For estimating the allowable compressive stress of long (slender) columns, the commonly used empirical relation is known as:

Introduction / Context:
Column design relies on empirical or semi-empirical formulas that account for instability (buckling) and material strength. Different formulas are favored for different slenderness regimes. The question asks for the named empirical relation typically associated with long columns (high slenderness ratio).

Given Data / Assumptions:

• Long columns: high slenderness, buckling-dominated behavior.
• Allowable stress derived by reducing Euler buckling stress with safety margins.
• Empirical naming conventions as used in traditional design texts.

Concept / Approach:
For long columns, the allowable stress is often approximated by a straight-line relationship versus slenderness ratio, termed the Straight line formula. By contrast, Johnson's parabolic (Parabolic) addresses intermediate columns, Rankine's formula blends crushing and buckling for all ranges, and Perry's formula is another widely used intermediate-range empirical relation.

Step-by-Step Solution:
Identify regime: long (slender) columns → buckling governs.Associate empirical naming: Straight line formula for long columns.Recognize other formulas target other regimes or provide unified blends.

Verification / Alternative check:
Classical design charts show linear decay of allowable stress with 1/(slenderness)^2 for long columns, consistent with the straight-line approximation on the Rankine–Gordon/Euler asymptote.

Why Other Options Are Wrong:

• Parabolic formula: Suits intermediate columns (Johnson).
• Perry's formula: An intermediate/unified approach, not specifically long-only.
• Rankine's formula: General formula across ranges; not the specific long-column empirical straight-line name.

Common Pitfalls:
Assuming one formula fits all slenderness without considering regime; confusing the names of Johnson (parabolic) and the straight-line approximation.

Final Answer:
Straight line formula