KEVOS® MAKEOS · Strength
Eccentrically Loaded Column (Secant Formula)
What is the peak stress in this column with an offset load, and what load factor brings it to yield?
Maximum stress in a column with an eccentric axial load by the secant formula, the Euler load, and the load factor at which the peak stress reaches yield.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Axial load | N | Compressive load. |
| Eccentricity of load | mm | Distance from the load line to the section centroid. |
| Area | mm² | Cross-section area. |
| Second moment of area (bending plane) | mm⁴ | Second moment of area about the axis of bending caused by the eccentricity. |
| Distance to extreme fibre | mm | Distance from the neutral axis to the compressed extreme fibre. |
| Length | mm | Unsupported length. |
| Effective length factor | Effective length factor for the end conditions. | |
| Elastic modulus | GPa | Elastic modulus from the material data sheet (steel is typically about 200 GPa). |
| Yield strength | MPa | Minimum yield strength from the material certificate or specification. |
| Required load factor | Factor of safety required by your design code, customer specification or company standard. |
Method and formulas
- r² = I ÷ A. σmax = P/A × [1 + (e c ÷ r²) sec((KL ÷ 2r) √(P ÷ A E))].
- Euler load P_E = π² E I ÷ (KL)².
- Load factor n: solve σmax(nP) = S_y by bisection (n P stays below P_E).
- Example: 40000 N at 10 mm eccentricity gives σmax = 72.884 MPa; yield is reached at 2.5431 × the load.
Limitations
- Elastic, prismatic, pin-ended (or effective-length) member with equal end eccentricities.
- Structural members must still be checked to the applicable code.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check columns, legs and struts loaded off-centre
- Allow for accidental eccentricity in a strut
- Compare with the concentric buckling result
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Maximum stress at the applied load
- Euler buckling load
- Load factor to first yield
Interpreting the result
- The secant formula is non-linear: doubling the load more than doubles the peak stress near buckling.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 281–287. Topic reference; no source tables or recommended settings are reproduced.
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