LECTURE #14 :3.11 MECHANICS OF MATERIALS F03INSTRUCTOR : Professor Christine OrtizOFFICE : 13-4022 PHONE : 452-3084WWW : http://web.mit.edu/cortiz/www• Review : Beam Bending 3 : Normal Stresses and Strains• Transformations of Stress and Strain IBeam Theory 3 : Normal Stresses and Strainsyσx(y)xσy=0NAMo(+ moment)σx(max)cσx(max)Txyz●N.A.compressiontensionmpnqdθρdxefyMoMoFlexure formula : where : normal stress in x-directioninternal bending momenty = vertical distance from NA axis(see Gere Chapter 12, Appendix D, p 321)I = moment of inertia of cross-sectoxxoMyIMσσ=−==()[]()[]maxmaxmaxmaxmaxmaxmaxional area()for rectangular beams : 2(),"flexural modulus"oxooxxMx yIhyMx yMyEI EIEIσεε=−==− =−=Beam Theory (Cont'd) : Shear Stresses and StrainsyzNA22 Derivation in Gere Section 5.8 : (rectangular cross section)24where : shear stress shear forceh = height of cross sectional areay= distance from NA (rectangular cross sectxyxyxyVhyIVτττ=− −==max3ion)2cross sectional areaVAA=−=ττττxy(y)x2-D (Plane) Stress Stateyzxy•Ox•Oττττxyσσσσxσσσσyττττxyσσσσxσσσσy00000xxyyx yσττσττττyxxy•Oxy•Ox’y’Given A State of Plane Stress, What Is The Equivalent Stress State On An Element Rotated By An Arbitrary Angle, θθθθ + CCWxy•Ox’y’xy•Ox’y’θθσxAoτxyAoσyAotanθτyxAotanθσx’ Ao/cosθτx’y’ Ao/cosθyxy•Ox’y’θθσxAoτxyAoσyAotanθτyxAotanθσx’ Ao/cosθτx’y’ Ao/cosθ''''''(1) tan tan(2) tan tanxyxyx x yxy x yxσστ θτ θττσσθθ=+ +=+ −yVariation of Stresses With θθθθSTRESS TRANSFORMATION EQUATIONSxy 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=0.8σσσσxxy 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
View Full Document