What is the centroid of a composite section?

Short Answer:

The centroid of a composite section is the geometric center of a shape made up of two or more simple areas, such as rectangles, triangles, or circles. It represents the average position of the entire area and is found by considering the centroids and areas of all individual parts. The centroid of a composite section is determined by taking the sum of the moments of each area about a reference axis and dividing it by the total area. It is used in structural and mechanical engineering to find neutral axes and analyze bending stresses.

Detailed Explanation :

Centroid of a Composite Section

In Engineering Mechanics, a composite section (or compound section) is a combination of two or more simple geometrical shapes joined together to form a single area. These sections are common in engineering structures such as beams, columns, frames, machine parts, and bridges.

The centroid of a composite section is the point at which the entire area of the combined section may be assumed to be concentrated. It acts as the center of gravity of the section if the material is uniform throughout. Since a composite section is made up of multiple shapes, the centroid is found by taking into account the centroids and areas of all the individual shapes.

Definition

The centroid of a composite section can be defined as:
“The point through which the total area of a composite figure is assumed to act, and it is obtained by taking the moment of each component area about a reference axis and dividing by the total area.”

Mathematically, the coordinates of the centroid are given by:

where,

  • = area of each individual shape,
  • = coordinates of the centroid of each area with respect to reference axes,
  • = coordinates of the centroid of the composite section.

Concept of a Composite Section

Composite sections are made by combining simple geometrical shapes like rectangles, triangles, semicircles, or trapeziums. These shapes are joined to form a more complex cross-section used in mechanical and structural designs.

Examples include:

  • I-section (used in beams)
  • T-section (used in structures)
  • L-section (used in angle members)
  • Channel section
  • Combined circular and rectangular sections

Each of these sections can be divided into simpler shapes for easier analysis.

Steps to Find the Centroid of a Composite Section

To determine the centroid of a composite section, the following systematic steps are followed:

  1. Divide the Section into Simple Shapes:
    Break down the composite section into known geometrical shapes such as rectangles, triangles, and circles.
  2. Determine the Area of Each Shape (A₁, A₂, A₃, …):
    Find the area of each simple shape using standard formulas, for example:

    • Rectangle:
    • Triangle:
    • Circle:
    • Semicircle:
  3. Locate the Centroid of Each Shape (x₁, y₁), (x₂, y₂), …:
    The centroid of each shape is found from basic geometry. For example:

    • Rectangle: at the center,
    • Triangle: from the base
    • Semicircle: at a distance from the flat base
  4. Choose Reference Axes:
    Usually, the base or edge of the composite shape is taken as the x-axis and y-axis for measurement.
  5. Find the Moment of Each Area about the Reference Axes:
    Multiply each area by the distance of its centroid from the reference axes:
  1. Compute the Total Area:
  1. Determine the Coordinates of the Centroid:
    Using the moment and total area, calculate:
  1. Locate the Centroid on the Drawing:
    Mark the point () on the figure. This point is the centroid of the composite section.

Example of a Composite Section

Example:
Find the centroid of a T-section having the following dimensions:

  • Flange: 150 mm wide and 20 mm thick
  • Web: 20 mm wide and 100 mm high

Solution:

  1. Divide the Section:
    The section consists of two rectangles:

    • Rectangle 1 (Flange): 150 × 20 mm
    • Rectangle 2 (Web): 20 × 100 mm
  2. Calculate the Areas:
  3. Find the Centroid of Each Rectangle (from the base):
    • Centroid of Rectangle 1 (flange):
    • Centroid of Rectangle 2 (web):
  4. Find the Moments (A × y):
    • For flange:
    • For web:
  5. Total Area:
  1. Centroid (ȳ):

So, the centroid of the T-section is located 86 mm from the base.

Importance of Finding Centroid of Composite Section

  1. Structural Analysis:
    Helps determine the location of the neutral axis in beams and columns.
  2. Design of Beams and Bridges:
    Centroidal location is used to compute bending stresses and deflection.
  3. Center of Gravity Calculation:
    For uniform materials, the centroid represents the center of gravity of the cross-section.
  4. Moment of Inertia:
    The centroid is needed as a reference axis for calculating the moment of inertia of the section.
  5. Machine Design:
    Used in designing structural parts such as connecting rods, brackets, and frames for balance and symmetry.

Key Notes

  • The centroid of a composite section depends only on geometry, not on material type.
  • It always lies within the boundary of the section if all component areas are positive.
  • The first moment of area about the centroidal axis is zero.
  • Reference axes should be chosen carefully for convenience in calculation.
Conclusion

In conclusion, the centroid of a composite section is the geometric center of a shape made up of multiple simple areas. It is found using the principle of moments, by taking the sum of the products of individual areas and their centroidal distances, divided by the total area. The concept is widely used in engineering design and analysis to determine the balance, strength, and stability of structures and machine components.