Unformatted text preview: The good news is … the ﬂexure formula sWll works pre=y well! Normal stress calculated from the ﬂexure formula is sWll fairly accurate even in the presence of shear forces and associated warping. • The ﬂexure formula can therefore be applied to beams subjected to nonuniform bending • Must sWll avoid locaWons where the beam abruptly changes shape or where there may be sharp disconWnuiWes in loading such as near the supports or close to a concentrated load. 2/22/13 M. Mello/Georgia Tech Aerospace 5 MOMENTS OF INERTIA and FLEXURAL RIGIDITY y Z y Pure bending in the y z plane Pure bending in the x z plane ~
M M M Z x y ~
M z ~
M z • relevant moment of inertia: Ix
• ﬂexural rigidity: EIx
2/22/13 • relevant moment of inertia: Iy
• ﬂexural rigidity: EIy
• Iy > Ix ! EIy > EIx M. Mello/Georgia Tech Aerospace 6 MOMENT OF INERTIA OF PLANE AREAS Ix =
Iy = 2/22/13 Z Z y 2 dA (12 9a) x2 dA (12 9b) M. Mello/Georgia Tech Aerospace 7 Moment of InerWa: A common and most useful example ²་ MOMENT OF INERTIA OF A RECTANGULAR CROSS SECTIONAL AREA (OF A BEAM) dx
dAx Ix = dAy Z y 2 dAy dAy = bdy
Ix = Z h/2 y 2 bdy
h /2 bh3
Ix =
12 Iy = Z x2 dAx dAx = hdx
Z b/2
Iy =
x2 hdx
b /2 hb3
Iy =
12 • Note that if h>b (as the diagram suggests), then Ix > Iy • Rectangular beam exhibits greater resistance to bending in the y z plane (increased ﬂexural rigidity) • i.e., EIx > EIy
2/22/13 M. Mello/Georgia Tech Aerospace 8 Moment of InerWa (IBB) of the rectangular cross secWonal with respect to BB dAy
IBB =
IBB =
IBB Z Z y 2 dAy
h y 2 bdy
0 bh3
=
3 (WITH RESPECT TO X AXIS) ²་ In general, the moment of inerWa increases as the reference axis is moved parallel to itself and away from the centroid. 2/22/13 M. Mello/Georgia Tech Aerospace 9 Example 12 3: Moment of inerWa of parabolic semisegment ✓ y = f ( x) = h 1 Area element: ✓ dA = ydx = h 1 2/22/13 2 ◆ x
dx
b2 x2
b2 ◆ DETERMINE: Ix and Iy , i.e., the moments of inerWa of the enWre area with respect to the x and y axes, respecWvely. Z x2 dA
• SOLVE for Iy ﬁrst : Iy = ✓
◆
Zb
x2
2hb3 2
I=
xh 1
dx = y
b2
15
0 • FOR Ix we can sWll integrate along x by taking advantage of the soluWon for the moment of a rectangular cross secWon (just determined on previous page). bh3
• i.e., recall I = (with r...
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 Spring '09
 ZHU
 Second moment of area, Ibeam, M., Mello/Georgia

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