# Chapter2 - AE 321 Practice Problems Chapter 2 Traction and...

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AE 321 – Practice Problems Chapter 2: Traction and Stress 1. Find the components of the traction on planes defined by n 1 = 1 2 , n 2 = 1 2 n 3 = 0 and n 1 = 1 2 n 2 = ! 1 n 3 = 0 for the following states of stress: (a) ! 11 = 12 = 21 = 0 13 = 31 = 0 22 = 23 = 32 = 0 33 = (b) = = = = = 0 = = = 0 = 0 2. The state of stress at a point P in a material is given by: ij [ ] = 20 2 1 2 " 15 2 1 2 3 # \$ % % & ( ( KPa (a) Compute the components of traction on the plane passing through P whose outward normal vector n makes equal angles with the coordinate axes. (b) Compute the normal and tangential components of traction on this plane. 3. Repeat the above exercise for the stress state: ij [ ] = 10 2 1 " 15 5 3 # \$ % % % & ( ( ( KPa 4. The state of stress at a point is given by σ ij . A plane is defined by the direction cosines of the normal (1/2, 1/2, 1/ 2). Under what conditions will the traction vector on this plane be parallel to the x 2 axis and have a unit magnitude?

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5. What are the conditions which a state of stress must satisfy for at least one surface to remain traction free? 6. Determine whether the following stress field is admissible in a body. Assume negligible body forces. ! xx = " 2 x 2 " 3 y 2 " 5 z Pa xy = z + 4 xy " 6 Pa yy = " 2 y 2 + 7 Pa xz = " 3 x + 2 y + 1 Pa zz = 4 x + y + 3 z " 5 Pa yz = 0 Pa 7. Determine the body forces for which the following stress field describes a state of equilibrium: ij " # \$ % = yz + 4 z 2 + 2 x 5 y + z xz + 3 y 8 x 3 sym . 2 xyz " # & & & \$ % kPa 8. Assume that the stress state in the interior of the body is given in a Cartesian coordinate system by: ij " # \$ % = & P 0 0 0 & P 0 0 0 & P " # \$ % ( ( ( kPa Note: This stress state is called isotropic or hydrostatic. (a) Does this stress state satisfy equilibrium? (b) Show that the normal stresses are – P and the shear stresses vanish for any other Cartesian coordinate system.
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Chapter2 - AE 321 Practice Problems Chapter 2 Traction and...

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