How is the centroid of a composite shape determined?

Explanation:

The centroid of a composite shape is the point at which the entire area (or mass) of the shape can be assumed to be concentrated. For composite sections (made of multiple simple shapes), the centroid is calculated using the principle of moments (or weighted average).

Step-by-Step Method:

Let the composite shape be divided into n standard geometric parts (rectangles, triangles, circles, etc.):

• Let A_i​ = area of the i-th shape

• Let x_i​,y_i​ = coordinates of the centroid of the i-th shape

Then the coordinates of the overall centroid are given by:

\(\bar x=\frac{\sum(A_ix_i)}{\sum A_i},\bar y=\frac{\sum(A_iy_i)}{\sum A_i}\)

Additional Information Parallel Axis Theorem

• This theorem is used to calculate the moment of inertia of a body about any axis parallel to an axis passing through its centroid.

• It is especially useful when the reference axis does not pass through the centroid of the shape.

• It accounts for the additional rotational inertia caused by the distance between the centroidal axis and the new axis.

• Commonly applied when calculating the moment of inertia of composite sections, or when the neutral axis lies away from the centroid.

Perpendicular Axis Theorem

• Applicable only to planar (2D) objects such as flat laminae or thin plates.

• It states that the moment of inertia about an axis perpendicular to the plane is equal to the sum of the moments of inertia about two mutually perpendicular axes in the plane, intersecting at the same point.

• This theorem is helpful in determining polar moment of inertia, which is important for analyzing torsion in circular sections.

• It is used mainly in symmetrical 2D shapes, like discs or thin plates, where analyzing in-plane and out-of-plane rotational behavior is important.