Understanding Buckling: Why Beams and Plates Fail Differently 

Beam Buckling vs. Plate Buckling Explanation

If you want to explore this topic deeper, please watch this lecture performed by SDC Verifier’s CEO and professor, Wouter van den Bos, on our YouTube channel:

Beam Buckling 

Image: Beam and Plate Buckling 

In a beam model, buckling analysis can be performed using beam elements. When analyzing beams, especially those designed as fixed boxes, the buckling length might theoretically be reduced (e.g., from center point to center point), but caution is advised in the design phase, as being slightly conservative may provide a necessary extra limit later on. 

A fundamental principle of beam buckling is that the lowest buckling shape is always the most critical one. Crucially, if a beam buckles and the material surpasses the yield limit at a certain location, the structure is generally considered to collapse, meaning there is no reserve capacity. 

Plate Buckling: Distinct Differences and the K Factor 

Image: Plate buckling visualization

Plate buckling occurs when a thin section’s edges are supported, typically by other plates in the perpendicular direction. If the edges are free, the results revert to those of standard Euler beam buckling. 

While plate buckling formulas utilize similar theoretical foundations to beam buckling, the behavior is significantly more complex. 

Critical Shapes and Geometry 

Unlike beams, the lowest buckling shape in plate buckling is not always the most critical. The critical risk is determined largely by the geometry ratio of the plate: 

1. Buckling Shape Factor (K): The buckling stress formula includes a factor K. This K factor depends on the resulting buckling shape. 

1. Width Dependence: Plate buckling risk primarily depends on the width (B) of the plate, rather than the length (A), assuming the length is not shorter than the width (A/B > 1). 

1. Square Buckling: A plate naturally tends to buckle in a square shape. If the length increases while the width remains constant, the most critical buckling shape changes (e.g., if the length is two times the width, the second buckling mode is the most critical). However, the overall buckling risk itself remains the same with an increase in length. 

Automating Beam and Plate Buckling Checks

Structural analysis software, SDC Verifier, provides engineers with powerful tools to automate the verification of beam and plate buckling according to international standards. 

• For beam buckling checks, SDC Verifier allows full compliance with international standards, including automated calculation of critical buckling shapes and stress comparisons.  

• For plate buckling checks, SDC Verifier calculates K-factors, effective widths, and post-buckling behavior automatically, handling various edge conditions and complex loading scenarios.  

• These tools save significant design time, reduce human error, and ensure structures meet international safety standards. 

Also, SDC Verifier provides Beam Member and Joint Checks App for large offshore and lattice-type structures by automatically detecting beam members and calculating buckling lengths in three directions (Y, Z, and torsional), fully independent of mesh density.

Comparing Buckling Limits and Yield Stress

All international standards (such as DNV, ABS, Lloyd’s Register, and Eurocode 3) compare the calculated Euler buckling limit (the stress at which the plate buckles) to the material’s yield stress σ_yield. 

If the buckling limit and the yield limit are assumed to be equal, this defines the maximum permissible width-to-thickness ratio (B/T). 

• For simply supported side plates, the buckling stiffness K is typically four. 

• Assuming a yield limit of 240, the required B/T ratio must be below 56 to prevent buckling from occurring before yielding. 

The Impact of High-Grade Steel 

When using higher grade steels, which possess a higher yield stress, the buckling phenomenon becomes more critical. To ensure the plate can achieve that higher yield limit before buckling, a lower maximum width-to-thickness ratio is necessary. 

For example, increasing the steel grade by 50% (e.g., from S235 to S360) means the designer can potentially reduce the plate thickness by 50%. But, it requires installing significantly more stiffeners (potentially 70% more), as the distance between the stiffeners must be reduced (e.g., from 56 times the thickness down to 46 times the thickness) to manage the increased buckling risk. 

The Role of Edge Conditions

Image: Comparison table of edges effect on plate buckling

The way the plate edges are supported drastically changes the buckling limit. 

• Simple Support (SS): This is the standard case, where there is no clamping at the edge, resulting in a K factor of four. 

• All Edges Clamped: Clamping all edges significantly reduces the buckling risk, increasing the buckling stiffness K to around seven. 

• Free Edges: Having a free edge results in a far lower stiffness. For a free edge, the stiffness might be approximately K=0.4, which is about ten times less than simple support. This severely restricts the maximum B/T ratio. If the B/T limit for simple support is 56, the limit for a free edge may drop to 18 times the thickness. 

Reserve Capacity and Effective Width

One of the most important practical differences between plate and beam behavior is the concept of reserve capacity. 

If a plate buckles at a stress level lower than its yield limit (i.e., it is a very wide plate), initial buckling occurs. However, the plate edges, which are typically connected to side plates, still retain stability and can reach the yield limit before they buckle. 

This post-buckling stability is quantified using the effective width. 

• Effective Width Definition: The effective width is the width portion of the plate that is assumed to carry the load up to the yield limit. 

• The effective width is determined by assuming the edges can reach yield while the central, buckled area carries little to no load. 

• For instance, if the limit where yielding and buckling coincide is 56 times the thickness (for S235 steel), then the effective width allows 28 times the thickness on each edge to reach the yield limit, even if the total plate is wider. 

Modifications to the effective width curve may be necessary to account for factors like initial plate imperfections or concentrated loads, such as those from a crane wheel. 

Combined Stress in Plate Design

Image: Effect of load type and combination

To assess the total risk, the standards calculate the risk in each direction, σ₁ and σ₂, and the shear risk, τ, and then combine these factors. 

The fundamental approach used across all major design standards is based on the idea of squaring the limits in each direction and ensuring their sum is below a certain safety threshold (often below one):

While different standards utilize varying safety factors and combination factors, the basic concept of comparing the buckling limit to the yield stress through these combined squared terms remains consistent.

Conclusion

Beams typically fail when the lowest buckling mode exceeds the material yield, offering little to no reserve capacity. Plates, on the other hand, can exhibit post-buckling stability, with factors such as geometry, edge conditions, and steel grade significantly influencing their behavior. Concepts like the buckling stiffness K, effective width, and combined stress criteria help engineers assess risk according to international standards. 

By recognizing these nuances, engineers can design safer, more efficient structures and make informed decisions about stiffener placement, material selection, and thickness. Software, like SDC Verifier, simplifies this process by automating buckling checks, calculating effective widths, and ensuring compliance with standards, ultimately saving time and reducing the risk of human error.