For a thin-walled metal pipe under internal pressure, if d is the inside diameter, p the internal pressure, f the permissible hoop tensile stress, and n the joint efficiency, what is the required wall thickness t?

Introduction / Context:
Design of thin-walled pressure pipes uses circumferential (hoop) stress formulas to ensure the shell thickness resists bursting due to internal pressure. Joint efficiency reduces the usable strength when longitudinal seams or joints are present.

Given Data / Assumptions:

• Thin cylinder assumption: wall thickness t ≪ diameter d.
• Internal pressure p (uniform).
• Allowable hoop stress f (permissible tensile stress).
• Joint efficiency n (0 < n ≤ 1).

Concept / Approach:
For a thin cylinder, hoop stress sigma_h satisfies sigma_h = (p * d) / (2 * t). Accounting for joint efficiency n, the effective allowable stress becomes f * n. Solve for t to meet sigma_h ≤ f * n.

Step-by-Step Solution:

Verification / Alternative check:
Longitudinal stress formula gives a smaller stress for thin cylinders; hoop stress governs thickness selection for most pipes under internal pressure.

Why Other Options Are Wrong:
Options b, c, d misplace factors 2 or n or invert terms, producing non-physical dependencies (e.g., thickness decreasing with pressure).

Common Pitfalls:
Forgetting to include joint efficiency; neglecting corrosion allowance; applying thin-cylinder formula to thick-walled pipes where Lame’s equations are required.

Final Answer:
t = (p * d) / (2 * f * n)