2009+13+Shear - Introduction to Solid Mechanics-Vm211...

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1 SJTU Introduction to Solid Mechanics-Vm211 Chapter 13 Transverse Shear Chapter 13 Transverse Shear SJTU Introduction to Solid Mechanics-Vm211 CHAPTER OUTLINE 1. Shear in Straight Members 2. The Shear Formula 3. Shear Stresses in Beams 4. Shear Flow in Built-up Members SJTU Introduction to Solid Mechanics-Vm211 SHEAR IN STRAIGHT MEMBERS ± Shear V is the result of a transverse shear-stress distribution that acts over the beam’s x-section. ± Due to complementary property of shear , associated longitudinal shear stresses also act along longitudinal planes of beam SJTU Introduction to Solid Mechanics-Vm211 SHEAR IN STRAIGHT MEMBERS ± As shown below, if top and bottom surfaces of each board are smooth and not bonded together, then application of load P will cause the boards to slide relative to one another. SJTU Introduction to Solid Mechanics-Vm211 SHEAR IN STRAIGHT MEMBERS ± However, if boards are bonded together, longitudinal shear stresses will develop and shear strains will distort x-section in a rather complex manner SJTU Introduction to Solid Mechanics-Vm211 SHEAR IN STRAIGHT MEMBERS ± As shown, when shear V is applied, the non-uniform shear-strain distribution over x-section will cause it to warp , i.e., not remain plane.
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2 SJTU Introduction to Solid Mechanics-Vm211 SHEAR IN STRAIGHT MEMBERS ± Recall that the flexure formula assumes that x- sections must remain plane and perpendicular to longitudinal axis of beam after deformation SJTU Introduction to Solid Mechanics-Vm211 SHEAR IN STRAIGHT MEMBERS ± This is violated when beam is subjected to both bending and shear, we assume the warping is so small that it can be neglected . This is true for a slender beam (small depth compared with its length) ± For transverse shear, shear-strain distribution throughout the depth of a beam cannot be easily expressed mathematically SJTU Introduction to Solid Mechanics-Vm211 SHEAR IN STRAIGHT MEMBERS ± Thus, we need to develop the formula for shear stress is indirectly using the flexure formula and relationship between moment and shear ( V = dM/dx ) SJTU Introduction to Solid Mechanics-Vm211 THE SHEAR FORMULA V, V + dV, w (x)?? SJTU Introduction to Solid Mechanics-Vm211 τ = VQ It 0 ()0 () ( ) ( ) 0 ( ) 11 x AA A F dA tdx Md M M ydA II dM I V It dx σσ τ ′′ = −−= + −− = = ∴= = ∫∫ Q = y dA = y’A’ A’ SJTU Introduction to Solid Mechanics-Vm211 THE SHEAR FORMULA ± Consider horizontal force equilibrium of upper portion of element dx with longitudinal shear stress uniformly acts over its bottom face with width t, using flexure formula and V = dM/dx , we get = VQ It
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3 SJTU Introduction to Solid Mechanics-Vm211 THE SHEAR FORMULA τ = VQ It = shear stress in member at the point located a distance y’ from the neutral axis. Assumed to be constant and
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This note was uploaded on 08/09/2011 for the course EE 211 taught by Professor Liuxila during the Summer '09 term at Shanghai Jiao Tong University.

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2009+13+Shear - Introduction to Solid Mechanics-Vm211...

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