Unformatted text preview: xc Ix = Ixc + Ad2
1
Iy = Iyc + Ad2
2
2/22/13 M. Mello/Georgia Tech Aerospace 14 Example: Return to the parabolic semisegment and compute moments of inerWa with respect to centroid • Problem is very easy if we apply the parallel axis theorem • Calculate the area (A) under the curve (this was done last lecture 2bh
for this very same cross secWon) A= 3 • Need the centroid coordinates (computed last lecture in EXAMPLE 12 1) 3b
2h
xc =
yc = 8
5 • Invoke the parallel axis theorem: Ix = xc + Ay 2 where I I y = I y c + Ax 2 CASE 17/ APPENDIX D IN GERE AND GOODNO: 3 Ixc 8bh
=
175 I yc 19hb3
=
480 2/22/13 16bh 3 2hb3
I x = = (determined in EXAMPLE 12 3) and Iy 105 • Hence, Ixc = Ix I yc = I x 15 16bh3
2
Ay =
105
2hb3
2
Ax =
15 M. Mello/Georgia Tech Aerospace ✓ ◆2
2bh 2h
3
5
✓ ◆2
2bh 3b
3
8
15 Now let’s examine how we can apply the parallel axis theorem to the following beam cross secWons (or composite areas) y0 Problem 4 : top Problem 5 : x0 side bo=om WIDE FLANGE SECTION CHANNEL SECTION 1. Make a simple table in order to keep track of the areas and centroids of each consWtuent secWon. 2. Label each secWon and calculate area of each individual secWon 3. Determine centroid locaWon within each secWon (in case of channel secWon tabulate Xc for each secWon). b
b b1
top
bottom
side
4. For example, in case of channel secWon : Xc = Xc
=
Xc
=
2
2 5. Determine Xc of the enWre secWon ((Yc) is obvious by symmetry) Xc =
2/22/13 P3 x i Ai
c
A i=1 (with respect to x0, y0 axes) M. Mello/Georgia Tech Aerospace 16 MOMENT OF INERTIA OF A CHANNEL SECTION Problem 5 : top side bo=om CHANNEL SECTION 6. Calculate moment of inerWa of each consWtuent rectangular secWon. Use: Iy = bh3 /12 7. The dimension “b” is always the side aligned the direcWon of moment vector (in this case the y axis)… 8. Next, set up (x,y) coordinate system with origin at the centroid. 9. Establish y values of all points of interest in the problem w.r.t centroid (you have to think about the direcWon in which the beam will bend). 10. Next, obtain distances (di) from the centroid of each individual secWon to the centroid of the composite secWon (again think about how the beam will bend). 11. Apply the parallel axis theorem to compute...
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 Spring '09
 ZHU
 Second moment of area, Ibeam, M., Mello/Georgia

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