4. Hydrostatic Forces on Surfaces in Fluid Mechanics (Civil Engineering)

What is Hydrostatic Force?

When a fluid is at rest, it exerts pressure on any surface in contact with it.
The total force exerted by a static fluid on a surface is called:

Hydrostatic Force

This force is very important in design of:

• Dams

• Gates

• Retaining walls

• Tanks

• Lock gates

---

2. Pressure Distribution on a Submerged Surface

We know:

$$p = \rho g h$$

So pressure increases linearly with depth.

Therefore, on a vertical surface:

• Pressure at top = small

• Pressure at bottom = large

Pressure diagram is triangular

---

3. Total Hydrostatic Force on a Plane Surface

For a plane surface submerged in a liquid:

$$F = \rho g A \bar{h}$$

Where:

Symbol	Meaning
$F$	Total hydrostatic force (N)
$\rho$	Density of fluid (kg/m³)
$g$	Acceleration due to gravity (9.81 m/s²)
$A$	Area of surface (m²)
$\bar{h}$	Depth of centroid of area from free surface (m)

Important:

Hydrostatic force = Pressure at centroid × Area

---

4. Center of Pressure

The center of pressure (C.P.) is the point where the total hydrostatic force acts.

It is always below the centroid because pressure increases with depth.

---

5. Depth of Center of Pressure for Vertical Plane Surface

$$h_{cp} = \bar{h} + \frac{I_G}{A \bar{h}}$$

Where:

Symbol	Meaning
$h_{cp}$	Depth of center of pressure
$\bar{h}$	Depth of centroid
$I_G$	Moment of inertia of area about centroidal axis
$A$	Area

---

6. Special Cases

---

(A) Vertical Rectangular Surface

Height = $h$, Width = $b$

Area:

$$A = b h$$

Centroid depth:

$$\bar{h} = \frac{h}{2}$$

Moment of inertia:

$$I_G = \frac{b h^3}{12}$$

Center of pressure:

$$h_{cp} = \frac{h}{2} + \frac{\frac{b h^3}{12}}{b h \times \frac{h}{2}}$$ $$h_{cp} = \frac{h}{2} + \frac{h}{6} = \frac{2h}{3}$$

---

(B) Vertical Triangular Surface (Apex at Top)

Centroid depth:

$$\bar{h} = \frac{2h}{3}$$

Moment of inertia:

$$I_G = \frac{b h^3}{36}$$

Center of pressure:

$$h_{cp} = \frac{2h}{3} + \frac{h}{6} = \frac{5h}{6}$$

---

7. Inclined Plane Surface

If a surface is inclined at angle $\theta$ with horizontal:

Total force:

$$F = \rho g A \bar{h}$$

(Same formula)

Center of pressure:

$$h_{cp} = \bar{h} + \frac{I_G \sin^2 \theta}{A \bar{h}}$$

---

8. Curved Surfaces

Hydrostatic force on curved surface is resolved into components:

---

(A) Horizontal Component $F_H$

Equal to hydrostatic force on vertical projection:

$$F_H = \rho g A_v \bar{h}_v$$

---

(B) Vertical Component $F_V$

Equal to weight of imaginary fluid above curved surface

$$F_V = \text{Weight of fluid}$$

---

Resultant Force

$$F = \sqrt{F_H^2 + F_V^2}$$

Direction:

$$\tan \theta = \frac{F_V}{F_H}$$

---

9. Numerical Example

A vertical rectangular gate 2 m high and 1 m wide is submerged in water.

Total force:

$$A = 2 \times 1 = 2 m^2$$ $$\bar{h} = 1 m$$ $$F = 1000 \times 9.81 \times 2 \times 1$$ $$F = 19.62 \, kN$$

---

Center of pressure:

$$h_{cp} = \frac{2}{3} \times 2 = 1.33 m$$

---

10. Applications in Civil Engineering

Used in design of:

• Gravity dams

• Spillway gates

• Sluice gates

• Reservoir walls

• Underground tanks