Flexural stiffness design using Miki’s diagram Flexural lamination parameters Boundaries of the domain What laminates have the same position on the Miki in-plane diagram as on the Miki flexural diagram?

Examples (0/90)s : 𝑧 0 =−2𝑡, 𝑧 1 =−𝑡, 𝑧 2 =0,h=4t 0 2 ± 45 : 𝑠 0 = 2 4 𝑡 3 − 𝑡 3 +8 𝑡 3 =0.875, 𝑠 90 = 2 4 𝑡 3 0+ 𝑡 3 =0.125 𝑊 1 ∗ =0.875cos 0 𝑜 +0.125cos 180 𝑜 =0.75 𝑊 3 ∗ =0.875cos 0 𝑜 +0.125cos 360 𝑜 =1 0 2 ± 45 :    𝑊 1 ∗ =0.875cos 0 𝑜 +0.125cos 90 𝑜 =0.875,    𝑊 3 ∗ =0.875cos 0 𝑜 +0.125cos 180 𝑜 =0.75

Stiffest laminate under lateral loads Recall displacement under sine load To find stiffest laminate we need to maximize S= 𝐷 11 +2 𝐷 12 +2 𝐷 66 𝑎 𝑏 2 + 𝐷 22 𝑎 𝑏 4 From Table 2.1 This implies that S is a linear function of the lamination parameters, and the stiffest laminate is an angle ply. Why?

Example 8.2.1a Design a 16-layer 20x15” laminated graphite epoxy plate to maximize its fundamental frequency. Material properties are: 𝐸 1 =18.5,  𝐸 2 =1.89,  𝐺 12 =0.93𝑀𝑠𝑖,  𝜈 12 =0.3, 𝑡=0.005", 𝜌=0.057𝑙 𝑏 𝑖 𝑛 3 Tsai-Pagano material properties (in Msi) are 𝑈 1 =8.3252,  𝑈 2 =8.3821,  𝑈 3 =1.9643,  𝑈 4 =2.5366,  𝑈 5 =2.8943

Normalized fundamental frequency Normalized frequency For our data For maximum frequency we want negative 𝑊 1 ∗ and negative 𝑊 3 ∗ , so angles near 60-deg. Why?

Maximization of frequency Iso-frequency contours on Diagram. Maximum where iso-frequency line is tangent to diagram Get Text suggests ±55.4 4𝑠 Can we do better? Should be omega21 in figure