Island Bend Dam — Spillway Discharge and Gate Force: 04-BS-7, December 2015, Q4

Question 4: Island Bend Dam — Spillway Discharge and Gate Force (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 (read from the attached Island Bend Dam drawing):

Quantity	Value
Full Supply Level, F.S.L.	RL 1185.67 m
Spillway crest RL	RL 1177.75 m
Radial gate size (width × height, width > height)	8.76 m × 8.23 m
Weir discharge coefficient C_d	0.80
Water density ρ	1000 kg/m³

Find. (a) discharge Q over the crest from one open gate; (b) horizontal hydrostatic force F on one closed radial gate.

Fig. Q4 — head over the crest H = F.S.L. − Crest RL = 1185.67 − 1177.75 = 7.92 m; the same 7.92 m head, applied over the gate's vertical projection, gives the closed-gate horizontal force.

Approach. (a) treat the raised gate opening as a rectangular (broad-crested) weir of length equal to the gate width; (b) the horizontal component of hydrostatic force on any submerged surface (flat or curved) equals the force on its vertical projection, computed exactly like a flat vertical gate.

Part (a) — Head over the crest. $$H = FSL - Crest = 1185.67 - 1177.75 = 7.92\ \text{m}$$
Part (a) — Rectangular weir discharge (gate width L = 8.76 m is the weir length; $C_d$ = 0.80): $$Q = C_d\frac{2}{3}\sqrt{2g}\,L\,H^{3/2} = 0.80\left(\frac{2}{3}\right)\sqrt{19.62}\,(8.76)(7.92)^{1.5}$$ $$Q = 0.80(0.6667)(4.429)(8.76)(22.29) = \boxed{461.3\ \text{m}^3/\text{s}}$$
Part (b) — Horizontal force on the closed gate. The gate's vertical projection is a rectangle of width 8.76 m with its top at the free surface (F.S.L.) and its bottom at the crest, i.e. height H = 7.92 m, so the centroid depth is H/2: $$F = \rho g \bar{h}_c A = 1000(9.81)\left(\frac{7.92}{2}\right)(8.76\times7.92)$$ $$F = 1000(9.81)(3.96)(69.38) = \boxed{2695\ \text{kN} = 2.695\ \text{MN}}$$

Quantity	Result
Head over crest H	7.92 m
(a) Discharge, one open gate	461.3 m³/s
(b) Horizontal force, one closed gate	2695 kN (2.70 MN)

Topic: Weir discharge and hydrostatic force on a radial (tainter) spillway gate.

Key relations: rectangular weir Q = C_d(2/3)√(2g)·L·H³⁄²; horizontal hydrostatic force F = ρg·h̄_c·A on the vertical projection.

Why this works. Once the gate is lifted clear of the water, the opening behaves exactly like a sharp/broad-crested weir; once it is closed, only the HORIZONTAL component of the curved gate's hydrostatic load matters for the pivot/strut design, and that component depends only on the vertical projected area — the curvature of the radial face cancels out of the horizontal sum.

Common pitfall. Using the full gate height (8.23 m) instead of the actual submerged depth (H = 7.92 m, since the gate top sits slightly above F.S.L.) inflates both the discharge and the force calculation.

Source: White, Fluid Mechanics, Ch. 2 (hydrostatic forces) and Ch. 3 (weir/flow-measurement formulas, per the paper's Reference Equations page).

Question 4 of 13: Island Bend Dam — Spillway Discharge and Gate Force

Question 4: Island Bend Dam — Spillway Discharge and Gate Force (5 marks)