Each gate of a lock is 6 m high and is supported by two hinges placed on the top and bottom of the gate. When the gates are closed, they make an angle of 120°. The width of the lock is 5 m. If the water levels are 4 m and 2 m on the upstream and downstream sides respectively, determine the magnitude of the forces on the hinges due to water pressure

Lock Gate Hinge Force Problem

Problem Statement

Each gate of a lock is 6 m high and is supported by two hinges placed on the top and bottom of the gate. When the gates are closed, they make an angle of 120°. The width of the lock is 5 m. If the water levels are 4 m and 2 m on the upstream and downstream sides respectively, determine the magnitude of the forces on the hinges due to water pressure.

Given Data

Height of each gate, $ H_{gate} = 6 \, \text{m} $
Width of lock, $ W_{lock} = 5 \, \text{m} $
Angle between gates = 120°
Upstream water level, $ H_1 = 4 \, \text{m} $
Downstream water level, $ H_2 = 2 \, \text{m} $

Diagram of Lock Gates

Plan and elevation view of the lock gates

Solution

1. Gate Geometry

The angle $ \theta $ for one gate with respect to the lock's centerline is:

$$ \theta = \frac{180^\circ - 120^\circ}{2} = 30^\circ $$

The width of each gate leaf ($l$) is calculated as:

$$ l = \frac{W_{lock} / 2}{\cos \theta} = \frac{5 / 2}{\cos 30^\circ} = \frac{2.5}{0.866} \approx 2.887 \, \text{m} $$

2. Forces on a Single Gate

Upstream Force ($F_1$):

$$ F_1 = \rho g A_1 \bar{h}_1 $$ $$ A_1 = H_1 \times l = 4 \times 2.887 = 11.548 \, \text{m}^2 $$ $$ \bar{h}_1 = H_1 / 2 = 4 / 2 = 2 \, \text{m} $$ $$ F_1 = 1000 \times 9.81 \times 11.548 \times 2 \approx 226571 \, \text{N} $$

This force acts at $ H_1 / 3 = 4/3 \approx 1.33 \, \text{m} $ from the bottom.

Downstream Force ($F_2$):

$$ F_2 = \rho g A_2 \bar{h}_2 $$ $$ A_2 = H_2 \times l = 2 \times 2.887 = 5.774 \, \text{m}^2 $$ $$ \bar{h}_2 = H_2 / 2 = 2 / 2 = 1 \, \text{m} $$ $$ F_2 = 1000 \times 9.81 \times 5.774 \times 1 \approx 56643 \, \text{N} $$

This force acts at $ H_2 / 3 = 2/3 \approx 0.67 \, \text{m} $ from the bottom.

Resultant Water Force ($F$):

$$ F = F_1 - F_2 = 226571 - 56643 = 169928 \, \text{N} $$

3. Location of Resultant Force ($x$)

Taking moments about the bottom of the gate:

$$ F \times x = (F_1 \times 1.33) - (F_2 \times 0.67) $$ $$ 169928 \times x = (226571 \times 1.33) - (56643 \times 0.67) $$ $$ 169928 \times x = 301339 - 37950 = 263389 $$ $$ x = \frac{263389}{169928} \approx 1.55 \, \text{m} $$

4. Hinge Reactions

The resultant water force $F$ is balanced by the reaction from the other gate ($P$) and the total reaction from the hinges ($R$). For miter gates, the hinge reaction is equal to the force between the gates.

$$ R = P = \frac{F}{2 \sin \theta} $$ $$ R = \frac{169928}{2 \sin 30^\circ} = \frac{169928}{2 \times 0.5} = 169928 \, \text{N} $$

Let $R_T$ and $R_B$ be the reactions at the top and bottom hinges.

$$ R_T + R_B = R = 169928 \, \text{N} $$

Taking moments about the bottom hinge to find $R_T$:

$$ R_T \times H_{gate} = R \times x $$ $$ R_T \times 6.0 = 169928 \times 1.55 $$ $$ R_T = \frac{263388.4}{6.0} \approx 43898 \, \text{N} $$

Now, we find $R_B$:

$$ R_B = R - R_T $$ $$ R_B = 169928 - 43898 = 126030 \, \text{N} $$

Final Result:

Force on the top hinge is $ R_T \approx 43898 \, \text{N} $ or $ 43.9 \, \text{kN} $.

Force on the bottom hinge is $ R_B = 126030 \, \text{N} $ or $ 126.03 \, \text{kN} $.

Explanation of Concepts

Forces on Lock Gates: The net force on a lock gate is the difference between the hydrostatic force from the higher upstream water level and the force from the lower downstream level. This net force acts perpendicular to the gate's surface.

Equilibrium of Gates: This net water force ($F$) is balanced by two other forces: a compressive reaction force ($P$) from the opposing gate where they meet, and a total reaction force from the hinges ($R$). For the standard miter gate design, the hinge reaction $R$ is equal in magnitude to $P$.

Hinge Force Distribution: The total hinge reaction ($R$) is distributed between the top and bottom hinges. To find the individual forces on each hinge, we use the principle of moments. By taking moments about one hinge, we can calculate the force on the other, ensuring the gate does not rotate.

Problem Statement

Given Data

Diagram of Lock Gates