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Unformatted text preview: secYon (also parabolic) ²་  By this analogy we may therefore use the same shear formula given by: ⌧= VQ Ib ²་  However, in the case of the circular cross secYon we have: ⇡r I= 4 4 Q = Ay = ✓ ⇡r2 2 ◆✓ 4r 3⇡ ◆ 2r3 = 3 (b = 2r) From Appendix D Gere and Goodno) 3/6/13 M. Mello/Georgia Tech Aerospace 30 Shear stress distribuYon along neutral axis of a beam with a circular cross secYon ²་  SubsYtuYng for I and Q in the general shear stress formula yields: ⇡r I= 4 ⌧max 4 Q = Ay = VQ = Ib ⌧max = ✓ ⇡r 2 ⌧max 2 ◆✓ 4r 3⇡ ◆ 2r3 = 3 (b = 2r) V (2r3 /3) =4 ⇡ r /4)(2r) ⌧max = 4V 3⇡ r 2 Maximum shear in circular cross- secYon is along diameter pq 4V 3A Beam with solid circular cross secYon ²་  Theory may also be extended to beams with hollow cross secYon ⇡ 4 2 3 ²་  In this case: = ( r 2 r 1 ) , Q = ( r 2 r 1 ) , and b = 2(r2 I 4 4 3 3 ⌧max VQ = Ib 3/6/13 ⌧max = 4V 3A ✓ 2 r2 + r2 r1 + 2 2 r2 + r 1 2 r1 r1 ) ◆ M. Mello/Georgia Tech Aerospace 31 SHEAR STRESSES IN WEBS OF BEAMS WITH FLANGES y ⌧xz >> ⌧xy in the flanges ⌧xy = ⌧yx ⌧yx ⌧zx z ⌧xz x ⌧xy ⌧xz = ⌧zx FLANGES EXHIBIT VERTICAL (y) and horizontal (z) shear y z FLANGES shear stress in the web act only the vertical direction WEB ⌧xy only ²་  In the case of nonuniform bending (when shear forces present), a beam with a wide- flange shape will exhibit a more complex shear stress distribuYon within the web and flange cross secYons. ²་  Stresses in flanges are calculated using the same technique used for finding the shear stresses in beams with rectangular cross secYons 3/6/13 M. Mello/Georgia Tech Aerospace 32 Shear stresses in the web of a wide- flange beam INVOKE THE SAME FORMULA WE HAVE SEEN FOR SHEAR STRESS IN A BEAM WITH RECTANGULAR CROSS- SECTIONAL PROFILE VQ but in this case b in the usual formula is the web thickness (t) ⌧ = Ib VQ and so, we write ⌧ = for shear stress in the web. 3/6/13 It M. Mello/Georgia Tech Aerospace 33 •  Dark shaded area is subdivided into two rectangles (corner fillets at b and c are ignored) ✓ h Upper flange area: A1 = b 2 y1 h1 2 ◆ ✓ h1 A2 = t 2 y1 ◆ (region between ef And the upper flange) y2 •  Calculate 1st moment of t...
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