Unformatted text preview: he cross secYonal area above y=y1 ✓
◆
✓
◆
h1
h/2 h1 /2
h1 /2 y1
Q = A1
+
+ A2 y 1 +
Q = A1 y 1 + A2 y 2
2
2
2
Q=
3/6/13 b2
(h
8 t
h2 ) + ( h2
1
81 M. Mello/Georgia Tech Aerospace 2
4y1 ) (5 42) 34 Shear stress in the web of a wide ﬂange beam ✓ ◆ ✓ h1
A2 = t
2 y1 ◆ y2 0 y1 h1
2 y 1 h 1 /2 h
Upper ﬂange area: A1 = b
2 bh3
• Calculate moment of inerYa: I = 12 (b t) h3 1 12 I= 1
(bh3
12 bh3 + th3 )
1
1 (subtracYon of rectangular air gaps from rectangle of dimensions b x h. VQ
V
2
2
2
2
• AND FINALLY, ⌧ = It = 8It b(h h1 ) + t(h1 4 y1 ) (shear stress is parabolic in y1) 3/6/13 M. Mello/Georgia Tech Aerospace 35 Shear stress in the web of a wide ﬂange beam ⌧min = y1 Vb
2
8It (h h 2 ) ; y 1 = ± h 1 /2
1 y2 ⌧max = 3/6/13 V
2
8It (bh M. Mello/Georgia Tech Aerospace bh2 + th2 ) ; y1 = 0
1
1 36 Final notes and remarks on wide ﬂange beams y1 y2 ²་ Shear force carried by the web: • integrate shear stress from –h1/2 to h1/2 • This leaves us with force per unit length along the thickness direcYon • Now just mulYply by thickness t to obtain total shear force carried by the web Vweb • This is typically 90 – 98% of the total shear force on a cross secYon th1
=
(2⌧max + ⌧min ) • Web resists most of the shear force! 3
• Consequently it is then common to esYmate the maximum shear force by: V (typically accurate to within of ⌧aver =
max shear force found using the th1 3/6/13 τmax formula M. Mello/Georgia Tech Aerospace 37 Finally, let’s return to the quesYon of the shear stress distribuYon in the upper and lower ﬂanges y ⌧yx
⌧zx z ⌧xy = ⌧yx
⌧xz x ⌧xy
⌧xz = ⌧zx Flanges exhibit vertical (y)
and horizontal (z) shear stresses
⌧xz >> ⌧xy in the ﬂanges y z FLANGES WEB ⌧xy only shear stress in the web act
only in the vertical direction 3/6/13 M. Mello/Georgia Tech Aerospace 38 LimitaYons of shear stress theory for wide ﬂange beams ²་ The theory we have examined is suitable for determining verYcal shear stresses in the web of a wide ﬂange beam. ²་ CalculaYng shear stresses in the ﬂange secYon is problemaYc since we cannot assume that the shear stresses are constant across the width of the ﬂange secYon. VQ
²་ So we can’t simply use the formula ⌧ = to calculate verYcal shear stresses in the web Ib VQ ²་ However, we can apply ⌧ = to calculate the horizontal shear stresses ⌧xz Ib
within the ﬂange secYon since these stresses are reasonably constant across the width of the ﬂange secYon. 3/6/13 M. Mello/Georgia Tech Aerospace 39...
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 Spring '09
 ZHU
 Force, Shear Stress, Stress, Shear strength

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