2009+15-2+StressStrain

2009+15-2+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 ± Equations for plane stress transformation have a graphical solution that is easy to remember and use. ± This approach will help you to “visualize” how the normal and shear stress components vary as the plane acted on is oriented in different directions. 15.4 MOHR’S CIRCLE: PLANE STRESS SJTU Introduction to Solid Mechanics-Vm211 ± Eqs 15-1 and 15-2 are rewritten as ± Parameter θ can be eliminated by squaring each eq. and adding them together. The result is () 10 - 15 2 cos 2 sin 2 ' ' θ τ σ xy y x y x + = xy y x y x y x x 2 2 ' ' 2 2 ' 2 2 + = + + 9 - 5 1 2 sin 2 cos 2 2 ' xy y x y x x + = + 15.4 MOHR’S CIRCLE: PLANE STRESS 1 - 5 1 2 sin 2 cos 2 2 ' xy y x y x x + + + = 2 - 5 1 2 cos 2 sin 2 ' ' xy y x y x + = SJTU Introduction to Solid Mechanics-Vm211 15.4 MOHR’S CIRCLE: PLANE STRESS ± If x , y , xy are known constants, thus we compact the Eq. as ( ) 12 - 15 2 2 where 11 - 5 1 2 2 2 ' ' 2 2 ' xy y x y x avg y x avg x R R + = + = = + xy y x y x y x x 2 2 ' ' 2 2 ' 2 2 + = + + SJTU Introduction to Solid Mechanics-Vm211 15.4 MOHR’S CIRCLE: PLANE STRESS ± Establish coordinate axes; positive to the right and positive downward, Eq. 15-11 represents a circle having radius R and center on the axis at point C ( avg , 0). This is called the Mohr’s Circle. 12 - 15 2 2 where 11 - 5 1 2 2 2 ' ' 2 2 ' xy y x y x avg y x avg x R R + = + = = + o SJTU Introduction to Solid Mechanics-Vm211 Case 1 ( x’ axis coincident with x axis) 1. = 0 ° 2. x’ = x 3. x’y’ = xy . ± Consider this as reference point A , and plot its coordinates A ( x , xy ). ± Apply Pythagoras theorem to shaded triangle to determine radius R . ± Using points C and A , the circle can now be drawn. 15.4 MOHR’S CIRCLE: PLANE STRESS o
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2 SJTU Introduction to Solid Mechanics-Vm211 Procedure for Analysis Construction of the circle 1. Establish coordinate system where abscissa represents the normal stress σ , (+ve to the right), and the ordinate represents shear stress τ , (+ve downward). 2. Use positive sign convention for x , y , xy , plot the center of the circle C , located on the axis at a distance avg = ( x + y )/2 from the origin. 15.4 MOHR’S CIRCLE: PLANE STRESS o SJTU Introduction to Solid Mechanics-Vm211 Procedure for Analysis Construction of the circle 3. Plot reference point A ( x , xy ). This point represents the normal and shear stress components on the element’s right-hand vertical face. (Since x’ axis coincides with x axis, θ = 0 . ) 15.4 MOHR’S CIRCLE: PLANE STRESS o SJTU
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This note was uploaded on 08/09/2011 for the course EE 211 taught by Professor Liuxila during the Summer '09 term at Shanghai Jiao Tong University.

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2009+15-2+StressStrain - Introduction to Solid...

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