Tangent Modulus Load⚠ unverified
Mechanical / Columns · Compute the inelastic buckling load using the tangent-modulus theory
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| E_t | Et | Pa | 1.0 | Tangent modulus of the material at the stress level of interest |
| I | I | m**4 | 1.0 | Minimum second moment of area of the cross-section |
| L | L | m | 1.0 | Unsupported length of the column |
| K | K | — | 1.0 | Effective-length factor accounting for end conditions. Default is 1.0 |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Pt | N | Tangent-modulus buckling load, in newtons (N) |
The science & history
Understanding the Parameters
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Tangent modulus $E_t$ — the instantaneous stiffness $\mathrm{d}\sigma/\mathrm{d}\varepsilon$ at the current stress. In the elastic range $E_t = E$ and the formula reduces to Euler; past the proportional limit the stress–strain curve flattens, $E_t$ drops sharply, and the buckling load collapses. $E_t$ is a material property that depends on the stress level, so the equation is implicit (the load sets the stress, which sets $E_t$).
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Section and length $I$, $L$, $K$ — enter exactly as in the Euler load: capacity $\propto I/(KL)^2$. The only change from Euler is swapping $E \to E_t$; all the geometric intuition carries over.
Derivation (Approaching a Proof)
Repeat the Euler eigenvalue derivation (Euler Buckling Load column) but track what happens to the material as it bends inelastically. Engesser's tangent-modulus assumption is that, at the instant of buckling, the whole section undergoes an additional increment of compressive strain (the column bows but the average stress keeps rising), so every fibre stiffens according to the same tangent slope $E_t$ of the stress–strain curve.
With the bending stiffness now $E_t I$ instead of $EI$, the governing equation is
$$E_t I\,\frac{\mathrm{d}^2 v}{\mathrm{d}x^2} + P\,v = 0,$$
identical in form to Euler's, so the same boundary conditions give
$$P_t = \frac{\pi^2 E_t I}{(KL)^2}.$$
The equation is implicit: $P_t/A = \sigma$ determines the stress, which fixes $E_t = E_t(\sigma)$ from the material curve, which in turn changes $P_t$ — so it is solved iteratively. Shanley (1947) later showed the true buckling load lies between the tangent-modulus load (lower bound) and the reduced- modulus load (Reduced Modulus Load, upper bound), with the tangent-modulus load being the load at which buckling initiates — making $P_t$ the practical design value.
Dimensional check. $[P_t] = \dfrac{\text{Pa}\cdot\text{m}^4}{\text{m}^2} = \text{N}$. ✓
History and Development
Friedrich Engesser proposed the tangent-modulus theory in 1889 to correct Euler's overprediction for inelastic columns, then revised it to the reduced-modulus theory after Jasinski's critique. The apparent paradox — which modulus is right? — stood for decades until F. R. Shanley (1947) resolved it: the tangent-modulus load is where buckling begins, and it is the correct engineering value. Modern inelastic column analysis and finite-element buckling checks rest on this insight.
Related Concepts: Reduced Modulus Load, Euler Buckling Load column, Intermediate Column Load, Critical Slenderness, Slenderness Ratio
Notes: Requires the tangent modulus at the operating stress (from the material's stress–strain curve), so it is solved iteratively. Gives the load at which inelastic buckling initiates — the practical lower- bound design value, below the reduced-modulus load.