Shear Stress in Beam • Transverse loading applied to a beam results in normal and shearing stresses in transverse sections. • When shearing stresses are exerted on the vertical faces of an element, equal stresses must be exerted on the horizontal faces. 28 March 2017 1 • Longitudinal shearing stresses must exist in any member subjected to transverse loading. for non-uniform bending, M acts on CπΆ and M + dM acts on Dπ·, consider dA at the distance y1 from N.A. the total horizontal force on CπΆ is π. π¦ ππ΄ πΌ πΉ1= ππΆ ππ΄ = Similarly πΉ2 = ππ· ππ΄ = π π + ππ . π¦ ππ΄ πΌ and the horizontal force on πΆ π·is πΉ3 = π b dx 28 March 2017 2 equation of equilibrium πΉ3 = πΉ2 − πΉ1 π b dx = π + ππ . π¦ ππ΄ − πΌ ππ π b dx = πΌ ππ 1 π = ππ₯ πΌ. π π. π¦ ππ΄ πΌ π¦ ππ΄ π¦ ππ΄ π = π πΌ. π π¦ ππ΄ denote Q = ∫y dA is the first moment of the cross section area above the level y (area CDπΆ π·) at which the shear stress π acts, then π = π. π πΌ. π 28 March 2017 shear stress formula 3 Notes For two equal rectangular beams of height h subjected to a concentrated load P, if no friction between the beams, each beam will be in compression above its N.A., the lower longitudinal line of the upper beam will slide w.r.t. the upper line of the lower beam. For a solid beam of height 2h, shear stress must exist along N.A. to prevent sliding, thus single beam of depth 2h will much stiffer and stronger than two separate beams each of depth h. 28 March 2017 4 Shear Stress in Beam of Rectangular Cross Section π. π π = πΌ. π for V, I, b are constants, π ~ Q β πΈ=π −π¦ 2 π β2 πΈ= − π¦2 2 4 β −π¦ 2 π¦+ 2 Then π. π π β2 π= = − π¦2 πΌ. π 2 πΌ 4 π=0 at y=± h/2, ππππ₯ occurs at y= 0 (N.A) Parabolic variation 28 March 2017 5 ππππ π. π π β2 = = − π¦2 πΌ. π 2 πΌ 4 ππππ π. π π β2 = = − 02 πΌ. π 2 πΌ 4 ππππ π. β2 = 8 πΌ ππππ π. β2 = π. β3 8 12 ππππ (at y=0) π. β3 π°= 12 ππππ π = 1.5 π΄ 12 π = 8 π. β ππππ = ππππ = π π΄ 3π 2 π. β = average shear stress ππππ (at y=0) = 1.5 ππππ 28 March 2017 6 A 14-ft long simply supported timber beam carries a 6-kip concentrated load at mid span, as shown in Figure. The cross-sectional dimensions of the timber are shown in Figure. (a) At section a–a, determine the magnitude of the shear stress in the beam at point H. (b) At section a–a, determine the magnitude of the shear stress in the beam at point K. (c) Determine the maximum horizontal shear stress that occurs in the beam at any location within the 14-ft span length. (d) Determine the maximum tension bending stress that occurs in the beam at any location within the 14-ft span length. 28 March 2017 7 28 March 2017 8 28 March 2017 9 If the T-beam is subjected to a vertical shear of V= 12 Kip, determine the maximum shear stress in the beam. Also, compute the shear-stress jump at the flange web junction AB. Sketch the variation of the shear-stress intenstiye over the entire cross section. 26 March 2018 19