# HW3 - AE 321 Homework#3 Due Friday Chapter 2 Traction and...

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AE 321 – Homework #3 Due: Friday September 26, 2008 Chapter 2: Traction and Stress 1. (a) Show that the following stress state is in equilibrium (assuming no body forces): ± ij ² ³ ´ µ = 3 x 2 + 3 y 2 zz 6 xy 34 x + y 3 y 2 0 sym .3 x + y z + 54 ² ³ · · · ´ µ ¸ ¸ ¸ kPa (b) Determine the principal stresses at location x=1/2, y=1, z=3/4. 2. The stress tensor at a point P referred to an (x, y, z) coordinate frame is: ij ² ³ ´ µ = 120 40 30 150 20 sym . 100 ² ³ ´ µ · · · kPa (a) Determine the stress components with respect to a rotated frame (x´, y´, z´) for which the matrix of direction cosines is ij [] = ² 0 0 00 1 ³ ´ µ µ µ · ¸ ¸ ¸ (b) Compute the principal stresses using both the (x, y, z) and (x´, y´, z´) axes as a basis. (c) Find the orientation of the principal planes with respect to the (x, y, z) axes. (d) Compute the stress invariants Q i for both frames and verify that they are equal. 3. Show that the principal values, S n , of the stress deviator S ( S ij = ij ² 13 kk ³ ij ) are given by the equation S n 3

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HW3 - AE 321 Homework#3 Due Friday Chapter 2 Traction and...

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