HW3solution - AE 321 Solution to Homework #3 Chapter 2:...

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AE 321 – Solution to Homework #3 Chapter 2: Traction and Stress 1. (a) To determine if the given state of stress is in equilibrium, we can to apply the equations of equilibrium to the stress tensor: x -direction 0 0 6 6 x x z y x xz xy xx Equilibrium is satisfied. y -direction 0 0 6 6 y y z y x yz yy yx Equilibrium is satisfied. z -direction 0 1 0 1 z y x zz zy zx Equilibrium is satisfied. (b) At 4 3 and , 1 , 2 1 z y x , the state of stress is ij 3 3 0 3 3 0 0 0 3 kPa The principal stresses are determined by solving the following equation 0 ij ij  (4.3) Then 3 3 0 3 3 0 0 0 3 3 3 2 9 0 Thus the characteristic equation is 6 3 0 From the characteristic equation we can easily see that the principal stresses are
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1 1 6 kPa 2 2 3 kPa 3 3 0 kPa 2. Given state of stress at point P as a function of the ( x, y, z ) coordinate frame ij 120 40 30 40 150 20 30 20 100 kPa (a) Determine the stress components with respect to a rotated frame (x’, y’, z’) for which the matrix of direction cosines is   1 0 0 0 2 1 2 3 0 2 3 2 1 ij . The components of stress tensor is obtained using the following transformation law       T ij lj ki kl R R or ' ' (3.18) Then ' kl 1 2 3 2 0 3 2 1 2 0 0 0 1 120 40 30 40 150 20 30 20 100 1 2 3 2 0 3 2 1 2 0 0 0 1 kPa After performing the multiplication, we found that the stress tensor in the rotated frame is
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' kl 107.85 32.99 2.32 32.99 162.13 35.98 2.32 35.98 100 kPa (b.1) Principal stress using the reference frame, i.e. ( x, y, z ) From equation (3.11) we have 120 40 30 40 150 20 30 20 100 0 120 150 100 400 40 40 100 600 30 800 30 150 0 After simplifying the characteristic equation to solve is given by 0 1505000 42100 370 2 3 The roots of the characteristic equation are 1 190.4 kPa , 2 102.42 kPa and 3 77.17 kPa * ij 190.4 0 0 0 102.42 0 0 0 77.17 kPa (b.2) Principal stress using the rotated frame, i.e. ( x’, y’, z’ ) From equation (3.11)
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0 100 98 . 35 32 . 2 98 . 35 13 . 162 99 . 32 32 . 2 99 . 32 85 . 107
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HW3solution - AE 321 Solution to Homework #3 Chapter 2:...

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