the element is Slender.

Let's see how this is applied in practice using a design example.

2. Classifying Compression Flanges

The most common elements you will classify are the girder flanges when they are in compression. This occurs in the top flanges of a tub girder during construction (before the deck is composite) and in the bottom flange in negative moment regions (e.g., over a pier).

The width-to-thickness ratio for an unstiffened flange element, like a top flange of a tub girder, is typically defined as:

λ_{f} = \frac{b _{f c}}{2 t _{f c}}

where $b_{f c}$ is the full width and $t_{f c}$ is the thickness of the compression flange. The "2" in the denominator accounts for the flange being a plate supported along one edge (the web) and free on the other, but symmetry in I-girders and box-girders effectively creates two such plates.

The limiting slenderness for a compact flange is given in AASHTO 6.10.8.2.2:

λ_{p f} = 0.38 \frac{E}{F _{y c}}

Let's see this in a worked example.

[PDF] aashto-lrfd design example horizontally curved steel box girder ...

This design example demonstrates a constructibility check for a tub girder, which involves classifying the top compression flange.

Please navigate to page D-8, section "Constructibility - Top Flange in Compression (continued)". Focus on the calculations for the local buckling resistance. Review the calculation of the slenderness ratio \lambda_f and the subsequent calculation of the limiting ratio \lambda_{pf}. Note the conclusion: since \lambda_f < \lambda_{pf}, the flange is classified as compact.

As you saw in the example, with $λ_{f} = 8.0$ and $λ_{p f} = 9.15$ , the flange is compact. The consequence, as stated in the text, is that the nominal flexural resistance for local buckling, $F_{n c}$ , can be taken as the full yield strength, $F_{y c}$ .

3. Classifying Stiffened and Unstiffened Webs and Flanges

The wide bottom flange of a tub girder and the deep webs present a more complex stability problem. They are often too slender to function effectively without stiffeners. This is where the noncompact and slender classifications become highly relevant.

Unstiffened Flanges (Potentially Slender)

Let's first consider an unstiffened bottom flange in a negative moment region. The width b is the full distance between the webs.

The following reading compares two design options for a bottom flange under compression: a thick, unstiffened plate versus a thinner, stiffened plate. Let's analyze the unstiffened case first.

[PDF] Three-Span Continuous Straight Composite Steel Tub Girder Bridge

This section of the AISC design example provides an excellent direct comparison between an unstiffened and a stiffened bottom flange, clearly demonstrating the different classifications.

Please read Section 8.10.1.1 Option A - Unstiffened Flange (page 94). Focus on the calculation of the flange slenderness ratio \lambda_f. Then, follow the calculation of the limiting slenderness ratio \lambda_r. The key takeaway is the comparison: Since λf = 69.18 > λr = 59.07. This classifies the unstiffened flange as slender.

The consequence of being slender is a significant reduction in capacity. The nominal resistance $F_{c b}$ drops to just 21.81 ksi, less than half the material's yield strength of 50 ksi.

Stiffened Flanges (Potentially Noncompact)

Now let's examine what happens when a longitudinal stiffener is added, as is common for tub girder bottom flanges over piers. The stiffener subdivides the wide plate into smaller, more stable sub-panels.

This image shows a steel girder with varying flange thicknesses. The flanges are thickest near the pier, where compressive stresses are highest, in part to maintain a non-slender (compact or noncompact) cross-section and prevent local buckling.

The key changes are:

The width b in the slenderness ratio is now the distance from the web to the stiffener, which is much smaller.
The plate buckling coefficient, k, is no longer a constant (like 4.0 for the unstiffened plate) but now depends on the stiffness provided by the longitudinal stiffener (Is). A stiffer stiffener increases k, which in turn increases the slenderness limits $λ_{p}$ and $λ_{r}$ .

The following reading demonstrates the full classification for a stiffened bottom flange.

[PDF] Three-Span Continuous Straight Composite Steel Tub Girder Bridge

This detailed calculation shows how adding a longitudinal stiffener changes the classification and resulting capacity of the bottom flange.

Please go to Section 8.7.1.1 Bottom Flange - Flexural Resistance in Compression - Stiffened Flange (starts on page 81). Notice how the slenderness ratio \lambda_f is now calculated with w being half the flange width. Review the calculation for the plate buckling coefficient k, which now depends on the stiffener's moment of inertia, Is. Follow the calculations for the limiting ratios $\lambda_p$ and $\lambda_r$. Observe the final comparison: Since λp < λf < λr. This classifies the stiffened flange as noncompact.

In this case, with $λ_{p} = 25.13 < λ_{f} = 27.18 < λ_{r} = 54.06$ , the element is noncompact. Its capacity, $F_{c b} = 48.79$ ksi, is only slightly below the yield strength. This demonstrates the efficiency of using stiffeners to control local buckling.

Classifying Webs

Webs are classified based on their slenderness, $D / t_{w}$ . For unstiffened webs, AASHTO 6.11.2.1.2 limits this ratio to $\leq 150$ . When a web is in flexure, we are most concerned with the portion in compression, $D_{c}$ . The slenderness of the compression portion of the web, $2 D_{c} / t_{w}$ , is compared against limits similar to those for flanges to determine if the web is slender. If it is, its contribution to the girder's flexural resistance is reduced via the web load-shedding factor, $R_{b}$ , which we saw in the previous reading. We will explore this factor in more detail when we design for flexure.

Conclusion

This lesson concludes our exploration of the fundamental principles of the AASHTO LRFD specification. You are now equipped with the tools to classify the primary components of a steel girder, which is the essential first step in determining its strength.

Key Takeaways:

The capacity of a steel compression element is governed by its stability against local buckling.
AASHTO classifies elements as compact, noncompact, or slender based on their width-to-thickness ratio, $λ$ .
Compact elements ( $λ \leq λ_{p}$ ) can reach their full plastic capacity.
Noncompact elements ( $λ_{p} < λ \leq λ_{r}$ ) can reach yield, but their capacity is limited by inelastic local buckling.
Slender elements ( $λ > λ_{r}$ ) buckle elastically before reaching yield, leading to a significant reduction in capacity.
Adding stiffeners is an effective strategy to reduce the effective width b and increase the buckling coefficient k, thereby improving the classification and increasing the resistance of an element.

This marks the end of Module 1. We have covered the LRFD philosophy, loads, load combinations, materials, resistance factors, and now, section classification. With this foundation, we are ready to begin Module 2, where we will start defining the specific geometry of steel trapezoidal tub girders, applying these principles to proportion their flanges and webs.

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