1. Laminates of Orthotropic plies
Section for in-plane loading only
Hooke’s law for k-th ply
Stress resultants

2. A Matrix
Basic equation is the same as in isotropic case, 𝑁=𝐴 𝜀 0 , but 16 and 26 terms
We now use the Tsai-Pagano “invariants”
Leading to lamination parameters

3. Effective properties From Q’s to A’s using lamination parameters

4. Hooke’s law with effective properties
Average stresses
Hooke’s law

5. Typical stiffness optimization problem
No stresses in individual plies, so no credible failure constraint.
When 0-deg, 90-deg and 45-deg plies are present this is reasonable even for strength.

6. Example 4.1.1
Graphite/epoxy 𝐸 1 =18.5𝑀𝑠𝑖,  𝐸 2 =1.89𝑀𝑠𝑖,   𝐺 12 =0.93𝑀𝑠𝑖,  𝜈 12 =0.3
Two load conditions
𝑁 𝑥 =10,000 𝑙 𝑏 𝑖 𝑛
𝑁 𝑥𝑦 =3,000 𝑙 𝑏 𝑖 𝑛
Allowable strains: Normal strains 0.4%, shear strain 0.006
Try 0 𝑛 , ±45 𝑛𝑠 , ± 𝑛𝑠

7. Sanity checks for first two laminates
For all-zero laminate 𝐸 𝑥 = 𝐸 1 =18.9𝑀𝑠𝑖,  𝐺 𝑥𝑦 =0.93𝑀𝑠𝑖
We find that to satisfy normal strain we need at least in, while to satisfy the shear constraint we need in. Are these numbers reasonable?
For ±45 laminate we find 𝐸 𝑥 =3.18𝑀𝑠𝑖,  𝐺 𝑥𝑦 =4.86​𝑀𝑠𝑖
We need in for normal strain constraint, and in for shear constraint. Are these reasonable?