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04-BS-7 · December 2016

Question 2 of 13: Middle-Third Stability Check on a Concrete Gravity Dam

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

Notes on this paper

04-BS-7 Mechanics of Fluids — December 2016 (National Examinations, three hours, closed book). Section A (Calculative) offers 9 questions and instructs "do seven"; Section B (Graphical and 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, 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 buoyancy/stability (Ch. 2), Bernoulli and control-volume momentum (Ch. 3), potential (inviscid) flow past a cylinder (Ch. 8), pipe friction and the Moody/Colebrook relation (Ch. 6), open-channel flow (Ch. 10), drag on immersed bodies and Stokes' law (Ch. 7); B. R. Munson et al., Fundamentals of Fluid Mechanics — actuator-disk propeller theory and variable-area tank draining.

Question 2: Middle-Third Stability Check on a Concrete Gravity Dam (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.

QuantityValue
Dam height (vertical upstream face) H20 m
Base width b15 m
Water depth hw18 m
Concrete density ρc2400 kg/m³
Cross-sectionright triangle: vertical face at the water side, hypotenuse to toe O

Find. The distance from O to where the resultant crosses the base, and whether it lies within the middle third (5–10 m from O).

18 m WL F_H R at 7.30 m from O O O 20 m base = 15 m
Fig. Q2 — dam cross-section: hydrostatic thrust F_H at hw/3 above the base, weight through the triangle's centroid, resultant crossing the base 7.30 m from toe O (within the shaded middle third, 5–10 m).

Approach. Compute the dam's self-weight (through the triangle's centroid) and the hydrostatic thrust from the 18 m water depth, take moments of both about the toe O, and locate where the net resultant crosses the base.

  1. Self-weight and its centroid distance from O. Cross-section area = ½(15)(20) = 150 m²/m width. $$W = \rho_c g A = 2400(9.81)(150) = \boxed{3531.6\ \text{kN/m}}$$ For a right triangle with the right angle at the base of the vertical face, the centroid sits at b/3 from that face, i.e. (b − b/3) = 10.0 m from the toe O.
  2. Hydrostatic thrust and its line of action. $$F_H = \tfrac12 \rho_w g h_w^2 = \tfrac12(1000)(9.81)(18^2) = \boxed{1589.2\ \text{kN/m}}$$ acting horizontally at hw/3 = 6.0 m above the base.
  3. Moments about the toe O. Stabilizing moment from the weight, overturning moment from the thrust: $$M_{stab} = W(10.0) = 35{,}316\ \text{kN}\cdot\text{m/m}, \qquad M_{OT} = F_H(6.0) = 9535.3\ \text{kN}\cdot\text{m/m}$$
  4. Locate the resultant and compare with the middle third. $$\bar{x}_O = \frac{M_{stab}-M_{OT}}{W} = \frac{35{,}316-9535.3}{3531.6} = \boxed{7.30\ \text{m from O}}$$ Middle third of the 15 m base spans 5.0–10.0 m from either end. Since 5.0 < 7.30 < 10.0, the resultant falls inside the middle third — the dam is safe.
QuantityResult
Self-weight W3531.6 kN/m
Hydrostatic thrust FH1589.2 kN/m
Resultant location from toe O7.30 m (middle third is 5–10 m)
Dam safe against cracking/separation?Yes