LECTURE 14 3 11 MECHANICS OF MATERIALS F03 INSTRUCTOR Professor Christine Ortiz OFFICE 13 4022 PHONE 452 3084 WWW http web mit edu cortiz www Review Beam Bending 3 Normal Stresses and Strains Transformations of Stress and Strain I Beam Theory 3 Normal Stresses and Strains compression Mo N A z d e p y dx n f q Mo y I where x normal stress in x direction Flexure formula x y y 0 M o internal bending moment y vertical distance from NA axis Mo x max c x y NA see Gere Chapter 12 Appendix D p 321 I moment of inertia of cross sectional area x max 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 x Mo m tension y EI x max T moment x Beam Theory Cont d Shear Stresses and Strains y x xy y NA z Derivation in Gere Section 5 8 V h2 xy rectangular cross section y 2 2I 4 where xy shear stress V shear force h height of cross sectional area y distance from NA xy rectangular cross section max A cross sectional area 3V 2A 2 D Plane Stress State y 0 xy x yx y 0 0 y 0 0 xy x x O z y y O yx xy x x Given A State of Plane Stress What Is The Equivalent Stress State On An Element Rotated By An Arbitrary Angle CCW y O x y y x O x y y x O x yy y x y Ao cos x x Ao cos x Ao xy Ao x O yx Aotan y Aotan y 1 x x yx tan x y tan y x y Ao cos x y x Ao cos x Ao xy Ao x O yx Aotan y Aotan xy x y 2 x tan tan Variation of Stresses With x y cos 2 xy sin 2 2 2 x y x y cos 2 y xy sin 2 2 2 x y sin 2 x y xy cos 2 2 x x y STRESS TRANSFORMATION EQUATIONS Variation of Stresses With For Y 0 2 x xy 0 8 x x y cos 2 xy sin 2 2 2 x y x y cos 2 y xy sin 2 2 2 x y sin 2 x y xy cos 2 2 x x y Principal Stresses and Angles x y cos 2 xy sin 2 2 2 x y x y cos 2 y xy sin 2 2 2 x y sin 2 x y xy cos 2 2 x x y Principal Stresses and Angles TAN 2 p 2 xy x y TAN 2 P 10 0 10 0 100 200 300 2 P
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