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04-BS-7 · December 2019

Question 10 of 13: Sliding Stability of Two L-Shaped Dams Under Uplift

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

Notes on this paper

04-BS-7 Mechanics of Fluids — December 2019 (National Examinations, three hours, closed book). Section A (Calculative) offers 9 questions and instructs “do seven”; Section B (Analytical) offers 4 questions and instructs “do three.” Every question is answered below (13 of 13), so students can use the full paper as a study resource. Constants used throughout (from the paper's own Constants page): g = 9.81 m/s², patm = 100 kPa, ρwater = 1000 kg/m³, SGglycerine = 1.26, SGmercury = 13.56, ρconcrete = 2400 kg/m³, ρair = 1.19 kg/m³ (20°C) / 1.21 kg/m³ (15°C), μwater = 1.0×10⁻³ N·s/m², μair = 1.8×10⁻⁵ N·s/m², Rair = 287 J/kg·K.

Reference texts: F. M. White, Fluid Mechanics, 8th ed. (McGraw-Hill) — fluid statics and manometry (Ch. 2), hydrostatic forces and the middle-third rule (Ch. 2), buoyancy and equilibrium (Ch. 2), dimensional analysis and drag (Ch. 5, 7), pipe friction and the Moody/Colebrook relation (Ch. 6), control-volume momentum and propeller/actuator-disk theory (Ch. 3, 11); J. D. Anderson, Fundamentals of Aerodynamics — wave/compressibility drag divergence (Ch. 5) for the Boeing 747 wind-tunnel chart used in Question 9.

Question 10: Sliding Stability of Two L-Shaped Dams Under Uplift (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.

Given. Dam A: a narrower L-shaped base (heel width comparable to the stem). Dam B: a wider L-shaped base with a longer heel extending upstream under the retained water. Both rest on a firm, non-seeping foundation, but full hydrostatic uplift pressure can develop across the ENTIRE base-foundation contact.

Dam A (narrow base) uplift triangle Dam B (wide base) same peak uplift, wider base
Both dams see the same peak uplift pressure at the heel (same water depth), but Dam B's wider base carries far more concrete weight, so uplift removes a smaller FRACTION of its net normal force.

Find. Which dam is more likely to slide, with the physical reason.

Dam A (the narrower-based dam) is more likely to slide. Sliding resistance is frictional, $F_{friction}=\mu N$, where $N$ is the NET normal force pressing the dam onto its foundation — the dam's own weight MINUS the total uplift force acting upward across the base. Because both dams retain the same water depth, the uplift pressure distribution (triangular, from full hydrostatic head at the heel down to zero at the toe, since there is no seepage loss under the wall) reaches the same PEAK value for both dams; the total uplift FORCE is that pressure distribution integrated over the base area.

Dam A's narrower base means its self-weight is comparatively small, so the uplift force — while numerically similar in peak pressure to Dam B's — removes a much larger FRACTION of Dam A's weight from the net normal force available for friction. Dam B's wider base carries proportionally more concrete weight (the base area, and hence the weight, grows with the SQUARE of the linear dimensions in a similar cross-section, while the uplift force at the same water depth grows only linearly with the extra base length), so uplift removes a smaller fraction of Dam B's normal force, leaving substantially more of it available to resist the same horizontal hydrostatic thrust (which depends only on water depth, not base width, and so is essentially the same for both dams).

With a smaller net normal force relative to the horizontal driving thrust, Dam A has less frictional resistance available and is therefore more likely to slide than the proportionally heavier Dam B.