Distributed Forces, Centroids, and Centers of Gravity - Theory & Concepts

In previous sections, we idealized forces as being concentrated at a single point. In reality, loads are often distributed over a line, an area, or a volume. To analyze these bodies effectively, we must find a single equivalent resultant force and determine its line of action.

The Earth exerts a gravitational force on every particle of a body. These forces form a parallel force system.

To find the coordinates ($\bar{x}, \bar{y}, \bar{z}$) of the CG, we equate the moment of the resultant weight about any axis to the sum of the moments of the weights of all the particles about that same axis (Varignon's Theorem).

If a body exists in a uniform gravitational field (where $g$ is constant everywhere), the Center of Gravity coincides with the Center of Mass.

The formulas for the centroid of a line, area, or volume are derived similarly to the center of mass, substituting the appropriate differential geometric property for mass.

Note: (Similar equations apply for $\bar{y}$ and $\bar{z}$.)

Many engineering structures are composed of simpler geometric shapes (rectangles, triangles, circles). The centroid of a composite body can be found without integration by breaking it down into these simple parts.

A composite area consists of a series of simpler, standard shapes (e.g., rectangles, triangles, circles) that are connected together. By breaking down a complex area into these simple parts, we can find the overall centroid without using calculus.

The Theorems of Pappus and Guldinus provide a simple way to calculate the surface area and volume of a body of revolution (a shape created by rotating a curve or an area around an axis). These theorems relate the area/volume directly to the centroid of the generating curve/area.

The surface area $A$ of a surface of revolution is equal to the product of the length $L$ of the generating curve and the distance traveled by the centroid of the curve during the rotation.

The volume $V$ of a solid of revolution is equal to the product of the generating area $A$ and the distance traveled by the centroid of the area during the rotation.

A common type of distributed force in structural analysis is a load distributed along the length of a beam, represented by a load function $w(x)$ (units: Force/Length, e.g., $\text{kN/m}$).

To solve equilibrium problems, this distributed load must be replaced by a single equivalent concentrated resultant force ($F_R$).

Magnitude: The magnitude of $F_R$ is equal to the total area under the load distribution diagram.

Location: The line of action of $F_R$ passes through the centroid ($\bar{x}$) of the area under the load diagram.

A prime application of centroids and distributed loads in civil engineering is calculating the pressure exerted by a fluid at rest on submerged surfaces (e.g., dams, tanks).

According to Pascal's Law, the pressure $p$ at a depth $z$ in a fluid with specific weight $\gamma$ is calculated by integrating the pressure over the area.

The pressure increases linearly with depth, forming a triangular or trapezoidal load distribution on vertical or inclined submerged flat plates. The point where this resultant force acts is called the Center of Pressure ($z_p$). It is always located below the centroid because the pressure is higher at the bottom of the plate.

For curved surfaces, it is easier to resolve the resultant force into horizontal and vertical components:

• Horizontal Component ($F_H$): Equal to the resultant force on the vertical projection of the curved surface, acting at the center of pressure of that vertical projection.
• Vertical Component ($F_V$): Equal to the weight of the fluid volume directly above the curved surface extending up to the free surface. The line of action passes through the centroid of that fluid volume.