Laminates of Orthotropic plies Section 2.4.3 for in-plane loading only Hooke’s law for k-th ply Stress resultants

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

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

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

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.

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 , ± 45 90 0 𝑛𝑠

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 0.135 in, while to satisfy the shear constraint we need 0.538 in. Are these numbers reasonable? For ±45 laminate we find 𝐸 𝑥 =3.18𝑀𝑠𝑖,  𝐺 𝑥𝑦 =4.86​𝑀𝑠𝑖 We need 0.787 in for normal strain constraint, and 0.102 in for shear constraint. Are these reasonable?