Loads on stairs — conversion from slope area to horizontal plan area: If a stair has tread T and rise R, and carries a uniformly distributed load w per square metre on its sloping surface, what is the equivalent load intensity per square metre when referred to the horizontal plan area of the stair?

Introduction / Context:
Stairs are often loaded and detailed on their sloping surface, but many design calculations and drawings refer loads to the horizontal plan area. A clear conversion between sloping-surface UDL and plan UDL is needed for consistent load take-off and design of supporting members.

Given Data / Assumptions:

• Tread = T (horizontal going per step).
• Rise = R (vertical per step).
• Uniform load on slope = w (kN/m^2 or similar).

Concept / Approach:
For one step, the sloping length equals √(T^2 + R^2). The plan (horizontal) projection for that step is T. Load per plan area equals load per slope area multiplied by the ratio of slope area to plan area, which reduces to the ratio of sloping length to tread for unit width.

Step-by-Step Solution:
Slope area per unit width of one step = √(T^2 + R^2).Plan area per unit width of one step = T.Equivalent plan UDL = w × (slope area / plan area) = w × √(T^2 + R^2) / T.

Verification / Alternative check:
Dimensional consistency holds: ratio is nondimensional, so the units remain as UDL. For small R (gentle stair), √(T^2 + R^2) ≈ T, giving plan load ≈ w, as expected. For steep stairs (larger R), plan intensity increases appropriately.

Why Other Options Are Wrong:
w × T / √(T^2 + R^2) and w × R / √(T^2 + R^2): These invert the correct ratio.w × T / R or w × √(T^2 + R^2) / R: Do not represent the projection relationship.

Common Pitfalls:

• Confusing plan load with line load on stringers; the latter requires separate derivation.
• Mixing total flight loads with per-step geometry.

Final Answer:
w × √(T^2 + R^2) / T