Bending and compression

Bending with compression check (stability), k_ij interaction Methods A + B, per EN 1993-1-1 §6.3.3

General data

N_Ed [kN]

M_y,Ed [kNm]

M_z,Ed [kNm]

Section type

Section

U (UPN) profiles are not available here: the interaction method (Annex A/B) and torsional buckling formulas apply to doubly-symmetric sections; for U (mono-symmetric) the bending–torsion coupling would not be covered.

Material

γ_M1

Class Class 4 requires effective properties (A_eff, W_eff — EN 1993-1-5) and is not covered.

1. Flexural buckling about y-y

L_cr,y [mm]

M_y shape

2. Flexural buckling about z-z

L_cr,z [mm]

M_z shape

3. Lateral-torsional buckling (LTB)

L_ET [mm]

M_y shape

National annex ▸

Parameter set

▸ Fill in the data on the left and press Calculate

Tool information

What this calculator checks

The calculator verifies the stability of a member under combined compression and bending — a beam-column — according to EN 1993-1-1 §6.3.3. It checks the two interaction expressions (6.61) and (6.62):

$\frac{N_{Ed}}{\chi_y N_{Rk}/\gamma_{M1}} + k_{yy}\frac{M_{y,Ed}}{\chi_{LT} M_{y,Rk}/\gamma_{M1}} + k_{yz}\frac{M_{z,Ed}}{M_{z,Rk}/\gamma_{M1}} \le 1$

$\frac{N_{Ed}}{\chi_z N_{Rk}/\gamma_{M1}} + k_{zy}\frac{M_{y,Ed}}{\chi_{LT} M_{y,Rk}/\gamma_{M1}} + k_{zz}\frac{M_{z,Ed}}{M_{z,Rk}/\gamma_{M1}} \le 1$

The first expression governs buckling about the major axis, the second about the minor axis; both must be satisfied.

Why it is not a simple sum

Compression and bending do not add linearly. Axial force amplifies the displacements caused by the moment (second-order effect), and the moment reduces the stability reserve in compression. The real interaction is captured by the factors $k_{ij}$, which depend on slenderness, on the moment diagram shape and on the level of axial load.

Each term combines an effect (compression, y-bending, z-bending) with its resistance reduced for instability: $\chi_y$, $\chi_z$ for compression, $\chi_{LT}$ for major-axis bending. So this check subsumes both flexural buckling and lateral-torsional buckling, plus their interaction.

The two methods for $k_

Eurocode gives two equivalent algorithms for the interaction factors:

Method A (Annex A) — more accurate, with auxiliary factors $C_{yy}$, $C_{yz}$, $C_{zy}$, $C_{zz}$
Method B (Annex B) — simpler, tabulated

The calculator runs both and reports the governing check = the larger of the two utilisations. The choice of method is left to the National Annex; showing both avoids the surprise of a local code mandating exactly the more unfavourable one.

Input data

$N_{Ed}$, $M_{y,Ed}$, $M_{z,Ed}$ — the axial force (kN) and the moments about the two axes (kNm).
Profile and section — the full properties, including the torsion constants for $\chi_{LT}$.
Cross-section class — decides the section moduli in $M_{Rk}$.
Buckling lengths $L_{cr,y}$, $L_{cr,z}$, $L_{ET}$ — for $\chi_y$, $\chi_z$ and the torsional part.
Moment diagram shape and end restraints — enter the equivalent moment factors ($C_1$, $k_c$) for $\chi_{LT}$ and the $k_{ij}$.
Steel grade and $\gamma_{M1}$ (recommended 1.0; the National Annex takes precedence).

Assumptions and limitations

Applies to members of constant, doubly symmetric (I/H) section.
For Class 4 the check uses effective properties.
Additional moments from local second-order effects ($N \cdot e_N$ for Class 4 sections) are treated per §6.3.3(4).
The calculation checks the isolated member with the given buckling lengths; it does not replace a global second-order analysis of the structure where that is required.

Flexural buckling — the compression component, $\chi_y$ and $\chi_z$
Lateral-torsional buckling — the bending component, $\chi_{LT}$
Bending and axial force — the $M$–$N$ interaction at section level (no instability)
Cross-section class — decides the properties in $N_{Rk}$ and $M_{Rk}$