# HW4 - Aerospace Structural Composites I Homework 4 Anirban...

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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 doesn’t depend on the thickness. A ply- thickness of 0.125mm is chosen for this example. The A-matrix is evaluated using the equations given below (same as hw 3. The function written for A-matrix 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 Q-matrix 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 graphite-epoxy 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 90-deg symmetric balanced laminates. For each point, write the stacking sequence. 45, -45, and 90-deg symmetric balanced laminates....
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HW4 - Aerospace Structural Composites I Homework 4 Anirban...

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