Unformatted text preview: of inerYa of rectangular secYon: I =
= bc3
12
3 3/6/13 M. Mello/Georgia Tech Aerospace “width” 12 SHEARING STRESSES τxy IN BBEAMS WITH RECTANGULAR CROSS SECTOION ⌧xy =
⌧xy VQ
Ib Now subsYtute for Q, I 3 c2 y 2
=
V
4 bc3 But area of the cross secYon is A=2bc and so, ✓
◆
3V
y2
3V
⌧xy =
1
⌧max =
(y=0) 2A
c2
2A Parabolic distribuYon of shear stress In a beam of rectangular cross secYon • DistribuYon of shearing stresses in a transverse secYon of a rectangular beam is parabolic • Shear stresses are zero at the top and bo=om y=+/ c (as it should be) • Maximum shearing stress in a beam of rectangular cross secYon is 50% larger than
simply assuming V/A. 3/6/13 M. Mello/Georgia Tech Aerospace 13 AlternaYve forms for τxy for a beam with rectangular cross secYon ⌧xy ✓
3V
=
1
2A since, I = bh3
12 and since, c = y2
c2 = ◆ (Derived on previous page) Ah2
12 h
2 we also have, ⌧xy = V
2I h2
4 y2 (5 36) MAXIMUM VALUE OF THE SHEAR STRESS: (Derived on previous page) ⌧max =
AlternaYve form ⌧max = 3V
2A (5 37) V h2
8I ²་ Preceding equaYons for shear stress can be used to calculate either VERTICAL shear stresses acYng on the cross secYons OR HORIZONTAL shear stresses acYng between layers of the beam 3/6/13 M. Mello/Georgia Tech Aerospace 14 LimitaYons of shear stress theory for beams ²་ LINEAR ELASTIC MATERIALS ONLY ²་ SMALL BEAM DEFLECTIONS ²་ FOR RECTANGULAR BEAMS REQUIRES b<<h ²་ For square beams (b=h) true max stress is 13% greater than what is predicted by (5 36) ²་ General theory not applicable to any type of cross secYon ²་ Edges of cross secYon must be parallel to y axis ²་ Shear stress must remain uniform across the width ²་ applies only to prismaYc beams (not to tapered beams) 3/6/13 M. Mello/Georgia Tech Aerospace 15 Eﬀects of shear strains Max shear strain (max warpage) ²་ Shear strain γ=τ/G varies parabolically over the for a beam with rectangular cross secYon. ²་ Originally plane cross secYons become warped.. ²་ IF SHEAR FORCE IS SAME AT EACH CROSS SECTION, THEN WARPING IS SAME AT EACH CROSS SECTION cross secYon become straight near surface where γ=0 ²་ STRETCHING AND SHORTENING LONGITUDINAL ELEMENTS DUE TO BENDING MOMENTS IS UNAFFECTED BY THE SHEAR STRAINS ²་...
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 Spring '09
 ZHU
 Force, Shear Stress, Stress, Shear strength

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