LECTURE 15 3 11 MECHANICS OF MATERIALS F03 INSTRUCTOR Professor Christine Ortiz OFFICE 13 4022 PHONE 452 3084 WWW http web mit edu cortiz www Review Transformations of Stress and Strain I Transformations of Stress and Strain II 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 Variation of Stresses With 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 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 xy sin 2 y 2 2 x y sin 2 xy cos 2 x y 2 x x y CW CCW 3 General Features of Stress Transformation 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 Principal Stresses and Angles 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 Principal Stresses and Angles tan 2 p 2 xy x y TAN 2 P 10 0 10 0 100 200 300 2 P Planes of Maximum Shear 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 Graphical Representation Principal Stresses and Angles y y x O x Graphical Represenation Planes of Maximum Shear Stress y y x O 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
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