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Unformatted text preview: Aerospace Structural Composites I Homework 4 Anirban Chaudhuri 1. Exercise 2 of Chapter 4, but with G xy required to be at least 20 GPa. Repeat Example 4.2.1 for a [ / /0 o ] S laminate. Example 4.2.1: Graphite epoxy laminate with E 1 = 128 GPa, E 2 = 13 GPa, G 12 = 6.4 GPa and 12 = 0.3. Design the laminate with maximum E x , subject to the requirement that G xy is at least 20 GPa and that xy is not larger than 1. Solution: The effective elastic constants are found using the equations given below. 2 11 22 12 22 12 66 22 1 1 x xy xy A A A E h A A G A h A Since the thickness is fixed the optimization problem doesnt depend on the thickness. A ply thickness of 0.125mm is chosen for this example. The Amatrix is evaluated using the equations given below (same as hw 3. The function written for Amatrix for 3 rd homework is used here). 1 1 2 1 3 1 4 1 cos2 sin 2 cos4 sin 4 A n A k k k n A k k k n A k k k n A k k k V h V t V t V t V t where, h (total height) =n*t The reduced stiffness matrix or the Qmatrix for the material is obtained by using the equation (1) [Eq. 2.4.8 in book]. 1 11 12 21 2 22 12 21 12 2 12 12 21 66 12 1 1 1 E Q E Q E Q Q G (1) The corresponding U values (material invariants) are given in equation (2) [Eq. 2.4.20 in book]. 1 1 1 2 2 1 2 6 6 2 1 1 2 2 3 1 1 2 2 1 2 6 6 4 1 1 2 2 1 2 6 6 5 1 1 2 2 1 2 6 6 1 (3 3 2 4 ) 8 1 ( ) 2 1 ( 2 4 ) 8 1 ( 6 4 ) 8 1 ( 2 4 ) 8 U Q Q Q Q U Q Q U Q Q Q Q U Q Q Q Q U Q Q Q Q (2) 1 2 3 11 1 2 3 22 4 3 1 12 5 3 2 66 3 2 3 16 4 26 2 3 1 2 1 2 A A A A A U U U A U U U V A U U V A U U V A V U U A V A U U The normalized plots for E x (normalized by E 1 ), G xy (normalized by 5G 12 ) and xy are plotted while varies between 0 o and 90 o (0 o 90 o ). From the figure 1 it can be seen that the requirement for xy (not larger than 1) is always satisfied and the requirement on G xy (at least 20 GPa) is satisfied in the range 30 o 60 o . This is the range that satisfies both the constraints. In this range the maximum E x is at = 30 o . The maximum value of the objective function at this point is E x = 79.92 GPa. The MATLAB code used is attached in Appendix I. Fig. 1: Elastic properties for graphiteepoxy laminate [ / /0 o ] S 2. Exercise 5, chapter 4 following the hint with h=0.34, h=0.28, and h=0.21. Draw the Miki diagram for each case, and indicate the points on the diagram that can be realized with only 0, 45, 45, and 90deg symmetric balanced laminates. For each point, write the stacking sequence. 45, 45, and 90deg symmetric balanced laminates....
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 Spring '08
 PETERIFJU

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