Reduced Modulus Load⚠ unverified
Mechanical / Columns · Compute the inelastic buckling load using the reduced-modulus theory
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| E_r | Er | Pa | 1.0 | Reduced (double) modulus of the material |
| 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 | Pr | N | Reduced-modulus buckling load, in newtons (N) |
The science & history
Understanding the Parameters
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Reduced modulus $E_r$ — a weighted average of the elastic modulus $E$ (governing the unloading side) and the tangent modulus $E_t$ (governing the loading side), with weights set by how the section geometry splits into loading and unloading zones. For a rectangular section it works out to $E_r = \dfrac{4 E E_t}{(\sqrt{E}+\sqrt{E_t})^2}$. Because unloading fibres are stiffer, $E_r > E_t$ always, so the reduced-modulus load exceeds the tangent-modulus load.
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Section and length $I$, $L$, $K$ — enter identically to Euler and tangent-modulus theory; the formula is the Euler expression with $E \to E_r$.
Derivation (Approaching a Proof)
Follow the Euler eigenvalue derivation but split the bending section into two zones at the instant of buckling under the reduced-modulus assumption: the axial load stays constant during the infinitesimal lateral deflection, so the column bends without net additional shortening. One side of the neutral axis therefore experiences increasing strain (loading, stiffness $E_t$) and the other decreasing strain (unloading, stiffness $E$).
Enforcing that the internal bending moment from this two-stiffness stress distribution balances the applied $P\,v$ yields the same eigenvalue equation with an effective flexural rigidity $E_r I$:
$$E_r I\,\frac{\mathrm{d}^2 v}{\mathrm{d}x^2} + P\,v = 0 \;\Longrightarrow\; P_r = \frac{\pi^2 E_r I}{(KL)^2},$$
where $E_r$ is fixed by requiring zero net change in axial force between the loading and unloading zones (this is what determines the neutral-axis position and hence the geometric weighting). The result $E_t < E_r < E$ places $P_r$ between the tangent-modulus and elastic Euler loads.
Shanley's resolution. The tangent-modulus load ($P_t$) is the load at which buckling initiates (some fibres are still elastic then, but begin to yield as it bows), while $P_r$ assumes constant load — an idealisation never quite realised. Real columns fail just above $P_t$ and below $P_r$; design uses the conservative $P_t$.
Dimensional check. $[P_r] = \dfrac{\text{Pa}\cdot\text{m}^4}{\text{m}^2} = \text{N}$. ✓
History and Development
Friedrich Engesser introduced the reduced-modulus concept in 1895 after Jasinski pointed out that his 1889 tangent-modulus theory ignored elastic unloading — for half a century the reduced-modulus load was believed to be the "correct" inelastic buckling load. F. R. Shanley (1946–47) overturned this, showing experimentally and theoretically that buckling initiates at the tangent-modulus load; the true maximum lies between $P_t$ and $P_r$. The episode is a classic case study in engineering theory self-correcting against experiment.
Related Concepts: Tangent Modulus Load, Euler Buckling Load column, Intermediate Column Load, Critical Slenderness, Slenderness Ratio
Notes: Theoretical upper bound on inelastic buckling; the tangent-modulus load is the practical (lower-bound) design value. $E_r$ depends on both $E$, $E_t$, and the section shape — supply the appropriate value for your cross-section.