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04-BS-7 · May 2013

Question 11 of 13: Which L-Shaped Dam Slides?

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 11: Which L-Shaped Dam Slides? (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.

[Figure not reproduced: Source figure, Question 11 (exam page 6): Dam A and Dam B. See the official exam paper or the cited reference text.]

Exam figure (page 6). Reservoir on the left of both dams. Dam A's base slab runs upstream, under the water. Dam B's runs downstream, on the dry side.

Reading the figure: the reservoir is on the left of both dams. Dam A has its vertical wall on the right, and its base slab (the "foot" of the L) extends upstream, under the reservoir, with water standing on top of it. Dam B has its wall on the left, facing the water, and its base slab extends downstream, on the dry side. Both dams hold the same depth of water. The horizontal hydrostatic thrust on each is therefore the same, $\tfrac12\rho g h^2$ per metre of dam. For Dam A this thrust is shared between the wall face above the slab and the upstream end of the slab, but the total over the full depth $h$ is unchanged.

Sliding resistance at the base is frictional: the largest horizontal force the foundation can resist before the dam slides is $F_{resist}=\mu N$, where $N$ is the total vertical force pressing the base onto the foundation and $\mu$ is the coefficient of friction between the base and the foundation. With no seepage under the base there is no uplift to subtract. Dam B's normal force is just its own self-weight. Dam A's normal force is its self-weight plus the weight of the whole column of reservoir water standing on its upstream slab. That is a substantial addition, because the slab is most of the footprint and the water over it is nearly full depth.

The driving (horizontal) force is the same for both dams, but Dam A's resisting (frictional) force is increased by the water over its slab. Dam A is therefore the more stable against sliding. Some illustrative numbers show the size of the effect. Take a 5000 N self-weight and $\mu=0.5$ for both dams, plus 1500 N of water over Dam A's slab. Dam B can resist only $0.5\times5000=2500\ \text{N}$, while Dam A can resist $0.5\times(5000+1500)=3250\ \text{N}$. The same thrust meets about 23% less resistance at Dam B.

Conclusion: Dam B (slab on the dry, downstream side, gaining nothing from the reservoir's weight) is the more likely to slide. Dam A's upstream slab carries the reservoir water, and that weight holds the dam down and in place. This is why gravity dams and cantilever retaining walls are given a heel that projects under the retained water or soil.