Centroids of Cross-Sectional Areas Click here for Moments of Inertia of Areas
In a
Nutshell: Two common methods can be used to located
the centroid – the method of integration and the
method of summation for composite areas where individual areas are joined
together. Note: If there is a void
(i.e. a hole such as a circular area), then the void has a negative
contribution. For the method of integration: A y_{cg} = ʃ
y dA and A z_{cg}
= ʃ z dA where A is the total area, y_{cg}
is the y-coordinate of the centroid, z_{cg} is the z-coordinate of the centroid, and (y,z) are the
coordinates to the element of area, dA |
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For the method of summation: Areas can be combined together to form “composite sectional areas”. The principle of first moments also applies to each “composite sectional area”. Use “summation” form.
For example: (Σ A_{i} )y_{cg} = Σ A_{i} y_{i} (Also, (Σ A_{i} )z_{cg} = Σ A_{i} z_{i} ) i.e. (area_{1} + area_{2}) y_{cg} = (z_{1} )area_{1} + (y_{2})area_{2} So y_{cg} = [ (y_{1} )area_{1} + (y_{2})area_{2} ] / (area_{1} + area_{2}) where y_{cg} is the y-component of the centroid of the composite area |
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