What is the first moment of area?

Short Answer:

The first moment of area is the product of an elemental area and the perpendicular distance of that area from a chosen reference axis. It represents how the area of a shape is distributed relative to an axis. The first moment of area is mainly used to find the centroid of a shape and to determine shear stress distribution in beams. Mathematically, it is given by , where is the distance of the elemental area from the axis. The unit of the first moment of area is or .

Detailed Explanation :

First Moment of Area

In Engineering Mechanics, the first moment of area is a fundamental concept that helps engineers determine the geometric center or centroid of a body and analyze the internal shear stress in structural members. It provides a measure of how the area of a body is distributed concerning a particular axis (either horizontal or vertical).

It is a geometric property that does not depend on the material of the object but only on its shape and dimensions. The first moment of area forms the basis for calculating centroids, neutral axes, and shear force distributions in mechanical and civil engineering structures.

Definition

The first moment of area is defined as:
“The algebraic sum of the products of each small element of an area and its perpendicular distance from a reference axis.”

Mathematically, it is expressed as:

where,

  • = First moment of area about the reference axis
  • = Perpendicular distance of the elemental area from the reference axis
  • = Small elemental area

Depending on the axis about which the moment is taken, we can write:

 

Physical Meaning

The first moment of area represents the spatial distribution of an area with respect to a reference axis. It tells us how far the area is distributed away from the axis and how much influence that distance has.

  • If the area is closer to the axis, the first moment of area is smaller.
  • If the area is farther from the axis, the first moment of area is larger.

It acts as a measure of geometric balance and is essential for finding the centroid, which is the point where the entire area could be considered to act.

Unit of First Moment of Area

The first moment of area is obtained by multiplying an area (mm² or m²) by a distance (mm or m), so its units are:

Formula for the First Moment of Area

For a continuous area:

For a discrete or composite area consisting of several parts:

where,

  • = Area of each part
  • = Distance of the centroid of each part from the reference axis

This equation is used when the shape is made of multiple geometric parts, such as rectangles, triangles, or circles combined together.

Application to Find the Centroid

The first moment of area is most commonly used to determine the centroid (geometric center) of a shape.

The centroid coordinates are found using:

or for a composite area:

Thus, the centroid is the point at which the total first moment of area about both axes becomes zero.

Examples of First Moment of Area

  1. For a Rectangle
    Consider a rectangle of base and height .
    The centroid of the rectangle lies at half its height, i.e., .
    Therefore, the first moment of area about the base is:
  1. For a Triangle
    For a triangle with base and height , the centroid lies at a height of from the base.
    Hence,
  1. For a Semicircle
    For a semicircle of radius , the centroid lies at a distance from the base.
    Hence,

Importance of First Moment of Area

  1. To Locate Centroids:
    The first moment of area helps determine the centroid or geometric center of plane figures and cross-sections.
  2. To Calculate Shear Stress in Beams:
    In strength of materials, the shear stress at a particular point in a beam is directly proportional to the first moment of area of the portion above (or below) that point. The formula is:

where,

    • = shear stress,
    • = shear force,
    • = first moment of area,
    • = second moment of area,
    • = width of beam section.
  2. In Structural Design:
    It helps in analyzing and designing beams, trusses, and columns, ensuring stability and strength.
  3. In Fluid Mechanics:
    The first moment of area is used to find the center of pressure on submerged surfaces.

Graphical Representation

If an area is divided into small elements , each located at a perpendicular distance from the x-axis, the first moment of area is obtained by summing all the products . This gives a measure of how the entire area’s position relates to the axis.

The greater the distance of the area from the axis, the larger its moment and thus its influence on the centroidal position.

Practical Example

Let’s consider a rectangular beam cross-section of width and height .
The area of the rectangle is .
The centroidal distance from the base is .
The first moment of area about the base is:

This value helps engineers determine shear stress and other related parameters.

Key Features

  • The first moment of area is linear with respect to distance.
  • It becomes zero when taken about the centroidal axis.
  • It is independent of the material property of the object.
Conclusion

In conclusion, the first moment of area is the product of an elemental area and its perpendicular distance from a chosen axis. It is a geometric property used to locate the centroid and analyze shear stress in structures. The first moment of area provides valuable insight into how an area is distributed with respect to an axis and is widely used in mechanical and civil engineering for design and stability analysis.