Hydrostatic Force on Submerged Surfaces

1. Combining the Components: The Resultant Force

Once you have calculated the horizontal and vertical force components, $F_H$ and $F_V$, finding the resultant force, ${\vec{F}}_R$, is a straightforward application of vector addition.

The magnitude of the resultant force is given by the Pythagorean theorem:

The direction of the resultant force is described by the angle, $\alpha$, it makes with the horizontal:

While calculating the force vector itself is simple, determining where it acts requires a more detailed look at the principle of moments.

2. The Line of Action on a Curved Surface

To find the line of action of the resultant force ${\vec{F}}_R$, we must ensure that the moment it creates about any point is identical to the net moment created by the distributed pressure field. By Varignon's theorem, this is equivalent to stating that the moment of the resultant is the sum of the moments of its components.

To apply this, you need the individual lines of action for $F_H$ and $F_V$:

1. Line of Action of $F_H$: The horizontal force acts through the center of pressure of the projected vertical area. We will derive how to find this point shortly.
2. Line of Action of $F_V$: The vertical force acts through the centroid of the volume of fluid directly above (or imaginarily above) the curved surface. Calculating this centroid may require integration, a skill you've developed in earlier modules.

Once you know the magnitudes and lines of action of $F_H$ and $F_V$, you can find the line of action of $F_R$ by equating moments about a convenient point (like a hinge or an origin).

A Crucial Shortcut for Circular Surfaces

There is an extremely useful special case. If the curved surface is an arc of a circle (or a section of a cylinder), every differential pressure force $d\vec{F}$ is normal to the surface and therefore points radially inward. Since all these elemental forces point towards the center of curvature, their vector sum—the resultant force ${\vec{F}}_R$—must also pass through the center of the circle. This simple geometric fact can often bypass the need for moment calculations.

3. Locating the Center of Pressure on a Plane Surface

As we just saw, finding the line of action for $F_H$ on a curved surface requires us to find the center of pressure (CP) on its vertical projection. Let's formally derive the method for finding the CP on any submerged plane surface.

The resultant force on a plane surface is $F_R=P_{c}A=(\rho g{h}_c)A$, where $h_c$ is the vertical depth of the area's centroid. The center of pressure, with coordinates $(x_p,y_p)$, is generally not at the centroid. The pressure is greater on the deeper parts of the surface, so the CP is shifted downwards from the centroid.

The following video provides an excellent and clear derivation of the formulas for the center of pressure coordinates.

The video derivations lead to these fundamental results:

• y-coordinate of CP: $y_p=y_c+\frac{I_{xx,c}}{y_{c}A}$
• x-coordinate of CP: $x_p=x_c+\frac{I_{xy,c}}{y_{c}A}$

Where:

• $(x_c,y_c)$ are the coordinates of the centroid of the plane area, with the y-axis pointing down along the plate from the free surface.
• $A$ is the area of the surface.
• $I_{xx,c}$ is the second moment of area about the centroidal axis parallel to the x-axis. This term is always positive, which confirms that $y_p>y_c$, meaning the center of pressure is always below the centroid.
• $I_{xy,c}$ is the product of inertia with respect to the centroidal axes. This term is zero for shapes that are symmetric about the y-axis (like rectangles and circles).

For a deeper, text-based derivation and worked examples, you can refer to the document "HYDROSTATICS (Total Pressure on Plane Surface)".

4. A Complete Example: Force and Line of Action

Let's apply these concepts to find both the resultant force and its line of action on a curved surface. The following reading selection works through an example for a parabolic wall, first by direct integration and then using the component method we've been developing. We will focus on the component method.

This example beautifully illustrates the full procedure:

1. Calculate Components: $F_H$ is found from the projected rectangular area, and $F_V$ is found by integrating to find the weight of the parabolic fluid volume.
2. Locate Component Lines of Action: The line of action for $F_H$ is at $H/3$ from the bottom of the projected rectangle. The line of action for $F_V$ is at the center of mass of the fluid volume, which is found to be at $x_{CM}=3L/8$.
3. Find Resultant Line of Action: By balancing the moments of the components against the moment of the resultant, the equation for the line of action is derived.

5. Tying it all together

Let's test your understanding with the problem from the image below. In the last lesson, you saw a diagram showing the derivation of the differential force components. This problem asks you to take the next step.

Your task:
Attempt to solve parts (a), (b), and (d) of this problem.

• Part (a) asks you to decompose the infinitesimal force $dF$ into its horizontal ($\delta F_H$) and vertical ($\delta F_V$) components. This is a recap of the direct integration setup.
• Part (b) asks you to integrate these components to derive the general formulas for $F_H$ and $F_V$. You should find that your results match the projection and weight methods we've discussed.
• Part (d) provides a specific parabolic shape, $f(x)=\frac{y_0}{x_0^2}{x}^2$. Use the formulas you derived to calculate the specific horizontal and vertical forces for this gate.

Take some time to work through this. It's an excellent exercise that bridges the fundamental calculus with the practical methods used in problem-solving.

Conclusion

In this lesson, you've unified your understanding of hydrostatic forces. You now know not only how to calculate the magnitude of the force on any surface—plane or curved—but also how to pinpoint its exact line of action.

Key Takeaways:

• The resultant force ${\vec{F}}_R$ is the vector sum of its horizontal (${\vec{F}}_H$) and vertical (${\vec{F}}_V$) components.
• The line of action of the resultant force on a curved surface is found by ensuring its moment equals the sum of the moments of its components.
• For a plane surface, the line of action is at the center of pressure (CP), which can be found using the formulas $y_p=y_c+\frac{I_{xx,c}}{y_{c}A}$ and $x_p=x_c+\frac{I_{xy,c}}{y_{c}A}$.
• The center of pressure is always located deeper than the centroid of the area.
• For a surface that is a circular arc, the resultant force always passes through the center of the circle.

With this knowledge, you are now equipped to analyze not just the forces on submerged objects, but also the torques they produce, which is crucial for determining stability. In our next lesson, we will apply these ideas to the concept of buoyancy, exploring Archimedes' principle and the conditions that determine whether an object will sink or float.