Stress Analysis - Sections (inc. Parallel axis theorem)

[Ubereem91]

im currently stuck on a stress analysis question, the question is the following:

for the section shown below, derive the following:
x, y, Ixx, Iyy, Ixy, theta, Iu and Iv
50
___ yy
|---| |
|---| |
|---| |
|---| +------ xx
|---|___________________
|-------------------------|
|______________________| 50
512


(note: numbers indicate length(mm) of sides) (also: ignore the dashes inside the section, only there to hold it together)

The left side of the section is 256mm in length.
In addition to the lengths of the sides, i am given a pair of axis in the diagram, which i think indicate the COG of the section (note that on the actual diagram, the axis pass through the section)

the question asks to derive x and y initially, and I am not entirely sure what these properties are and how to get them. I am also not sure what the xx and yy axis indicate, I am assuming the COG of the section? (centre of gravity)

also, Ixx = Ina + Ah^2 (na=neutral axis)
= (bd^3)/12 + Ah^2

(bd^3)/12 is fine, but i don't know the distance between the xx axis/yy axis in relation to the actual section.

if anybody could give me a kickstart id be very grateful, cheers!

 

[SteamKing]

Since you have all of the dimensions of the section except the location of the center of gravity of the cross section with respect to the sides, why don't you try to find x & y first (these usually have a horizontal line over them or are called x-bar and y-bar.) The axes passing thru x-bar and y-bar parallel to the sides are also conveniently the neutral axes of the section. When you calculate Ixx and Iyy, make sure these quantities are referred to the neutral axes (Hint: this is where the parallel axis theorem isused.)

 

[Ubereem91]

you said why don't you try and find x and y first, where do i start? what are these actual properties?

thanks for the help by the way

 

[SteamKing]

You are given the lengths of the sides and the thickness of the section. Can you calculate the cross sectional area? Once you calculate the cross sectional area, you can calculate the first moment of area about a convenient axis (I would chose a trial axis with the origin at the heel of the angle). You can find x-bar and y-bar from this information. When you calculate the second moments of area (I would use the same trial axis), the inertias should be transferred from the trial axis to the axes with the c.g. of the section as the origin. Because this section does not have an axis of symmetry, Ixy will not equal 0. Once Ixx, Iyy, and Ixy are known, then you must find the principal second moments of inertia Iu and Iv. For this calculation, Mohr's circle may be used.

 

[pongo38]

Divide the section into two rectangles and draw up a table with the following headings:
part number n, Area A, distance x of centroid of this part from the reference axis yy, Ax
Then sum the A and sum the Ax columns. SumAx / sum A gives you xbar. Same idea for ybar. The table then extends to have columns h, Ah^2, bd^3/12, sum of Ah^2 + bd^3/12.
Add up this last column to get Ixx. That should get you started.