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Center of area, combined profiles

This exercise is about the following questions:

  • How to calculate the common centroid of composited areas?
  • How to determine the centroid of a rectangle?


For the cross-section of two adjacent profiles, the common center of area is to be determined.

Given: a = 50 mm, b = 22 mm, c = 70 mm, d = 20 mm.

Compound faces
Compound faces


Since the adjacent profiles are purely rectangular cross-sections, the individual centroids can be easily determined.

Individual centres of area
Individual centres of area
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Both surfaces are mirror-symmetrical about the x-axis, so it is sufficient to consider the x-coordinate. The general equation for the x-coordinate of the centre of area is:

\[ \tag{1} x_S = \frac{\sum A_i \cdot x_i}{\sum A_i} \]

If there are several areas, it is advisable to list the individual areas and the relevant coordinates of the individual centre points in a table:

\[ i \]\[ A_i \]\[ x_i \, in \, mm \]\[ A_i \cdot x_i\, in \,mm^3 \]
\[ \sum \]2.06056.260

\[ \tag{2} x_S = \frac{56260mm^3}{2060mm^2} \]

\[ \tag{3} x_S = 27,3 mm \]

And for the sake of completeness:

\[ \tag{4} y_S = 0 mm \]