Hydrostatic Forces on Submerged Planes

Hello! Welcome back to our course on Mechanical Engineering fundamentals.

In the last lesson, we learned how pressure increases with depth in a static fluid using the hydrostatic equation, $\Delta P=\rho gh$. Now, we'll build on that to address a critical engineering question: if you submerge a surface in a fluid, what is the total force exerted on it by the fluid pressure, and where exactly does that force act?

Your learning outcome for today is to calculate hydrostatic forces and the center of pressure on submerged plane surfaces. This is a direct application of what we just learned and is essential for designing structures that interact with fluids—from the hull of an aircraft flying through the air (which is a fluid) to the walls of its fuel tanks.

1. Pressure, Force, and the Center of Pressure

From our previous lesson, we know that pressure on a submerged surface isn't uniform; it increases with depth. This means that the bottom part of a submerged gate experiences more pressure than the top part.

The key challenge is twofold:

1. Magnitude: Find the total or resultant force ($F_R$) by summing up the pressure over the entire area.
2. Location: Find the point where this single resultant force effectively acts. This point is called the center of pressure (CP).

As the image suggests, because the pressure is higher on the lower parts of the surface, the center of pressure (CP) will always be located deeper than the geometric center, or centroid (C), of the surface. This distinction is crucial for stability and structural analysis.

2. Deriving the Formulas for Force and Location

To find the formulas for the resultant force and the center of pressure, we use integration. While you've expressed a preference for a formula-based approach, seeing where the formulas come from can provide a deeper understanding. The following video does an excellent job of deriving these key equations in a clear, step-by-step manner.

3. Summary of Key Formulas

Let's distill the information from the video into the core formulas you'll need.

Resultant Force, $F_R$

The magnitude of the resultant hydrostatic force on a submerged plane surface is the product of the pressure at the centroid of the area and the area of the surface.

$F_R=P_{c}A$

Where:

• $P_c$ is the gauge pressure at the centroid of the surface.
• $A$ is the total area of the submerged surface.

Since $P_c=\gamma h_c$ (where $\gamma$ is the specific weight of the fluid and $h_c$ is the vertical depth from the free surface to the centroid), we can also write:

$F_R=\gamma h_{c}A$

This is an elegant result: to find the magnitude of the complex distributed force, you only need to know the pressure at one point—the centroid.

Center of Pressure, ($x_R$, $y_R$)

The resultant force $F_R$ acts at the center of pressure (CP), with coordinates ($x_R,y_R$). The coordinate system's origin is at the free surface, with the y-axis pointing down along the inclined plane.

The y-coordinate, $y_R$, is given by:

$y_R=y_c+\frac{I_{xc}}{y_{c}A}$

Where:

• $y_c$ is the distance along the inclined plane from the free surface to the centroid.
• $I_{xc}$ is the area moment of inertia of the surface about its own centroidal x-axis.
• $A$ is the area of the surface.

Notice the term $\frac{I_{xc}}{y_{c}A}$ is always positive. This confirms that $y_R$ is always greater than $y_c$, meaning the center of pressure is always below the centroid.

The x-coordinate, $x_R$, is given by:

$x_R=x_c+\frac{I_{xyc}}{y_{c}A}$

Where:

• $x_c$ is the x-coordinate of the centroid.
• $I_{xyc}$ is the product of inertia of the area.
• For surfaces that are symmetric about the y-axis (like rectangles, circles, triangles), $I_{xyc}=0$, which simplifies the location to $x_R=x_c$. In many introductory problems, this will be the case.

Geometric Properties

To use these formulas, you need the geometric properties of common shapes (Area, Centroid, Moment of Inertia). This information is typically found in tables.

4. Application: Analyzing a Hinged Gate

Now let's apply these concepts to a complete problem. The real power of finding the center of pressure comes when you need to analyze the moments on a submerged object, such as determining the force needed to keep a hinged gate closed.

The next segment of the video you watched earlier provides an excellent, detailed walkthrough of this type of problem.

This example perfectly illustrates why we need the center of pressure. If you were to incorrectly apply $F_R$ at the centroid, your moment calculation ($\text{Force}\times \text{distance}$) would be wrong, leading to an incorrect design.

5. An Alternative View: The Pressure Prism

For simpler geometries like rectangular gates, there's a highly intuitive way to visualize the hydrostatic force: the pressure prism.

Imagine the pressure distribution on the surface as a 3D volume (a "prism").

• For a fully submerged vertical rectangle, the pressure distribution is a triangle. The resultant force is the area of this triangle (in a 2D view), and it acts at the triangle's centroid (1/3 of the way up from the base).
• For a partially submerged gate, the pressure distribution is a trapezoid. You can find the resultant force and location by treating it as a combination of a rectangle and a triangle.

This method gives the same result as the formulas but can be more visual. The following short video explains this concept well.

Conclusion

In this lesson, you've bridged the gap from knowing the pressure at a point to calculating the total force over an entire surface. This is a fundamental skill in fluid mechanics and structural engineering.

Key Takeaways:

• The resultant hydrostatic force on a submerged plane is the pressure at the centroid times the area: $F_R=P_{c}A$.
• This force acts at the center of pressure (CP), not the centroid.
• The CP is always located below the centroid, at a distance given by the formula $y_R=y_c+\frac{I_{xc}}{y_{c}A}$.
• Finding the correct location of the force is critical for performing accurate moment calculations, which are essential for analyzing the stability of gates, dams, and tanks.

Next Steps:
We've now covered pressure in a static fluid and the forces on submerged planes. In our next lesson, we will explore another famous consequence of hydrostatic pressure: buoyancy. You will learn to apply Archimedes' principle to solve problems of buoyancy and determine the stability of submerged and floating bodies, a topic of great importance in the design of ships and aircraft.