HW04_Solution - EML 4507 FEA Design Fall 2010 HW04 Solution...

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EML 4507 FEA & Design Fall 2010 HW04 Solution 23. A strain rosette consisting of three strain gages was used to measure the strains at a point in a thin- walled plate. The measured strains in the three gages are: A = 0.001, B = −0.0006, and C = 0.0007. Not that Gage C is at 45 o with respect to the x -axis. (a) Determine the complete state of strains and stresses (all six components) at that point. Assume E = 70 GPa, and = 0.3. (b) What are the principal strains and their directions? (c) What are the principal stresses and their directions? (d) Show that the principal strains and stresses satisfy the stress-strain relations. Solution: (a) From figure it is obvious xx = A = 0.001 and yy = B = −0.0006. Shear strain can be found using the transformation relation in Eq. (1.50). The 2-D version of Eq. (1.50) becomes 2 2 nn xx x yy y xy x y n n n n where n x = cos(45 o ) and n y = sin(45 o ). Thus, 2 2 (45 ) cos 45 sin 45 sin45cos45 0.0007 C nn xx yy xy By solving the above equation, we obtain xy = 0.001. Since the strain rosette only measures plane stress state, zz is unknown. But, there is no shear strain in the z-direction, xz = yz = 0. In order to calculate the unknown stress zz , we use the constitutive relation for plane stress. Since the plate is in a state of plane stress, zz = xz = yz = 0. Other stresses can be obtained from stress-strain relations for plane stress conditions as shown below: 2 1 63.1 MPa 1 23.1 1 13.5 MPa x x y y xy xy E G For plane stress condition the through-the-thickness strain is obtained from Eq. (1.59), as 0.000171 zz xx yy E x y A B C
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(b) For a state of plane stress, zz = −0.000171 is a principal stress and the z -axis (0,0,1) is the corresponding principal strain direction. The other two principal strains can be found from the eigen value problem in 2D strain state: 0 [ ]{ } 0 xx xy x xy yy y n n I n Two principal strains are calculated from the condition that the determinant of the coefficient matrix is zero: 2 ( )( ) 0 xx
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