LECTURE #15 :3.11 MECHANICS OF MATERIALS F03INSTRUCTOR : Professor Christine OrtizOFFICE : 13-4022 PHONE : 452-3084WWW : http://web.mit.edu/cortiz/www• Review : Transformations of Stress and Strain I• Transformations of Stress and Strain II1. Plane Stress Rotations : Given a state of plane stress, what is the equivalent stress state on an element rotated by an arbitrary angle, θθθθ + CCW defined relative to x-axis• original coordinate axis : x/y, stresses in original coordinate axis : σx, σy, τxy,θ• new coordinate axis : x’/y’, stresses in new coordinate axis : σx’, σy’, τx’y’ ,θ• Consider a free body diagram of a “wedge” where a cut is made along an inclined plane where the transformed stresses are desired :Review Lecture #14 : Stress Transformations Ix•Oσσσσxττττxyσσσσyyxy•Ox’y’σσσσx’ττττx’y’σσσσy’θθθθθθθθx•Oσσσσxττττxyσσσσyyxy•Ox’y’σσσσx’ττττx’y’σσσσy’θθθθθθθθxy•Ox’y’σσσσx’ττττx’y’σσσσy’θθθθθθθθxy•Ox’y’θθσx Aoτxy Aoσy Aotanθτyx Aotanθσx’ Ao/cosθτx’y’ Ao/cosθxy•Ox’y’θθσx Aoτxy Aoσy Aotanθτyx Aotanθσx’ Ao/cosθτx’y’ Ao/cosθEquations of Static Equilibrium, Geometry, Trig ⇒STRESS TRANSFORMATION EQUATIONS :•stresses vary continuously as the axis is rotated :xy xyx' xyxy xyy' xyxyx'y' xy()cos(2)sin(2 )22()cos(2)(sin(2 )22()sin(2)cos(2 )2σσ σσ θστθσσ σσ θστθσσ θττθ+−+−−=+ +=− −=− +Variation of Stresses With θθθθxy xyx' xyxy xyy' xyxyx'y' xy()cos(2)sin(2 )22()cos(2)(sin(2 )22()sin(2)cos(2 )2σσ σσ θστθσσ σσ θστθσσ θττθ+−+−−=+ +=− −=− +Variation of Stresses With θθθθFor σσσσY=0.2σσσσx, ττττxy=0.8σσσσxCCW +CW-xy xyx' xyxy xyy' xyxyx'y' xy()cos(2)sin(2 )22()cos(2)(sin(2 )22()sin(2)cos(2 )2σσ σσ θστθσσ σσ θστθσσ θττθ+−+−−=+ +=− −=− +3 General Features of Stress Transformationxy xyx' xyxy xyy' xyxyx'y' xy()cos(2)sin(2 )22()cos(2)(sin(2 )22()sin(2)cos(2 )2σσ σσ θστθσσ σσ θστθσσ θττθ+−+−−=+ +=− −=− +Principal Stresses and Anglesxy xyx' xyxy xyy' xyxyx'y' xy()cos(2)sin(2 )22()cos(2)(sin(2 )22()sin(2)cos(2 )2σσ σσ θστθσσ σσ θστθσσ θττθ+−+−−=+ +=− −=− +-100100 100 200 300TAN(2θP)2θPPrincipal Stresses and Anglesxypxy2tan(2 )τθσσ−=Planes of Maximum Shearxy xyx' xyxy xyy' xyxyx'y' xy()cos(2)sin(2 )22()cos(2)(sin(2 )22()sin(2)cos(2 )2σσ σσ θστθσσ σσ θστθσσ θττθ+−+−−=+ +=− −=− +Graphical Representation :Principal Stresses and Anglesxy•Ox’y’Graphical Represenation: Planes of Maximum Shear Stressxy•Ox’y’Derivation of General Equation for Principal Stressesxy xyx' xyxy xyy' xyxypxy()cos(2)sin(2 )22()cos(2)(sin(2 )222tan(2 )σσ σσ θστθσσ σσ θστθτθσσ+−+−−=+ +=−
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