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LECTURE+24+COE-3001-A

# Along neutral axis of a beam with circular

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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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