Perforated plate with tubes through holes: If hole diameter is dn on pitch ph and each hole carries a tube of inner diameter dt, what is the ligament efficiency?

Introduction / Context:
Ligament efficiency measures the fraction of solid material left between adjacent openings and is important in mechanical integrity of tube sheets and perforated plates. When tubes pass through holes, the relevant dimension that removes load-bearing material is the tube outside/inside diameter as applicable.

Given Data / Assumptions:

• Hole diameter denoted dn, hole pitch ph.
• Each hole contains a tube of inner diameter dt (the opening through the plate is effectively dt for fluid passage and ligament calculation based on the free gap).
• Simple linear pitch in one direction considered.

Concept / Approach:
Ligament efficiency along one pitch is defined as the ratio of the net ligament width to the pitch: η = (ph - effective opening width) / ph. When a tube passes through the hole, the effective opening replacing solid material is the tube diameter used in the stress path—here specified as dt. Hence η = (ph - dt) / ph.

Step-by-Step Solution:

Verification / Alternative check:
In tube-sheet design, similar relations use tube outside diameter when structural metal is removed to accommodate the tube. Here, the problem states dt explicitly for the opening dimension, justifying the result.

Why Other Options Are Wrong:

• (a) uses dn (bare hole) and ignores the tube; not aligned to the stated geometry.
• (b) introduces an incorrect mixed term dn . dt / 2.
• (d) normalizes by dn instead of pitch, which is not the standard definition.

Common Pitfalls:
Confusing the appropriate diameter (hole vs. tube). Always use the dimension that removes load-carrying ligament in the direction being evaluated.

Final Answer:
(ph - dt) / ph