Calculating moment of inertia about z axis

[cherry]

Homework Statement

The couple M is applied to a beam of the cross section shown in a plane forming an angle with the vertical. Determine the stress at (a) point A, (b) point B, (c) point D.

Relevant Equations

I_rectangle = (1/12)bh^3

Hi, I'm a little confused on calculating the moment of inertia about the z-axis.
For calculating the moment of inertia about the y-axis, I did the following (verified to be correct):
I_y = 1/12 * 80 * 90^3 + 1/12 * 80 * 30^3

I did the same for the z-axis but it turned out to be wrong.
The correct equation for I_z is 1/3 * 90 * 60^3 + 1/3 * 60 * 20^3 + 1/3 * 30 * 100^3

This is my first time seeing a second moment of inertia equation like that with a denominator of 3.
If someone could post an explanation regarding the equation for I_z, I'd greatly appreciate it.

Thanks!

 

[erobz]

Homework Statement: The couple M is applied to a beam of the cross section shown in a plane forming an angle with the vertical. Determine the stress at (a) point A, (b) point B, (c) point D.
Relevant Equations: I_rectangle = (1/12)bh^3

View attachment 357146
Hi, I'm a little confused on calculating the moment of inertia about the z-axis.
For calculating the moment of inertia about the y-axis, I did the following (verified to be correct):
I_y = 1/12 * 80 * 90^3 + 1/12 * 80 * 30^3

I did the same for the z-axis but it turned out to be wrong.
The correct equation for I_z is 1/3 * 90 * 60^3 + 1/3 * 60 * 20^3 + 1/3 * 30 * 100^3

This is my first time seeing a second moment of inertia equation like that with a denominator of 3.
If someone could post an explanation regarding the equation for I_z, I'd greatly appreciate it.

Thanks!

Please go ahead and show your work for $I_z$. Also see LaTeX Guide to format the equations.

 

[Steve4Physics]

Relevant Equations: I_rectangle = (1/12)bh^3

The above equation gives the (area or second) moment of inertia of a rectangle (measuring $b \times h$) about an axis when:

a) the axis passes through the rectangle's centroid and lies in the plane of the rectangle;

b) the sides of length $b$ are parallel to the axis (so the sides of length $h$ are perpendicular to the axis).

If the above conditions do not apply, you need a different formula.

 

[vela]

If someone could post an explanation regarding the equation for Iz, I'd greatly appreciate it.

I suggest reviewing the parallel-axis theorem to understand the factor of 1/3.

 

[Lnewqban]

Hi, I'm a little confused on calculating the moment of inertia about the z-axis.

Please, see:
http://hyperphysics.phy-astr.gsu.edu/hbase/parax.html#pax

https://engcourses-uofa.ca/books/statics/moments-of-inertia-of-area/composite-areas/

 

[Steve4Physics]

We've not heard back from the OP. However, note that the 'official' answer given in Post #2 as:

The correct equation for Iz is 1/3 * 90 * 60^3 + 1/3 * 60 * 20^3 + 1/3 * 30 * 100^3

can be written down immediately.

We consider the shape broken into several pieces such that the z-axis lies along an edge of each piece:

The standard equation for the second moment of inertia when the axis lies along the edge of a rectangle is $I = \frac 13 bh^3$ where $b$ is the length of the side along the axis. (The formula, if not supplied, can be derived using the parallel axis theorem.)

To get the 'official' answer (above) the two yellow rectangles are treated as a single one measuring 60 mm x 20 mm.