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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.

## Solution

Since the adjacent profiles are purely rectangular cross-sections, the individual centroids can be easily determined. Individual centres of area $\require{cancel}$

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$
11.1001112.100
29604644.160
$\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:

$\tag{4} y_S = 0 mm$