Sliding Stability of Two Triangular Dams: 04-BS-7, December 2018, Q10

Question 10: Sliding Stability of Two Triangular Dams (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: sloping upstream (water) face, vertical downstream (dry) face. Dam B: vertical upstream (water) face, sloping downstream (dry) face. Both dams have the same base friction coefficient $\mu$ and rest on a firm, flat, non-seeping foundation.

Dam A's sloping upstream face gives the hydrostatic thrust a downward vertical component (added to the weight); Dam B's vertical upstream face gives a purely horizontal thrust with no such addition.

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

Dam B (the vertical-upstream-face dam) is the one most likely to slide. Sliding resistance at the base is frictional, $F_{friction}=\mu N$, where $N$ is the total NORMAL (vertical) force pressing the dam onto its foundation. For Dam B, the upstream face is vertical, so the hydrostatic pressure acts everywhere horizontally — it contributes nothing to $N$, which is therefore just the dam's own weight, $N_B=W_B$. The full horizontal hydrostatic thrust $F_H$ must then be resisted by friction from that weight alone, $\mu W_B$, with no help from the water itself.

Dam A's upstream face is sloped, so the hydrostatic pressure — which always acts perpendicular to the wetted surface — is no longer purely horizontal. Resolving it into components, the horizontal component is (for the same water depth) essentially the same driving force as Dam B's, but there is now ALSO a downward vertical component, because the pressure pushes perpendicular to a surface that is tilted. This vertical component adds directly to the dam's own weight in the normal-force balance, $N_A=W_A+F_{V}$, which increases the available frictional resistance $\mu N_A$ beyond $\mu W_A$ alone — the water itself is, in effect, helping to hold Dam A down onto its foundation. With a larger resisting force available against essentially the same horizontal driving force, Dam A is therefore LESS likely to slide, and by the same argument, Dam B is more likely to slide.

Concept & Theory

Topic. Sliding stability of a gravity structure under hydrostatic loading.

Key relations. Frictional resistance $F_{friction}=\mu N$; hydrostatic pressure acts perpendicular to the wetted surface, so a sloped face develops both horizontal and vertical force components while a vertical face develops horizontal force only.

Why this works. This is the same principle exploited by battered (sloped) retaining-wall and dam faces in practice: tilting the water-side face inward over the water increases the weight of water "sitting on" the structure's own footprint, which increases the normal force and hence the frictional resistance available to resist sliding, at essentially no cost in horizontal thrust (which depends only on depth, not face angle).

Common pitfall. Assuming both dams see identical hydrostatic loading because they are the "same shape, just mirrored" — the loading direction relative to the WETTED FACE, not the dam's overall silhouette, is what determines whether a vertical force component exists.

Source. White, Fluid Mechanics, Ch. 2 (Hydrostatic forces on plane surfaces; sliding and overturning stability).

Question 10 of 13: Sliding Stability of Two Triangular Dams

Question 10: Sliding Stability of Two Triangular Dams (5 marks)