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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Full Document
 Spring '09
 ZHU
 Force, Shear Stress, Stress, Shear strength

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