Library / Hydraulics, open channels and fluid power / HYD-03
Hydraulics, open channels and fluid power
Bottom-hinge moment on a vertical rectangular gate
Moment of the hydrostatic load about a hinge at the bottom edge of a rectangular gate of width b, wetted from the free surface down to depth h, with atmospheric pressure on the other side.
- Ref
- HYD-03
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- M [N·m]
A rectangular gate wetted from the free surface to a bottom hinge at depth h carries a triangular pressure distribution. The total force is ½ρ g b h², acting at one third of the depth above the hinge, so the moment about the hinge is M = ρ g b h³/6. The moment rises with the cube of depth, so a small rise in water level has a large effect.
Moment of the hydrostatic load about a hinge at the bottom edge of a rectangular gate of width b, wetted from the free surface down to depth h, with atmospheric pressure on the other side.
Calculator
- Total hydrostatic force, ½ ρ g b h²88,290 N
- Height of the line of action above the hinge, h/31 m
- Moment if the water rises by 10 %117,500 N·m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The water pushes with 88,290 N at 1 m above the hinge, giving a moment of 88,290 N·m.
- A 10 % rise in depth raises the moment by 33 %, because it grows with the cube of the depth.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N·m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| h Water depth at the gate | 0.1 to 30 m | 3.27 | 8.829e7 | 8.829e7 | +10% | +33.1 % |
| b Gate width | 0.1 to 50 m | 4,415 | 2,207,000 | 2,203,000 | +10% | +10 % |
| ρ Liquid density | 500 to 14,000 kg/m³ | 44,150 | 1,236,000 | 1,192,000 | +10% | +10 % |
| g Gravitational acceleration | 1 to 30 m/s² | 9,000 | 270,000 | 261,000 | +10% | +10 % |
Sweep of Water depth at the gate
| Water depth at the gate (m) | Hinge moment (N·m) |
|---|---|
| 0.1 | 3.27 |
| 3.837 | 184,800 |
| 7.575 | 1,421,000 |
| 11.31 | 4,734,000 |
| 15.05 | 1.115e7 |
| 18.79 | 2.168e7 |
| 22.52 | 3.737e7 |
| 26.26 | 5.923e7 |
| 30 | 8.829e7 |
Sweep one variable
Variables and how each one contributes
Hinge moment
What the equation gives you. Everything else feeds it.
Liquid density
The liquid density, in kg/m³. Raises the moment in direct proportion. Raising it by 10 % (to 1,100) moves the result by +10 %, from 88,290 to 97,120 N·m.
Gravitational acceleration
The gravitational acceleration, in m/s². Raises the moment in direct proportion. Raising it by 10 % (to 10.79) moves the result by +10 %, from 88,290 to 97,120 N·m.
Gate width
The gate width, in m. Raises the moment in direct proportion. Raising it by 10 % (to 2.2) moves the result by +10 %, from 88,290 to 97,120 N·m.
Water depth at the gate
The water depth at the gate, in m, from the free surface to the hinge. Raises the moment with the cube of the depth. Raising it by 10 % (to 3.3) moves the result by +33.1 %, from 88,290 to 117,500 N·m.
Worked example with the values above
- Cube the depth: h³ = 3³ = 27 m³.
- ρ g b = 1,000 × 9.81 × 2 = 19,620.
- M = 19,620 × 27 / 6 = 88,290 N·m.
Assumptions and limits
- A liquid at rest with constant density and atmospheric pressure at the free surface.
- Gauge pressure is used, with atmospheric pressure acting on the other side of the surface.
Common mistakes
- Measuring depth from the wrong datum, instead of from the free surface to the centroid.
- Using the plate length where the vertical depth is needed.
- Using absolute pressure, which adds a force that atmospheric pressure on the other side cancels.
Statistics and uncertainty
The moment varies with the cube of the depth, so a 1 % error in depth gives about a 3 % error in the hinge moment. Water level changes with waves and surge can matter much more than the fixed geometry.
Where it is used
- Sizing gates, hatches, windows and tank walls.
- Reading pressure differences from manometers and finding the surface shape of rotating liquids.
Maths you need
- Hydrostatic pressure: p = ρ g h, increasing linearly with depth
- Centroid: the geometric centre of an area
- Second moment of area: I, in m⁴, about an axis through the centroid