When the geometrical centre of area (centroid) of a column does not correspond to the load path, the situation introduces eccentric loading, which can lead to additional bending moments along with axial compression.

Here’s how you can approach the buckling analysis in this case:

1. Determine Eccentricity (e):

• The eccentricity is the distance between the line of action of the load and the centroid of the column’s cross-section.
• Eccentricity introduces bending in addition to axial compression, creating a combined load scenario.

2. Calculate the Total Stress:

• The total stress at any point on the cross-section is the sum of the direct axial stress due to the compressive load and the bending stress due to the eccentricity.

Where:

• = axial load
• = cross-sectional area of the column
• = moment due to the eccentricity
• = section modulus (depends on the cross-sectional shape)

3. Account for Buckling Using Euler’s Formula:

In the case of eccentric loading, you still apply Euler’s formula for critical buckling load:

However, because of the eccentricity, the effective load becomes a combination of axial load and the additional moments caused by eccentricity.

4. Use the Interaction Equation:

When both axial compression and bending are present, it’s important to check the combined effect of these two. One way to handle this is to use an interaction equation for combined axial and bending stress:

Where:

• is the critical axial load from Euler’s buckling formula.
• is the critical moment that would cause bending failure (yielding or buckling in the lateral direction).

5. Consider Secondary Bending (P-Δ effect):

Due to eccentric loading, the column will bend, and as it bends, additional moments (secondary moments) arise from the axial load acting on the deflected shape of the column. This phenomenon is called the P-Δ effect. You may need to perform a second-order analysis or use the magnification factor to account for this.

The magnification factor can be applied to account for the increase in bending moment:

Where is a function of the slenderness of the column and load level.

6. Check for Local Buckling:

If the cross-section of the column is not compact (thin-walled sections, for example), local buckling could occur before overall buckling. Ensure that the local buckling checks are performed depending on the type of column cross-section (e.g., wide flange, hollow circular).

7. Perform Numerical Analysis (If Needed):

If the geometry and loading are complex, or the column is subject to significant deflection, consider using finite element analysis (FEA) software to perform a nonlinear buckling analysis, taking into account eccentricity and P-Δ effects more accurately.