2009+15-4+StressStrain

2009+15-4+StressStrain - Introduction to Solid...

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1 SJTU Introduction to Solid Mechanics-Vm211 Chapter 15 Stress and Strain Transformation Chapter 15 Stress and Strain Transformation SJTU Introduction to Solid Mechanics-Vm211 Normal and shear strains 15.7 GENERAL EQNS OF PLANE- STRAIN TRANSFORMATION () 21 15 2 sin 2 2 cos 2 2 ' + + + = θ γ ε xy y x y x x 22 15 2 cos 2 2 sin 2 2 ' ' + = xy y x y x 23 - 15 2 sin 2 2 cos 2 2 ' xy y x y x y + = 1 - 5 1 2 sin 2 cos 2 2 ' τ σ xy y x y x x + + + = 2 - 5 1 2 cos 2 sin 2 ' ' xy y x y x + = 3 - 5 1 2 sin 2 cos 2 2 ' xy y x y x y + = SJTU Introduction to Solid Mechanics-Vm211 15.8 MOHR’S CIRCLE PLANE STRAIN ± Advantage of using Mohr’s circle for plane strain transformation is to see graphically how the normal and shear strain components at a point vary from one orientation of the element to the next. SJTU Introduction to Solid Mechanics-Vm211 15.8 MOHR’S CIRCLE PLANE STRAIN ± Eliminate parameter θ in Eqs. 15-21 and 15-22 and rewrite the result as 21 15 2 sin 2 2 cos 2 2 ' + + + = xy y x y x x 22 15 2 cos 2 2 sin 2 2 ' ' + = xy y x y x 9 2 - 15 2 2 2 2 avg R xy x = + SJTU Introduction to Solid Mechanics-Vm211 15.8 MOHR’S CIRCLE PLANE STRAIN ± Eliminate parameter θ in Eqs. 15-21 and 15-22 and rewrite the result as 2 2 avg 2 2 2 avg 2 2 2 where 9 2 - 15 2 + = + = = + xy y x y x xy x R R It represents the Eq. of Mohr’s circle for strain with a center on the axis at point C ( avg , 0 ) and a radius R. SJTU Introduction to Solid Mechanics-Vm211 15.8 MOHR’S CIRCLE PLANE STRAIN Procedure for Analysis Construction of the circle ± Establish a coordinate system such that the abscissa represents the normal strain , with positive to the right, and the ordinate represents half the value of the shear strain, /2, with positive downward.
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2 SJTU Introduction to Solid Mechanics-Vm211 15.8 MOHR’S CIRCLE PLANE STRAIN Procedure for Analysis Construction of the circle ± Using positive sign convention for ε x , y , and γ xy , determine the center of the circle C , which is located on the axis at a distance avg = ( x + v )/2 from the origin. SJTU Introduction to Solid Mechanics-Vm211 15.8 MOHR’S CIRCLE PLANE STRAIN Procedure for Analysis Construction of the circle ± Plot the reference point A having coordinates ( x , xy /2). This point represents the case for which the x’ axis coincides with the x axis. Hence θ = 0 ° . SJTU Introduction to Solid Mechanics-Vm211 15.8 MOHR’S CIRCLE PLANE STRAIN Procedure for Analysis Construction of the circle ± Connect point A with center C of the circle and from the shaded triangle determine the radius R of the circle.
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2009+15-4+StressStrain - Introduction to Solid...

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