Chapter13

Chapter13 - 1-Mechanics & Materials 1FAMU FSU...

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Unformatted text preview: 1-Mechanics & Materials 1FAMU FSU College of EngineeringDepartment of Mechanical EngineeringChapter 13 Chapter 13 Transverse ShearTransverse Shear2Transverse ShearTransverse ShearTransverse loads generate bending moments and shear forces. Bending moments bbending stresses through the depth of the beam. Shear forces btransverse shear-stresses distributed through the beam. 3 Consider a typical beam section with a transverse load. The top and bottom surfaces of the beam carry no load, hence the shear stresses must be zero here. Transverse ShearTransverse Shear4Shear FormulaShear FormulaTo determine the shear stress distribution consider a loaded beam:Consider a FBD of the element dx with the bending moment stress distribution only: 5 Summing the forces horizontally on this element, the stresses due to the bending moments only form a couple, therefore the force resultant is equal to zero. Shear FormulaShear Formula6 Consider now a segment of this element from a distance yabove the Neutral Axis to the top of the element. For it to be in equilibrium,a shear stress xymust be present. Shear FormulaShear Formula7Let the width (into the page) of the section at a distance y from theNA be a function of y and call it t'.Applying the horizontal equilibrium equation, gives: 21=+-=tdytdytdyFxyyyxyyxxtoptopShear FormulaShear Formula8Substituting for the magnitude of the stresses using flexure formula gives: Simplifying and dividing by dx and t gives:Considering the relation between shear and bending moment)(=++-tdxtdyIydMMtdyIyMxyyyxxyyxtoptop=topyyxxyytdyItdxdM1dxdMVxx==topyyxxyytdyItVShear FormulaShear Formula9First Moment of Area, QFirst Moment of Area, QThe integral Represents the first moment of area A about the Neutral axis.This quantity is termed as QThe centroid is given as So the first Moment of Area is given asThe units for Q are (length)3; m3, in3, etc.....
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Chapter13 - 1-Mechanics & Materials 1FAMU FSU...

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