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Unformatted text preview: AE 321 – Homework #7 Solution 1. The figure below shows both the Cartesian coordinates z y x , , and the cylindrical coordinates z r , , used to define the boundary conditions of the hollow circular cylinder. (i) Boundary conditions in Cartesian coordinates (a) Fixed end surface: z and b r a (remember 2 2 2 z y x r ) z y x u u (b) Free end surface: L z and b r a s Cauchy z e n ' ˆ zz z yz y xz x P T T T (c) Outer lateral surface: L z and 2 2 2 b y x y x e b y e b x n ˆ ˆ Then T x P x b q y b xx x b xy y b T y P y b q x b xy x b yy y b T z xz x b yz y b (d) Inner lateral surface: L z and 2 2 2 a y x . Following the same procedure as in part (c) n x a ˆ e x y a ˆ e y T x P i x a xx x a xy y a T y P i y a xy x a yy y a T z xz x a yz y a (ii) Boundary conditions in cylindrical coordinates (a) Fixed end surface: z u r u u z (b) Free end surface: L z and b r a z e n ˆ T r rz T z T z P zz (c) Outer lateral surface: L z and b r r e n ˆ T r P rr T q r T z rz (d) Inner lateral surface: L z and a r n ˆ e r T r P i rr T r r T z rz rz 2. Since this is an internal body moment, the equations concerning the principle of conservation of linear momentum will not be affected. Therefore the derivation of Cauchy’s equation and the equations of motion will remained unchanged. However the equation corresponding to the conservation of angular momentum will have one additional term related to the presence of an internal body moment (much in analogy to an internal body force appearing in the equations of motion). Therefore the principle of conservation of angular momentum will reduce to ijk jk w i V dV 0 ijk jk w i . Therefore we can no longer be guaranteed to have a symmetric stress tensor! 3. Since the solution for the tensile case is known (i.e. the stress field through the infinite plate is known), the stress in any direction can be determined easily as follows (see Figure 5.1): In the z y x , , coordinates system: yy and yz xz xy zz xx (5.1) In the ' , ' , ' z y x coordinates system (using transformation or Mohr’s circle): 2 2 ' ' sin sin yy x x (5.2) 2 2 ' ' cos cos yy y y (5.3) cos sin cos sin ' ' yy y x (5.4) Graphically, this stress field can be represented as shown in Figure 5.2. Graphically, this stress field can be represented as shown in Figure 5....
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 Spring '08
 Dutton,J
 stress tensor, xz Cauchy

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