Sliding Friction Required to Hold a Concrete Dam: 04-BS-7, December 2013, Q3

Question 3: Sliding Friction Required to Hold a Concrete 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.

Quantity	Value
Concrete density, ρ_c	2400 kg/m³
Top width	2 m
Base width	6 m
Dam height	5 m
Water depth, h	4 m (vertical upstream face; no uplift)

Trapezoidal dam cross-section, unit length into the page; vertical upstream face carries the full hydrostatic thrust horizontally.

Find. Minimum coefficient of friction μ between the dam base and its foundation.

Approach. Because the upstream face is vertical, the hydrostatic resultant is purely horizontal; compute it and compare against the friction resistance available from the dam's own self-weight (no uplift, so the full weight bears on the base).

Hydrostatic thrust. Per unit length, $F=\tfrac12\rho g h^{2}=\tfrac12(1000)(9.81)(4)^{2}=\boxed{78{,}480\ \text{N} = 78.48\ \text{kN}}$.
Dam weight. Trapezoidal cross-sectional area $A=\tfrac12(2+6)(5)=20\ \text{m}^2$, so $$W = \rho_c g A = 2400\times9.81\times20 = \boxed{470{,}880\ \text{N} = 470.9\ \text{kN}}$$
Minimum friction coefficient. With no uplift the base normal force equals the full weight, so sliding equilibrium ($\mu W = F$) requires $$\mu_{\min} = \frac{F}{W} = \frac{78{,}480}{470{,}880} = \boxed{0.167}$$

Quantity	Value
Hydrostatic thrust, F	78.48 kN
Dam self-weight, W	470.9 kN
Minimum friction coefficient, μ_min	0.167

Topic: Hydrostatic force on a plane surface and sliding stability.

Key relations: resultant hydrostatic force on a vertical submerged face, $F=\tfrac12\rho g h^2$ per unit width; sliding limit $\mu_{\min}=F/N$.

Why this works: a vertical face produces a purely horizontal resultant (no vertical pressure component), so the entire normal reaction at the base comes from self-weight alone once uplift is excluded — the friction condition is then a direct ratio of driving to resisting force.

Common pitfall: using the full dam height (5 m) instead of the actual water depth (4 m) in the hydrostatic-force formula, or forgetting that a sloped (non-vertical) face would add a vertical force component that changes N.

Source: Crowe, Elger & Roberson, Engineering Fluid Mechanics, Ch. 3 (hydrostatic forces on plane surfaces).

Question 3 of 13: Sliding Friction Required to Hold a Concrete Dam

Question 3: Sliding Friction Required to Hold a Concrete Dam (5 marks)