"CESE 656 – Advanced Mechanicsfor Structural EngineersCivil Structural EngineeringUniversity of Alabama at BirminghamShear Center Review of Shear Flow fromUndergraduate Mechanics2 Shear Center CESE 656 IntroductionTransverse loading applied to a beam resultsin normal and shearing stresses intransverse sections.Distribution of normal and shearing stressessatisfies( ) F = s dA = 0 M = y t - z t dA = 0 x x x xz xy ? ? F = t dA = -V M = z s dA = 0 y xy y x ? ? F = t dA = 0 M = ( - y s ) = M z xz z x ? ? When shearing stresses are exerted on thevertical faces of an element, equal stressesmust be exerted on the horizontal facesLongitudinal shearing stresses must exist inany member subjected to transverseloading.1- 3 Shear Center CESE 656 Shear FlowConsider prismatic beamFor equilibrium of beam elementF = 0 = ?H + ( s - s )dA ? x ? D C A M - M D C ?H = y dA ? I A Note,Q = y dA ? A dM M - M = ?x = V ?x D C dx Substituting,VQ ?H = ?x I ?H VQ q = = = shear flow ?x I 1- 4 Shear Center CESE 656 Shear Flow Shear flow,?H VQ q = = = shear flow ?x I whereQ = y dA ? A = first moment of area above y 1 2 I = y dA ? A + A' = second moment of full cross section Same result found for lower area' ' ?H VQ ' ' q = = = -q ?x I ' Q + Q = 0 = first moment with respectto neutral axis ' ?H = - ?H 1- 5 Shear Center CESE 656 Determination of the Shearing Stress in aBeamThe average shearing stress on the horizontalface of the element is obtained by dividing theshearing force on the element by the area ofthe face.?H q ?x VQ ?x t = = = ave ?A ?A I t ?x VQ = It On the upper and lower surfaces of the beam,t = 0.It follows that t = 0 on the upper andyx xy lower edges of the transverse sections.If the width of the beam is comparable or largerelative to its depth, the shearing stresses at D1 and D are significantly higher than at D.2 1- 6 Shear Center CESE 656 Shearing Stresses t in Common Types of BeamsxyFor a narrow rectangular beam,2 ? ? VQ 3V y ? ? t = = 1 - xy 2 ? ? Ib 2 A c ? ? 3V t = max 2 A For American Standard (S-beam)and wide-flange (W-beam) beamsVQ t = ave It V t = max A web 1- 7 Shear Center CESE 656 Shear Flow in Thin-Walled Members?? ?? ? ? = , and? =? ? ? ? V must act along an axis of symmetry or along a principal plane for non- symmetric sections? The cutting plane on which q is determined must be perpendicular to afree surface? The shear flow determined is parallel to a free surface? The shear flow perpendicular to a free surface is negligible (thin walls)8 Shear Center CESE 656 Shear Flow in Thin-Walled MembersConsider a segment of a wide-flange beamsubjected to the vertical shear V.The longitudinal shear force on the elementisVQ ?H = ?x I The corresponding shear stress is?H VQ t = t ˜ = zx xz t ?x It Previously found a similar expression forthe shearing stress in the webVQ t = xy It t ˜ 0 NOTE: in the flangesxy t ˜ 0 in the webxz 1- 9 Shear Center CESE 656 Shearing Stresses in Thin-Walled MembersThe variation of shear flow across thesection depends only on the variation ofthe first moment.VQ q = tt = I For a box beam, q grows smoothly fromzero at A to a maximum at C and C’ andthen decreases back to zero at E.The sense of q in the horizontal portionsof the section may be deduced from thesense in the vertical portions or thesense of the shear V.1- 10 Shear Center CESE 656 "