Real-time EN 1993-1-1 combined axial-bending check for steel beam-columns. Plot your design point on the P-M interaction envelope and get instant PASS/FAIL judgment.
Parameters
Column length L5.0 m
Cross-section
Material
End conditions
Axial force N800 kN
Bending moment My120 kN·m
Bending moment Mz30 kN·m
Check formula (EN 1993-1-1)
$$\frac{N_{Ed}}{N_{b,Rd}}+ k_{yy}\frac{M_{y,Ed}}{M_{b,Rd}}\leq 1.0$$
Reduction factor: $\chi = \frac{1}{\Phi + \sqrt{\Phi^2 - \bar{\lambda}^2}}$
Slenderness: $\bar{\lambda}= \sqrt{\frac{A f_y}{N_{cr}}}$
PASS — η = 0.00
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Ncr Euler critical load (kN)
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Slenderness λ̄
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Axial utilization N/Nb,Rd
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Bending utilization M/Mb,Rd
P-M Interaction Diagram
Buckling resistance vs. slenderness
What is Beam-Column Buckling?
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What exactly is a "beam-column"? Is it just a column that's also a beam?
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Basically, yes! It's a structural member that has to resist both an axial compressive force (like a column) and a bending moment (like a beam). The tricky part is they interact. The axial load makes the member more prone to bending failure, and the bending makes it more prone to buckling. Try the simulator: set the axial load high and see how little bending moment it can take before failing.
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Wait, really? So the "P-M Interaction Diagram" is like a map of safe combinations?
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Exactly! It's a failure envelope. Any combination of axial load (P) and moment (M) that plots inside the curve is safe. Outside, it fails. In practice, your design point from the sliders—your specific $N_{Ed}$ and $M_{y,Ed}$—must lie inside. Move the sliders and watch your point move in real-time; if it turns red, you've exceeded the limit.
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What's that $k_{yy}$ factor in the formula? It looks like an extra safety multiplier.
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Good eye. It's an interaction factor that accounts for how the moment's shape and the member's slenderness amplify the combined effect. A common case is a column in a building frame with moments at the ends. Change the "Moment Distribution" dropdown in the simulator—from uniform to triangular—and watch the $k_{yy}$ value and the interaction curve change. It's not just a safety factor; it's physics.
Physical Model & Key Equations
The core of the check is the interaction formula, which ensures the combined stress from axial load and bending doesn't cause instability. The European standard EN 1993-1-1 uses this linear interaction check.
$N_{Ed}, M_{y,Ed}$ = Design axial force and bending moment (your inputs in the simulator). $N_{b,Rd}, M_{b,Rd}$ = Design buckling resistance for compression and lateral-torsional buckling for bending. $k_{yy}$ = Interaction factor that depends on moment distribution and member slenderness.
The buckling resistance $N_{b,Rd}$ is reduced from the pure squash load by a factor $\chi$, which depends on the member's slenderness $\bar{\lambda}$. This captures how long, slender columns fail at much lower loads.
$\chi$ = Reduction factor (between 0 and 1). $\bar{\lambda}$ = Non-dimensional slenderness. $A f_y$ = Squash load (cross-section area × yield strength). $N_{cr}$ = Elastic critical buckling load (Euler load). Try increasing the "Effective Length" in the simulator: watch $\bar{\lambda}$ increase and $\chi$ drop, shrinking the safe zone on the diagram.
Real-World Applications
Building Frame Columns: In multi-storey steel buildings, columns are rarely under pure axial load. They experience significant bending moments from floor beams, wind loads, and imperfect alignment. Engineers use P-M diagrams to verify every column, especially corner columns which have biaxial bending.
Crane Runway Girders: The vertical girder of an overhead crane is a classic beam-column. It carries the vertical weight of the crane and load (axial compression) while also resisting the horizontal thrust and eccentric loads from movement (bending). Failure is often a buckling instability.
Offshore Platform Legs: The legs of jacket-type offshore platforms are subjected to enormous axial loads from the platform weight and environmental forces, combined with large bending moments from waves and currents. The interaction check is critical for safety in these harsh environments.
Industrial Storage Racks: The uprights in pallet racking are slender beam-columns. They carry vertical pallet loads (axial force) and moments from the eccentricity of the loads and potential impacts from forklifts. Fast, accurate P-M checks allow for optimizing material use while ensuring safety against collapse.
Common Misconceptions and Points to Note
When you start using this kind of tool, there are a few common pitfalls. The first is the assumption that strengthening the cross-section solves everything. It's true that increasing the flange width or web thickness of an H-section will improve its sectional properties. However, if the member length L remains long, the buckling resistance $N_{b,Rd}$ will hardly improve. For example, for a 5m long member, changing from an H-200x200 to an H-250x250 will increase the Euler buckling load proportionally to the moment of inertia, but if the slenderness ratio $\bar{\lambda}$ is still large, the buckling reduction factor $\chi$ won't increase as much as you might think. Effective buckling countermeasures require considering both "section strengthening" and "re-evaluating support conditions (effectively shortening the length)".
The second is misidentifying the "main player" between axial force and bending moment. Because the tool lets you adjust N and M independently, you might tend to think of their limit values separately. But in actual structures, for instance in a column with eccentric loading, axial force and bending moment are proportional. In your simulation, unless you evaluate by moving points on the graph along the expected path of the design load (e.g., a case where bending increases under constant axial force), your verification won't be realistic.
The third is forgetting to consider local buckling. This tool deals with member buckling (global buckling). However, parts composed of plate elements, like the flanges and web of an H-section, can experience "local buckling" where the plate itself buckles under high compressive stress. EN1993 prevents this through limits on width-to-thickness ratios. Even if the tool shows sufficient strength, if the section's width-to-thickness ratio exceeds the code limit, that section cannot be used. You must always check for both global and local buckling.
Related Engineering Fields
The concept of the P-M interaction diagram isn't limited to beam-columns. The interaction between axial force and bending moment is rooted in fundamental concepts of mechanics of materials. It's the first step from the combination of simple bar tension and bending towards more complex problems.
A directly connected field is Reinforced Concrete (RC) structures. Although the calculation procedures differ from steel, RC columns also simultaneously resist compressive axial force and bending. Their sectional strength is represented by an "interaction diagram" determined by the balance between the concrete compression block and the tension/compression in the reinforcement. While the shape is different, the underlying idea of defining a safe region is the same. Understanding the steel P-M interaction will help you grasp the meaning of RC interaction diagrams more intuitively.
Going a step further connects you to the world of nonlinear structural analysis. The interaction formulas from EN1993 used in this tool are actually a simplified method with a somewhat "linear" approximation. For a more precise evaluation, you would perform an elastoplastic analysis considering geometric nonlinearity (large deformations) and material nonlinearity (yielding). The resulting ultimate strength of the member typically envelops this P-M interaction curve. In other words, this curve acts as a "map" towards more advanced analysis.
For Further Learning
If you become interested in the calculation principles behind this tool, consider taking the next step. Start with the derivation of "Euler Buckling". Solving the differential equation $$EI \frac{d^2 y}{dx^2} = -P y$$ with boundary conditions yields the famous formula $P_{cr} = \frac{\pi^2 EI}{L^2}$. Through this derivation, you can intuitively understand that buckling is a problem of "stability of equilibrium".
Next, explore the "why" behind the EN1993 formulas. Why isn't a simple addition like $\frac{N}{N_{Rd}} + \frac{M}{M_{Rd}} \leq 1$ sufficient? How does the interaction factor $k_{yy}$ account for the effects of second-order moments ($P-\Delta$ effect) or initial imperfections (initial curvature)? Reading the background documents or commentaries to the code reveals the history of how this "conservative simplified formula" was developed based on countless experiments and numerical analyses.
Ultimately, learn about the extension of buckling modes. This tool primarily deals with flexural buckling about the weak axis (y-axis), but in reality, buckling about the strong axis (z-axis) and lateral-torsional buckling combining twisting and bending are also crucial. Especially for deep beams or cases where load is applied to only one flange, lateral-torsional buckling can become dominant. Understanding that a single phenomenon like buckling has diverse "faces" depending on the member shape and load direction will significantly broaden your perspective in structural design.