Increases as the reference axis is moved

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: espect to x- axis) x 3 •  and so for the differenWal area element this translates to 1 y3 3 dx dIx = dxy = 3 3 M. Mello/Georgia Tech Aerospace 10 Example 12- 3: Moment of inerWa of parabolic semisegment (complete Ix derivaWon) ✓ x2 b2 y = f ( x) = h 1 Z b Z ◆ b y3 Ix = dIx = dx 3 0 0 ◆3 Z b 3✓ h x2 dx Ix = 1 3 b2 0 16bh3 Ix = 105 Moment of inerWa of rectangular differenWal area element 1 y3 3 dIx = dxy = dx 3 3 2/22/13 M. Mello/Georgia Tech Aerospace 11 MOMENT OF INERTIA OF A COMPOSITE AREA ²་  Here are some common shapes which are oGen encountered when analyzing stresses in beams Hollow rectangular beam Wide flange secWon Channel- secWon beam ²་  MOMENT OF INERTIA OF A COMPOSITE BEAM = SUM OF THE MOMENTS OF INERTIA OF ITS PARTS WITH RESPECT TO THE SAME AXIS ²་  IN THE CASE OF THE HOLLOW BEAM ABOVE: ²་  COMPUTE THE MOMENT OF INERTIA WITH RESPECT TO THE AXIS PASSING THROUGH THE CENTROID ²་  TAKE THE ALGEBRAIC SUM OF OF THE MOMENTS OF INERTIA OF THE OUTER AND INNER RECTANGLES ²་  USING RESULTS FROM THE PREVIOUS EXAMPLE WE HAVE: Ix = 2/22/13 outer Ix inner Ix bh3 Ix = 12 b 1 h3 1 12 M. Mello/Georgia Tech Aerospace Note: A similar approach may be applied for Iy 12 MOMENT OF INERTIA OF A COMPOSITE AREA ²་  For the wide- flange secWon it is oGen necessary to compute the moment of inerWa with respect to the XX or YY axis, depending upon how the beam is loaded. ²་  Although we certainly know how to compute the moment of inerWa of the 3 consWtuent rectangular secWons, what it is not immediately clear is how to incorporate the physical distance of the two outer secWons from the centroid locaWon. ²་  We have a similar problem in the case of the channel secWon… ²་  The determinaWon of moment of inerWa in these cases is achieved with the aid of a special theorem known as the parallel axis theorem. We’ll come back to these two cases… WIDE FLANGE SECTION 2/22/13 Channel- secWon beam M. Mello/Georgia Tech Aerospace 13 PARALLEL- AXIS THEOREM FOR MOMENTS OF INERTIA •  xC,yC axes have origin at the centroid •  Parallel set of x,y axes have origin at point O •  Consider: 0 Z Z Z Z Ix = (y + d1 )2 dA = y 2 dA + 2d1 ydA + d2 dA 1 Ixc ARBITRARY SHAPE WITH CENTROID C A First moment of the area with respect to...
View Full Document

This note was uploaded on 09/19/2013 for the course CEE 3001 taught by Professor Zhu during the Spring '09 term at Georgia Tech.

Ask a homework question - tutors are online