## Beam Centroid

Calculate the centroid for the I-Beam cross section below.

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__**Hint**

**Hint**

We can determine the centroid by breaking down the cross section into three segments, and setting a datum at the beam's bottom face (you could easily set the datum elsewhere like the top, or broken the segments into different areas):

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__**Hint 2**

**Hint 2**

To find the centroid:

$$$\bar{y}=\frac{\sum A_iy_i}{\sum A_i}$$$

where
$$A_i$$
is the individual segment's area, and
$$y_i$$
is the individual segment's centroid distance from a reference line or datum.

We can determine the centroid by breaking down the cross section into three segments, and setting a datum at the beam's bottom face (you could easily set the datum elsewhere like the top, or broken the segments into different areas):

To find the centroid, we use the formula:

$$$\bar{y}=\frac{\sum A_iy_i}{\sum A_i}$$$

where
$$A_i$$
is the individual segment's area, and
$$y_i$$
is the individual segment's centroid distance from a reference line or datum.

Segment 1:

$$$A_1=150\times 10=1500$$$

$$$y_1=10+400+\frac{10}{2}=415$$$

Segment 2:

$$$A_2=30\times 400=12,000$$$

$$$y_2=10+\frac{400}{2}=210$$$

Segment 3:

$$$A_3=10\times 200=2000$$$

$$$y_3=\frac{10}{2}=5$$$

Thus,

$$$\bar{y}=\frac{A_1y_1+A_2y_2+A_3y_3}{A_1+A_2+A_3}$$$

$$$\bar{y}=\frac{(1,500)(415)+(12,000)(210)+(2,000)(5)}{(1,500+12,000+2,000)}$$$

$$$\bar{y}=\frac{622,500+2,520,000+10,000}{15,500}=\frac{3,152,500}{15,500}=203.4$$$

203.4