This document examines the calculation of moment resistance for concrete columns subjected to biaxial bending and axial load, as defined in Eurocode EN 1992-1-1. It first presents the Exact Solution, which resolves arbitrary cross-section shapes through an iteratively tilted neutral axis, and then the Simplified Method (Clause 5.8.9), which reduces the problem to two uniaxial checks combined by a power-sum criterion. To illustrate their relative accuracy and ease of use, a square section is analyzed under varying axial force ratios (NEd/NRd) across all load directions, highlighting when the simplified approach provides sufficient approximation and when the full exact solution is preferable.
Exact Solution
To use the exact solution for biaxial bending, with or without normal force, the neutral axis is tilted compared with the cross-section’s main axis. The neutral axis is perpendicular to the direction of deformation of the cross-section.
This method might be complex to compute, due to the tilted geometry, and unknown angle of neutral axis. An iterative approach can be used for computing the biaxial moment capacity of arbitrarily shaped reinforced concrete cross-sections with this method.
- Choose an angle for the Neutral axis.
- Determine MyRd and MzRd.
- Check if the direction of the moment resistance and the moment loads are equal.
- Repeat 1-3 until the correct neutral axis angle has been found
Simplified Method (EN 1992-1-1 5.8.9)
Biaxial bending combined with normal forces in concrete columns is a complex problem. Eurocode EN 1992-1-1 5.8.9 provides a simplified method, where the problem is divided into two uniaxial bending calculations. The combined utilization is then found using the simplified criterion:
\left(\frac{\text{M}_\text{z,Ed}}{\text{M}_\text{z,Rd}}\right)^\text{a} + \left(\frac{\text{M}_\text{y,Ed}}{\text{M}_\text{y,Rd}}\right)^\text{a} \leq 1.0The exponent a is between 1 – 2 and depend on the cross-section shape and the ratio between NEd/NRd.
| Circular or elliptical cross-section | Rectangular cross-section | |||
|---|---|---|---|---|
| NEd/NRd | - | 0.1 | 0.7 | 1.0 |
| a | 2.0 | 1.0 | 1.5 | 2.0 |
With linear interpolation for intermediate values
Example
The capacity of a square concrete section with a resulting moment load in different directions 0°-360° is compared with the Exact solution and the Simplified Method (EN 1992-1-1 5.8.9).
Below is a plot of the capacities for different ratios of NEd/NRd. With low ratios of NEd/NRd, the simplified method greatly underestimates the capacity for biaxial bending. However, for increasing NEd/NRd ratios the simplified method is sufficiently accurate.
