Secant Formula Stress⚠ unverified

Understanding the Parameters

• Axial load $P$ — enters three ways: the direct stress $P/A$, the eccentric moment $Pe$, and the amplification argument. This triple role is what makes a beam-column nonlinear: the axial load both causes and amplifies the bending.

• Eccentricity $e$ and applied moment $M$ — the two sources of primary bending. $Pe$ is the moment from the off-centre load; $M$ is any additional applied end moment. Their sum $(M+Pe)$ is the first-order moment that the secant then amplifies.

• Extreme-fibre distance $c$ and inertia $I$ — convert the amplified moment into a bending stress via the flexure formula $\sigma = Mc/I$ (see Beam Bending Stress).

• The secant argument — $\text{arg} = \dfrac{KL}{2}\sqrt{P/EI}$ is half the column's "buckling phase." At $P = P_{cr} = \pi^2 EI/(KL)^2$ the argument equals exactly $\pi/2$, where $\sec\to\infty$. So the formula predicts unbounded stress at the Euler load — the mathematical signature of buckling.

Registry note: the calculator lists $\text{arg}$ implicitly; it is the standard beam-column phase $\tfrac{KL}{2}\sqrt{P/EI}$ (equivalently $\tfrac{L}{2r}\sqrt{P/EA}$). Confirm the exact argument and whether $K$ is applied in the ToolBox implementation. Flagged in Known Issues.

Derivation (Approaching a Proof)

Model a pin-ended column of length $L$ loaded by $P$ at eccentricity $e$. The lateral deflection $v(x)$ obeys the beam-column equation with the load acting through its own eccentricity:

$$EI\,v'' + P v = -P e.$$

This is a linear ODE with constant forcing. Its solution, applying the symmetric boundary conditions, gives the mid-height deflection

$$v_{\max} = e\left[\sec\!\left(\frac{L}{2}\sqrt{\frac{P}{EI}}\right) - 1\right].$$

The maximum bending moment at mid-height is $M_{\max} = P(e + v_{\max}) = P e\,\sec(\text{arg})$ — the eccentric moment $Pe$ amplified by the secant factor. Adding the direct axial stress $P/A$ and the bending stress $M_{\max}c/I$ (with any extra applied $M$ folded into the eccentric term) gives

$$\sigma_{\max} = \frac{P}{A} + \frac{(M + Pe)\,c\,\sec(\text{arg})}{I}.$$

The secant amplification $\sec(\text{arg})$ is the exact (not approximate) moment-magnification factor for a beam-column; it reduces to the familiar $1/(1 - P/P_{cr})$ approximation for small loads and diverges at $P = P_{cr}$.

Dimensional check. $\left[\dfrac{(M+Pe)c}{I}\right] = \dfrac{\text{N}\cdot\text{m}\cdot\text{m}}{\text{m}^4} = \text{Pa}$; $\sec$ is dimensionless; $[P/A] = \text{Pa}$. ✓

History and Development

The secant formula descends from the beam-column theory developed in the 19th and early 20th centuries (Ayrton–Perry, Timoshenko). It was central to allowable-stress column design — AISC used a secant-based column formula for decades — because it rationally accounts for the unavoidable initial eccentricity of real columns through an equivalent eccentricity ratio $ec/r^2$. Modern codes have largely replaced it with empirical column curves, but the secant formula remains the clearest analytical bridge between bending and buckling.

Related Concepts: Beam Column Moment, Euler Buckling Load column, Beam Bending Stress, Slenderness Ratio, Interaction Formula, Factor of Safety

Notes: The secant argument reaches $\pi/2$ at the Euler load, where $\sigma_{\max}\to\infty$ — restrict use to $P < P_{cr}$. The classic form uses the eccentricity ratio $ec/r^2$; confirm the exact argument used by the calculator (see registry note).