LECTURE 16 3 11 MECHANICS OF MATERIALS F03 INSTRUCTOR Professor Christine Ortiz OFFICE 13 4022 PHONE 452 3084 WWW http web mit edu cortiz www REVIEW STRESS TRANFORMATIONS II SAMPLE PROBLEMS INTRODUCION TO MOHR S CIRCLE Review Lecture 14 Stress Transformations I 1 Plane Stress Rotations Given a state of plane stress what is the equivalent stress state on an element rotated by an arbitrary angle CCW defined relative to x axis original coordinate axis x y stresses in original coordinate axis x y xy new coordinate axis x y stresses in new coordinate axis x y x y y y y y xy O x y x x y O x x x Consider a free body diagram of a wedge where a cut is made along an inclined plane where the transformed stresses are desired Equations of Static Equilibrium Geometry Trig y y x y Ao cos x x Ao cos x Ao xy Ao x O yx Aotan y Aotan x y cos 2 xy sin 2 2 2 x y x y cos 2 xy sin 2 y 2 2 x y sin 2 xy cos 2 x y 2 x x y STRESS TRANSFORMATION EQUATIONS stresses vary continuously as the axis is rotated Principal and Maximum Shear Principal Stresses 1 2 x y x y y 2 2 xy 2 2 1 max imum normal stress 2 min imum normal stress y p1 1 y max Principal Angles Planes 2 xy tan 2 p x 2 x min p2 x O x y p1 1 p2 2 y Maximum Shear Stresses x y 2 max min xy 2 max max imum shear stress min min imum shear stress 2 y Planes Angles of Maximum Shear x y tan 2 s 2 xy s1 max s2 min y x y max x O Relationships Between Principal and Shear s p 45o p1 and p2 90o apart mutually perpendicular planes s1 and s2 90o apart mutually perpendicular planes 1 2 max 2 x s1 x Derivation of General Equation for Principal Stresses x y cos 2 xy sin 2 2 2 x y x y cos 2 y xy sin 2 2 2 2 xy tan 2 p x x y x y Sample Problem 1 Stresses on an Inclined Element An element in plane stress is subjected to x 16 000 psi y 6000 psi xy 4000 psi Find and draw the stresses acting on an element inclined at an angle 45o STRESS TRANSFORMATION EQUATIONS y O x y cos 2 xy sin 2 2 2 x y x y cos 2 xy sin 2 y 2 2 x y sin 2 xy cos 2 x y 2 x x z face or x y plane x y Sample Problem 1 Stresses on an Inclined Element An element in plane stress is subjected to x 16 000 psi y 6000 psi xy 4000 psi Find and draw the stresses acting on an element inclined at an angle 45o STRESS TRANSFORMATION EQUATIONS x y cos 2 xy sin 2 2 2 x y x y cos 2 xy sin 2 y 2 2 x y sin 2 xy cos 2 x y 2 x x y Sample Problem 1 Stresses on an Inclined Element y y x O x Sample Problem 2 Principal and Maximum Shear Stresses An element in plane stress is subjected to x 12300 psi y 4200 psi xy 4700 psi a Determine the principal stresses and show them on a sketch of a properly oriented element b Determine the maximum shear stresses and show them on a sketch of a properly oriented element y O x z face or x y plane Sample Problem 2 Principal and Maximum Shear Stresses An element in plane stress is subjected to x 12300 psi y 4200 psi xy 4700 psi a Determine the principal stresses and show them on a sketch of a properly oriented element b Determine the maximum shear stresses and show them on a sketch of a properly oriented element x x y 2 x y cos 2 xy sin 2 2 Sample Problem 2 Principal Stresses y x y O x Sample Problem 2 Principal and Maximum Shear Stresses An element in plane stress is subjected to x 12300 psi y 4200 psi xy 4700 psi a Determine the principal stresses and show them on a sketch of a properly oriented element b Determine the maximum shear stresses and show them on a sketch of a properly oriented element x y tan 2 s 2 xy x y 2 max min xy 2 2 Sample Problem 2 Maximum Shear Stresses y y x O x y cos 2 xy sin 2 2 2 x y x y cos 2 xy sin 2 y 2 2 x y sin 2 xy cos 2 x y 2 x x y x MOHRS CIRCLE Stress Transformation Equations x y cos 2 xy sin 2 1 2 2 x y x y cos 2 xy sin 2 2 y 2 2 x y sin 2 xy cos 2 3 x y 2 x x y MOHRS CIRCLE x ave 2 x2 y R 2 ave x y 2 x y 2 R xy 2 2 CONSTRUCTION OF MOHR S CIRCLE x ave 2 x2 y R 2 ave x y 2 x y 2 R xy 2 2 Given stress state y y y O xy x z face or x y plane x x O C
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