The Rectangular Homogeneous Gate Shown Below Is

Introduction
A rectangular homogeneous gate is a fundamental concept in fluid mechanics and hydraulic engineering, often used to illustrate how fluids exert forces on submerged surfaces. The gate is characterized by a uniform cross‑section, constant thickness, and material density, which allows analysts to treat it as a single, rigid body when calculating hydrostatic loads. Understanding the behavior of such a gate is essential for designing spillways, sluice gates, canal locks, and any structure where water pressure acts on a flat, rectangular plate. This article explores the theory behind the rectangular homogeneous gate, derives the hydrostatic force and its point of action, examines the moment about typical supports, and provides a worked example to reinforce the concepts.

What Is a Rectangular Homogeneous Gate?

A rectangular homogeneous gate is a flat plate whose shape is a rectangle, whose thickness is uniform, and whose material is of constant density throughout. The term homogeneous emphasizes that the gate’s weight is evenly distributed, so its centroid coincides with its geometric center. In most textbook problems the gate is assumed to be rigid, incompressible, and free of deformation under the fluid load.

Key attributes that simplify analysis:

• Shape: Rectangle of width b (horizontal dimension) and height h (vertical dimension).
• Orientation: One side of the rectangle is often hinged or supported along a horizontal edge (the top or bottom), while the opposite edge is free to move under water pressure.
• Uniform density (ρₛ): Enables the gate’s weight to be represented by a single force acting at its centroid.
• Negligible friction at the hinge: Allows the moment equilibrium to be considered solely due to fluid pressure and gate weight.

These assumptions make the rectangular homogeneous gate an ideal model for teaching the principles of hydrostatics and for preliminary design calculations.

Physical Description and Assumptions

Before diving into equations, it is helpful to visualize the typical configuration:

1. Fluid: Usually water, with density ρ (≈ 1000 kg/m³) and gravitational acceleration g (≈ 9.81 m/s²).
2. Gate position: Submerged vertically or inclined; the top edge may be at the free surface or at a certain depth y₀ below it.
3. Support: A hinge or pin located along the top edge (or sometimes the bottom edge) that permits rotation but resists translation.
4. Homogeneity: The gate’s mass per unit area m = ρₛ·t (where t is thickness) is constant, so the weight W = m·g·A acts at the gate’s geometric center. Because the pressure in a static fluid varies linearly with depth, the resultant force on a flat surface is not uniformly distributed. The gate’s homogeneity, however, lets us treat its weight as a single point load, simplifying the moment balance.

Hydrostatic Force on the Gate

Pressure Distribution

In a fluid at rest, the pressure at a depth y measured downward from the free surface is:

[ p(y) = \rho , g , y ]

For a rectangular gate of width b and height h, whose top edge is at depth y₀, the pressure varies linearly from pₜ = ρ g y₀ at the top to p_b = ρ g (y₀ + h) at the bottom.

Resultant Hydrostatic Force

The total force F exerted by the fluid on one side of the gate is obtained by integrating pressure over the area:

[ F = \int_{A} p , dA = \int_{y₀}^{y₀+h} (\rho g y) , b , dy ]

Carrying out the integration:

[ \begin{aligned} F &= \rho g b \int_{y₀}^{y₀+h} y , dy \ &= \rho g b \left[ \frac{y^{2}}{2} \right]_{y₀}^{y₀+h} \ &= \frac{\rho g b}{2} \left[ (y₀+h)^{2} - y₀^{2} \right] \ &= \rho g b \left( y₀ h + \frac{h^{2}}{2} \right) \ &= \rho g , A , \bar{y} \end{aligned} ]

where A = b·h is the gate area and (\bar{y} = y₀ + \frac{h}{2}) is the depth of the centroid. Thus, the magnitude of the hydrostatic force equals the pressure at the centroid multiplied by the area—a result known as the centroidal pressure theorem.

Center of Pressure

Although the resultant force acts as if it were applied at the centroid for magnitude calculations, its actual line of action is lower because pressure increases with depth. The center of pressure (CP) is the point where the total moment of the pressure distribution about any axis equals the moment of the resultant force. For a vertical rectangular gate, the CP depth y_{cp} measured from the free surface is:

[y_{cp} = \bar{y} + \frac{I_{x}}{\bar{y} A} ]

where Iₓ is the second moment of area about the horizontal axis through the centroid. For a rectangle:

[ I_{x} = \frac{b h^{3}}{12} ]

Substituting gives:

[y_{cp} = y₀ + \frac{h}{2} + \frac{b h^{3}}{12 , (y₀ + h/2) , b h} = y₀ + \frac{h}{2} + \frac{h^{2}}{12 , (y₀ + h/2)} ]

The CP is always below the centroid, and the offset decreases as the gate is submerged deeper (large y₀ makes the fraction small).

Moment About the Hinge or Support

General Moment Equation

If the gate is hinged along its top edge (depth y₀), the hydrostatic force creates a moment that tends to rotate the gate open. The moment *

Moment About the Hinge or Support

The hydrostatic force acting at the center of pressure (CP) creates a moment about the hinge point (top edge at depth y₀). This moment tends to rotate the gate open, opposing the restoring moment provided by the gate's weight (treated as a single point load acting at the gate's center of gravity).

The magnitude of the hydrostatic moment M about the hinge is:

[ M = F \cdot (y_{cp} - y_0) ]

Substituting the hydrostatic force F = ρ g A \bar{y} and the CP depth y_{cp} = y_0 + \frac{h}{2} + \frac{h^2}{12(y_0 + h/2)}:

[ M = \rho g A \bar{y} \left( \frac{h^2}{12(y_0 + h/2)} \right) ]

For a rectangular gate, this simplifies to:

[ M = \rho g b h \left( y_0 + \frac{h}{2} \right) \left( \frac{h^2}{12(y_0 + h/2)} \right) = \frac{\rho g b h^3}{12} ]

This result shows that the hydrostatic moment depends only on the gate's geometry (height h and width b) and fluid density, not on the depth y₀ of the top edge. This is because the CP depth offset compensates for the varying pressure distribution.

Equilibrium Considerations

For gate stability:

• If M exceeds the restoring moment from the gate's weight, the gate opens.
• If the restoring moment dominates, the gate remains closed.
• The net moment determines whether the gate rotates or remains stationary.

Conclusion

The hydrostatic force on a submerged gate is characterized by its magnitude, direction, and point of application. The resultant force acts at the center of pressure, which lies below the centroid due to the linear pressure gradient. For a rectangular gate, the hydrostatic moment about the hinge depends solely on the gate's geometry and fluid density, simplifying design calculations. Understanding these principles is crucial for ensuring the safe and stable operation of hydraulic structures like gates, valves, and locks, where hydrostatic forces must be accurately predicted to prevent unintended movement or failure.