LECTURE #13 :3.11 MECHANICS OF MATERIALS F03INSTRUCTOR : Professor Christine OrtizOFFICE : 13-4022 PHONE : 452-3084WWW : http://web.mit.edu/cortiz/www• Review : Torsion and Beam Bending 1&2• Beam Bending 3 : Stresses in Beams1. Torsion : loading of a bar by an external torque or twisting moment that tends to produce rotation about the longitudinal (z) axis of the bar2. Pure Torsion :ASSUMPTIONS :•every cross-section of the bar is identical and subjected to same internal torque•all cross-sections remain plane, the same shape, and the radii remain straight as they rotate around the longitudinal axis•if the angle of rotation is small, the change in length and radius can be neglected •only twisting and pure shear stresses, no bending or warping (assume bar is weightless)•linear elastic, isotropicParameters :L=bar lengthφ=angle of twist at end of barγφz=shear strainδ =shear deformationr=radial coordinatez=axial coordinateR=cross-sectional area radiusT=torqueτφz=shear stress©2001 Brooks/Cole, a division of Thomson Learning, Inc. ThomsonLearning™is a trademark used herein under license.©2001 Brooks/Cole, a division of Thomson Learning, Inc. ThomsonLearning™is a trademark used herein under license.©2001 Brooks/Cole, a division of Thomson Learning, Inc. ThomsonLearning™is a trademark used herein under license.©2001 Brooks/Cole, a division of Thomson Learning, Inc. ThomsonLearning™is a trademark used herein under license.τφz(r)= τmax(r/R)Tcross-sectionalareaτmaxzz=0z=Lφrr=Rr=0γSOLID BAR=double arrow notation©2001 Brooks/Cole, a division of Thomson Learning, Inc. ThomsonLearning™is a trademark used herein under license.©2001 Brooks/Cole, a division of Thomson Learning, Inc. ThomsonLearning™is a trademark used herein under license.©2001 Brooks/Cole, a division of Thomson Learning, Inc. ThomsonLearning™is a trademark used herein under license.©2001 Brooks/Cole, a division of Thomson Learning, Inc. ThomsonLearning™is a trademark used herein under license.Review Lecture 10 : Torsionδrigid supportzzz1. Geometrical Statement : d 2. Shear Strain Displacment Relation : =rdzd 3. Constitutive Law : = =Grdz4. Equations of Static Equilibrium (Rotational) :TLconstant T : = JG[ radiansrdGφφφδφφγφτγφπ=oo4444oi2=180, 1 rad=57.3] J(m) = polar moment of inertia, rsolid bar, circular cross-sectional area : J=2(r-r) hollow bar : J=2JGJG(Nm)="torsional rigidity", k(Nm)=="torsionaLππ( )zfzfl stiffness" Tr5. Shear Stresses : =JTr ==modulus of rupture in torsionJφφττDERIVATION:Review Lecture 10 : Torsion©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™is a trademark used herein under license.©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™is a trademark used herein under license.©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™is a trademark used herein under license.I. Basics : BEAM : structural member subjected to lateral loads, i.e. forces or moments having their vectors perpendicular to the axis of the bar nomenclature :L=length or span, b=width, h=heightNA=neutral axis passes through centroid (x,y=0) of cross-sectionI=moment of inertia of cross section about NAReview : Beam TheoryxyzLbhNAwebflangeflange1. simply supported2. overhangingPA ByA=0 yB=0PA ByA=0 yB=0PAByA=0, θA=0I-beamcross-sectionrectangularcross-sectionII. Types of Supports and Boundary ConditionsNAwebflange T-beamcross-section3. cantilever34rectangularcircularbhrI=, I= 124EI=flexural modulusπReview : Beam Theory (Cont'd)III. Types of Loading0•a0•a0•aPo0•aslope=k0•aLLLLLMoPoconcentrated moment or coupleconcentrated loaddistributed, uniform loadlinearly varying loadparabolically varying loadIV. Successive Integration Method / Shear and Bending Moment Diagrams :V. Sign Conventions :Review : Beam Theory (Cont’d)(+)VV(-)VV(+)(-)M MMMcompressiontensioncompressiontension1122123dVq(x) q=- q= loading function dxdMV(x) =-q(x)dx+C V=V=shear forcedxdM(x) =V(x)dx+Cx +C M= M=bending momentCx1(x) =M(x)dx+ +Cx+C EI5dxθθ∫∫∫321234dV =EI =curvature=slope of y-displacement curvedxCxC xy(x) =q(x)dx + ++Cx +C y=vertical displacement65θθ∫Beam 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"oxooxxMxyIhyMxyMyEIEIEIσεε=−==−=−=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=−=τxy(y)xBeam Theory 3 : Stresses and StrainsxyzyzBeam Theory 3 : Stresses and Strainsxyzyz?Beam Theory 3 : Stresses and Strainsxyz?ON.A.compressiontensionmpnqdθρdxefyMoMoBeam Theory 3 : Relate Stress to Momentyσx(y)xσy=0NAMo(+
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