LECTURE 13 3 11 MECHANICS OF MATERIALS F03 INSTRUCTOR Professor Christine Ortiz OFFICE 13 4022 PHONE 452 3084 WWW http web mit edu cortiz www Review Torsion and Beam Bending 1 2 Beam Bending 3 Stresses in Beams 1 Torsion loading of a bar by an external torque or twisting moment that tends to produce rotation about the longitudinal z axis of the bar 2 Pure Torsion double arrow notation 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 z L linear elastic isotropic z 0 z r max r R Parameters L bar length angle of twist at end of bar z shear strain shear deformation r radial coordinate z axial coordinate R cross sectional area radius T torque z shear stress max z r 0 rigid support T cross sectional area r r R SOLID BAR 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 Torsion Review Lecture 10 Torsion DERIVATION 1 Geometrical Statement rd 2 Shear Strain Displacment Relation z r d dz d dz 4 Equations of Static Equilibrium Rotational TL constant T JG o radians 180 1 rad 57 3o J m 4 polar moment of inertia 3 Constitutive Law z G z Gr r4 solid bar circular cross sectional area J 2 4 4 r r hollow bar J o i 2 JG JG Nm 2 torsional rigidity k Nm torsiona l stiffness L Tr 5 Shear Stresses z J z f TJf r modulus of rupture in torsion 2001 Brooks Cole a division of Thomson Learning Inc Thomson Learning is a trademark used herein under license Review Beam Theory I Basics BEAM structural member subjected to lateral loads i e forces or moments having their vectors perpendicular to the axis of the bar rectangular cross section y nomenclature z h x flange web I beam cross section b L section about NA web NA NA flange flange L length or span b width h height NA neutral axis passes through centroid x y 0 of cross section I moment of inertia of cross T beam cross section bh 3 r4 Irectangular Icircular 12 4 EI flexural modulus II Types of Supports and Boundary Conditions 1 simply supported 3 cantilever P A A B yA 0 yB 0 2 overhanging A yA 0 yB 0 B P yA 0 A 0 P B Review Beam Theory Cont d III Types of Loading 0 a Mo concentrated moment or couple L Po concentrated load L 0 distributed uniform load P o a a 0 L linearly varying load slope k a 0 L parabolically varying load a 0 L Review Beam Theory Cont d IV Successive Integration Method Shear and Bending Moment Diagrams dV q q loading function dx dM V V shear force dx q x V x q x dx C1 M x V x dx C1x C2 1 C1x 2 x M x dx C2 x C3 EI 5 d M M bending moment dx dV EI curvature slope of y displacement curve dx C1x 3 C 2 x 2 y x q x dx C 3x C 4 6 5 y vertical displacement V Sign Conventions M V V tension compression M tension M V V compression M Beam Theory 3 Normal Stresses and Strains Mo compression N A M o x max ymax I h 2 M o x max ymax for rectangular beams ymax Mo y x max EI EI flexural modulus x EI x Mo y dx tension n M y Flexure formula x o I where x normal stress in x direction M o internal bending moment y vertical distance from NA axis see Gere Chapter 12 Appendix D p 321 I moment of inertia of cross sectional area x max z d m e y p f q y y 0 Mo x max c x y NA x max T moment x Beam Theory Cont d Shear Stresses and Strains Derivation in Gere Section 5 8 V h2 xy rectangular cross section y 2 2I 4 where xy shear stress y x V shear force xy y NA h height of cross sectional area y distance from NA xy rectangular cross sect ion max A cross sectional area z 3V 2A Beam Theory 3 Stresses and Strains y y z x z Beam Theory 3 Stresses and Strains y y z x z Beam Theory 3 Stresses and Strains O Mo N A y z d compression Mo m e tension n p f y dx q x Beam Theory 3 Relate Stress to Moment y y 0 Mo x max c x y NA x max T moment x
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