In many pracycal cases of interest we can

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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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This note was uploaded on 09/19/2013 for the course CEE 3001 taught by Professor Zhu during the Spring '09 term at Georgia Institute of Technology.

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