Allowable-stress column design – identifying the secant formula applicability For steel columns with slenderness ratio up to about 160, which expression represents the “secant formula” approach for determining allowable compressive stress (symbolically in terms of Euler parameter), as used in elastic design practice?

Introduction / Context:
The secant formula is a classical elastic-stability expression that accounts for column deflection under eccentric loading. It refines allowable stress beyond simple Euler or straight-line formulas and is applicable up to moderate slenderness (often quoted up to about 160).

Given Data / Assumptions:

• Elastic behavior, pinned or equivalent end conditions through effective length L_e.
• Eccentricity e at load application producing bending plus compression.
• Symbols: σ_y (yield stress), n (factor of safety), P (axial load), A (area), E (modulus), r (radius of gyration).

Concept / Approach:
For an eccentrically loaded column, the maximum stress at extreme fiber is influenced by second-order effects. The secant formula relates the bending magnification to sec( k ) where k = (L_e / r) * sqrt( P / (AE) ). In allowable-stress design, σ_allow is obtained by ensuring the amplified combined stress does not exceed σ_y / n.

Step-by-Step Solution:
Write extreme-fiber stress: σ_max = (P/A) + (Pe / Z) * sec(k)Impose σ_max ≤ σ_y / n → rearrange for an equivalent allowable compressive stress expression.The compact symbolic option matching this dependency is the listed secant form with the magnifier sec(k).

Verification / Alternative check:
At very small k (stocky columns), sec(k) ≈ 1, reducing to direct compression with modest eccentricity. As slenderness grows, sec(k) increases, reducing allowable stress—consistent with stability behavior.

Why Other Options Are Wrong:

• Pure σ_y/fs or constant fractions of σ_y ignore slenderness and second-order effects.
• Linear knock-down forms without the secant term do not capture the geometric nonlinearity inherent in buckling.

Common Pitfalls:
Using the secant formula beyond its intended λ range or without correct effective length; misapplying e as initial crookedness instead of load eccentricity.

Final Answer:
σ_allow = (σ_y / n) * [1 + ( (Pe)/(AE) ) sec( (L_e/r) * sqrt(P/(A*E)) ) ]^−1