BIZOMAT 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. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Axial load
- What it is
- The axial compressive load on the column, applied parallel to its axis but offset from the centroid by the eccentricity.
- Units and format
- Enter a number in N: newtons. The form accepts 0.000001 N or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter newtons; the tool reports the Euler load in kilonewtons. It must be below the Euler load or no result is given.
What this means: Eccentricity of load
- What it is
- The sideways distance from the line of action of the load to the centroid of the section, for example a bracket or seat that loads the column off-centre.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- More eccentricity adds bending, raising peak stress and lowering the load factor to yield.
- Get it right
- Measure in the bending plane that matches the second moment of area you enter.
What this means: Area
- What it is
- The cross-sectional area of the column, used for the direct stress and, with the second moment of area, the radius of gyration.
- Units and format
- Enter a number in mm²: square millimetres. The form accepts 0.000001 mm² to 100000000 mm². Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1000 mm² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the net steel area of the actual section, from section tables or a section calculation.
What this means: Second moment of area (bending plane)
- What it is
- The second moment of area of the section about the axis it bends around because of the eccentricity.
- Units and format
- Enter a number in mm⁴: millimetres to the fourth power (second moment of area). The form accepts 0.000001 mm⁴ or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 400000 mm⁴ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Choose the axis that matches the offset direction; it is not always the strong axis.
What this means: Distance to extreme fibre
- What it is
- The distance from the neutral axis to the outermost fibre on the compressed side, where the peak stress occurs.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 25 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- For an unsymmetrical section, measure to the face on the side the load is offset towards.
What this means: Length
- What it is
- The unsupported length of the column between points that are restrained against sideways movement.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2000 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A longer column lowers the Euler load and increases the secant amplification of the bending stress.
What this means: Effective length factor
- What it is
- The multiplier that turns the actual length into the effective buckling length according to how the ends are held (pinned, fixed or free).
- Units and format
- Enter a plain number with no unit. The form accepts 0.1 to 5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The effective length is K × L; the Euler load falls with the square of it.
- Get it right
- Take it from your structural code for the actual end restraint; a pinned-pinned column uses 1.
What this means: Elastic modulus
- What it is
- The elastic (Young’s) modulus of the column material, used for the Euler load and the secant amplification.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 1000 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter gigapascals, not megapascals.
What this means: Yield strength
- What it is
- The minimum yield strength of the column material; the tool finds the load multiplier at which the peak secant stress reaches it.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 100000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 250 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the specified minimum or certificate value, not a typical value.
What this means: Required load factor
- What it is
- The minimum multiple of the applied load, before first yield, that your design code or specification requires; the calculated load factor is checked against it.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 20. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- This is a load factor, not a stress ratio: the peak stress rises faster than the load because of the secant term.
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.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Crooked members and offset loads
What it covers: How initial crookedness and off-centre loads turn the sudden buckle into growing bending: the amplification factor, the secant formula, wind on a tank stand and a detail that moved the load line.
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