Y10 6 derivayon of shear formula cont

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Unformatted text preview: as a free body 0 H 0 H since H = Z 0 Q= ydA = Q a0 0 Q + Q = 0 (equals moment of entire area with respect to centroid (which must be zero) 3/6/13 M. Mello/Georgia Tech Aerospace 7 DerivaYon of shear formula (cont.) “shear flow” Horizontal shear per unit length: H 3/6/13 H VQ = x I Q = 1st moment with respect to the neutral axis of the porYon of the cross secYon located above (or below) the point at which shear flow is computed I = Centroidal moment of inerYa of the enYre cross secYon 0 M. Mello/Georgia Tech Aerospace 8 DeterminaYon of the shearing stresses in a beam •  V = shear force acYng on the secYon A=t x ⌧ave = ⌧ave = 3/6/13 H VQ x = A Itx VQ It •  t = WIDTH OF THE BEAM AT THE HEIGHT (y1) OF INTEREST •  Q = 1st moment of area (a) or area (a’) as depicted in the figures •  I = moment of inerYa of the cross- secYonal area M. Mello/Georgia Tech Aerospace 9 DeterminaYon of the shearing stresses in a beam Recall, ⌧xy = ⌧yx • Shearing stresses ⌧xy and ⌧yx exerted respectively on a transverse plane passing through D’ are equal •⌧yx = 0 on upper and lower surfaces of the beam • Hence, ⌧xy = 0 along upper and lower edges of the beam • Even though Q is maximum for y = 0, we cannot conclude that ⌧ave will be maximum along the neutral axis 3/6/13 M. Mello/Georgia Tech Aerospace 10 DeterminaYon of the shearing stresses in a beam (cont.) 0 0 0 •  ⌧xy is technically largest at D 1 and D 2 compared to D •  But if width of the beam remains small compared to depth, the shearing stress varies 0 0 only slightly along the line D1 D2 •  Hence, in many pracYcal cases of interest we can use ⌧ ave to compute ⌧ xy at 0 0 D any point along 1 D2 Recall, 3/6/13 ⌧xy = ⌧yx M. Mello/Georgia Tech Aerospace 11 SHEARING STRESSES τxy IN COMMON TYPES OF BEAMS “depth” •  Advanced elasYcity theory shows that for a beam of rectangular cross secNon of width “b” and depth “h”, as long as b < h/4, the value of the shearing stress at C1 and C2 will not exceed by more than 80% the average shearing stress computed along the neutral axis. VQ ⌧xy = •  Hence, (5- 35) Ib 1 y = (c + y ) 2 •  Compute 1st moment of upper dark shaded shaded area Q = Ay 1 Q = b( c y ) · ( c + y ) 2 “width” 12 2 Q = b(c y) 2 bh3 2 •  Moment...
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