Define first moment of area.

Short Answer:

The first moment of area is the product of the area of a small section and the distance of its centroid from a chosen reference axis. It is used to determine the centroid of a shape and to calculate shear stress in beams. Mathematically, it is expressed as , where  is the area and  is the distance from the reference axis.

In simple words, the first moment of area helps in finding how the area of a shape is distributed relative to an axis. It represents how far and how much area exists from the axis and is an important concept in bending and shear stress calculations.

Detailed Explanation:

First Moment of Area

The first moment of area is a geometrical property of a shape that describes the distribution of area about a specific axis. It is defined as the summation or integration of the product of a small elemental area and the perpendicular distance of its centroid from a reference axis.

Mathematically, the first moment of area about an axis is given by:

Where,

  •  = first moment of area about the axis
  •  = distance of the small area element  from the reference axis
  •  = small element of area

The first moment of area is expressed in mm³ or m³, depending on the units used.

Concept of First Moment of Area

The first moment of area helps in understanding how the area of a body is spread about an axis. It gives an idea of how much influence the area has on that axis due to its position.

If a shape has an area located far from the axis, its first moment of area is large, even if the area itself is small. Similarly, if the area is close to the axis, its first moment of area is smaller. Hence, it depends on both — magnitude of area and its distance from the axis.

Mathematical Expression

Consider a small area element  located at a distance  from the reference axis.
Then, the first moment of area about the axis is given by:

For discrete areas, it is written as:

Where,

  •  = small part of total area
  •  = distance of that small part from the axis

This equation shows that the first moment of area is a sum of the moments of small areas about the given axis.

First Moment of Area About the Centroidal Axis

When the first moment of area is taken about the centroidal axis, it becomes zero.
This happens because the positive and negative areas (on both sides of the centroid) are equal and cancel each other.

That is,

Hence, the centroid of a section is the point where the first moment of area about that axis is zero.

Applications of First Moment of Area

  1. Finding Centroid:
    The centroid of a shape can be located by using the first moment of area. For a shape with several parts, the centroidal distance is given by:

where  is the centroidal distance from the reference axis.

  1. Calculation of Shear Stress in Beams:
    The shear stress at any point in a beam under load is directly related to the first moment of area.
    The shear stress formula is given by:

where,

    •  = shear stress
    •  = shear force
    •  = first moment of area above or below the point of interest
    •  = moment of inertia of the whole section
    •  = width of the beam at that point

Hence,  plays a key role in determining how shear stress is distributed in beams.

  1. Structural Design:
    Engineers use the first moment of area to design beams and cross-sections, ensuring that they carry loads efficiently without excessive deflection or failure.
  2. Composite Sections:
    In combined or composite sections, the first moment of area helps in determining the position of the neutral axis and centroid.

Unit and Dimensions

  • SI Unit: mm³ or m³
  • Dimensional Formula: [L³]

Since it is a product of area (L²) and distance (L), its dimension becomes cubic in nature.

Physical Meaning

The first moment of area can be visualized as the “weighted average” of an area’s position with respect to an axis. The farther the area is located from the reference axis, the greater its influence or moment. Hence, it not only represents the size of the area but also its distribution with respect to the axis.

For example, in a beam, the shear force acting at a certain point depends on how the area above or below that point is distributed, and this is directly measured through the first moment of area.

Example

Consider a simple rectangular section of width  and depth .
To find the first moment of area about the neutral axis:

This value can be used in the shear stress formula to find the stress distribution across the section.

Conclusion:

The first moment of area represents the combined effect of an area and its distance from a reference axis. It is an essential parameter in mechanics of materials used to locate the centroid and calculate shear stress in beams. The concept helps engineers understand how forces and stresses act within structural members, ensuring safe and efficient designs.