Empirical column formula identification: Name the empirical formula commonly used to calculate the allowable (permissible) stress for long columns in structural design.

Introduction / Context:
Real columns deviate from Euler’s ideal; empirical or semi-empirical formulas are used to compute allowable stresses, especially for intermediate and long columns in practice. Recognizing which expression is traditionally adopted is key for code-level calculations.

Given Data / Assumptions:

• Design for allowable stress, not ultimate load.
• Column may be intermediate/long with imperfections and residuals.
• Elastic–inelastic transition captured empirically.

Concept / Approach:

The Rankine–Gordon formula blends crushing (short-column) and Euler buckling (long-column) effects into one relation for permissible stress or load. It is widely cited as an empirical basis for allowable stress in long columns, predating modern code curves and Perry–Robertson refinements.

Step-by-Step Solution:

Verification / Alternative check:

Handbooks present Rankine–Gordon in the form 1/P_allow = 1/P_crush + 1/P_Euler, or equivalent stress forms, illustrating the blend.

Why Other Options Are Wrong:

Johnson straight-line/parabolic apply primarily to intermediate columns; Perry’s is another method but Rankine–Gordon is the classic empirical name historically associated with long columns in exams.

Common Pitfalls:

Assuming Euler alone for design; ignoring empirical limits that account for imperfections and non-ideal behavior.

Final Answer:

Rankine–Gordon formula