LECTURE+24+COE-3001-A

# Y10 6 derivayon of shear formula cont

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

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online