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04-BS-7 · May 2013

Question 12 of 13: Barge Crossing an Aqueduct

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

04-BS-7 Mechanics of Fluids — National Examination, 2013-May. Three (3) hours duration, closed book. Section A (Calculative, 9 questions, do 7) and Section B (Analytical, 4 questions, do 3); every question is answered below regardless of the exam's "do N of M" instruction, so the set is a complete study resource.

Reference texts: Crowe, C.T., Elger, D.F. & Roberson, J.A., Engineering Fluid Mechanics (the exam's own Moody chart and drag-coefficient chart are reproduced from this text); Douglas, J.F., Gasiorek, J.M., Swaffield, J.A. & Jack, L.B., Fluid Mechanics; White, F.M., Fluid Mechanics.

Check — assumptions used across this paper:
  • Where a question does not restate an ambient temperature, air is taken at the ISA sea-level standard of 15°C, giving ρair = 1.21 kg/m³ (the Constants table's 15°C value) — used in Q5, Q6, Q7 and Q9.
  • Q8's Moody chart and Q9's sphere drag-coefficient chart are supplied as attachments. Both are solved via the equations the charts themselves plot: the Colebrook–White equation for Q8 (the Moody chart is a graphical solution of Colebrook–White) and the Morrison (2013) curve-fit for sphere drag versus Reynolds number for Q9 (which reproduces the published "Sphere" curve to within a few percent over this Re range).
  • Q1, Q3 and Q11 depend on their figures; the relevant crop of the exam page is reproduced beside each, and every reading used is taken from it.

Question 12: Barge Crossing an Aqueduct (5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

This question is a direct application of Archimedes' principle to a structural-loading question in disguise. The aqueduct's pillars must support the total weight of everything the trough carries: the water itself, plus the barge whenever it is present. The question is whether that total changes when the barge arrives.

A freely floating barge, by definition, displaces exactly its own weight in water: $W_{barge}=\rho_w g V_{disp}$, where $V_{disp}$ is the submerged volume (length × width × draught). Before the barge arrives, the trough over the pillars is full of water up to its working level. When the barge sails in, it pushes aside (displaces) a volume of water equal to $V_{disp}=15\times3\times1.2=54\ \text{m}^3$ — that displaced water does not vanish, it is simply no longer occupying the space the barge's hull now fills (in an open canal it would raise the level slightly elsewhere; here we're only asked about the weight on the pillars, which is unaffected by where the displaced water physically sits).

Crucially, the WEIGHT removed from the trough (the 54 m³ of water no longer there) is exactly equal to the WEIGHT added by the barge itself, because the barge's draught adjusts until it displaces precisely its own weight:

$$W_{barge} = \rho_w g V_{disp} = 1000\times9.81\times54 = 529{,}740\ \text{N} = 529.7\ \text{kN}$$

This is exactly the weight of water the barge's hull displaced. The net load transmitted to the pillars — (water remaining) + (barge) — is therefore identical, term for term, to the load before the barge arrived (water alone, un-displaced). The canal being only 5 m wide and 2 m deep (rather than open water) does not change this conclusion; it only matters if the barge were to run aground or the water level were externally constrained, neither of which applies here.

Conclusion: the compressive force on the aqueduct pillars does not change ($\Delta F = 0$) as the barge passes over — a floating vessel never adds net weight to whatever it floats upon, because it always displaces its own weight in fluid.