STRESS TRANSFORMATION Zdeněk Padovec

Stress transformation y L unidirectional lamina EL, ET, GLT, νLT, α= 0°, 15°, 90° 𝜀 𝑥 𝜀 𝑦 𝛾 𝑥𝑦 =??? σx T α x

Stress transformation isotropic material 𝜀 𝑥 𝜀 𝑦 𝛾 𝑥𝑦 = 1 𝐸 − 𝜐 𝐸 0 − 𝜐 𝐸 1 𝐸 0 0 0 1 𝐺 𝜎 𝑥 𝜎 𝑦 𝜎 𝑥𝑦 = 1 𝐸 − 𝜐 𝐸 0 − 𝜐 𝐸 1 𝐸 0 0 0 1 𝐺 𝜎 𝑥 0 0 , 𝜀 𝑥 = 𝜎 𝑥 𝐸 , 𝜀 𝑦 =−𝜐 𝜀 𝑥 , 𝛾 𝑥𝑦 =0, 𝐺= 1 2 1+𝜈 . result is independent on angle orientation α

Stress transformation composite material compliance matrix in coordinate system LT 𝑆= 1 𝐸 𝐿 − 𝜐 𝑇𝐿 𝐸 𝑇 0 − 𝜐 𝐿𝑇 𝐸 𝐿 1 𝐸 𝑇 0 0 0 1 𝐺 𝐿𝑇 transformation to coordinate system xy 𝑆 ´ = 𝑇 𝜀 −1 𝑆 𝑇 𝜎

Stress transformation transformation to coordinate system xy 𝑇 𝜀 = cos 2 𝜃 sin 2 𝜃 sin𝜃cos𝜃 sin 2 𝜃 cos 2 𝜃 −sin𝜃cos𝜃 −2sin𝜃cos𝜃 2sin𝜃cos𝜃 cos 2 𝜃− sin 2 𝜃 𝑇 𝜎 = cos 2 𝜃 sin 2 𝜃 2sin𝜃cos𝜃 sin 2 𝜃 cos 2 𝜃 −2sin𝜃cos𝜃 −sin𝜃cos𝜃 sin𝜃cos𝜃 cos 2 𝜃− sin 2 𝜃 Hooke´s law 𝜀 𝑥 𝜀 𝑦 𝛾 𝑥𝑦 = 𝑆 11 ´ 𝑆 12 ´ 0 𝑆 21 ´ 𝑆 22 ´ 0 0 0 𝑆 66 ´ 𝜎 𝑥 𝜎 𝑦 𝜎 𝑥𝑦

Stress transformation 𝑆 11 ´ = 𝑆 11 cos 4 𝛼+ 𝑆 22 sin 4 𝛼+ 2 𝑆 12 + 𝑆 66 sin 2 𝛼 cos 2 𝛼 𝑆 12 ´ = 𝑆 21 ´ = 𝑆 11 + 𝑆 22 − 𝑆 66 sin 2 𝛼 cos 2 𝛼+ 𝑆 12 (cos 4 𝛼+ sin 4 𝛼) 𝑆 22 ´ = 𝑆 11 sin 4 𝛼+ 𝑆 22 cos 4 𝛼+ 2 𝑆 12 + 𝑆 66 sin 2 𝛼 cos 2 𝛼 𝑆 66 ´ =2 2 𝑆 11 + 𝑆 22 − 2𝑆 12 − 𝑆 66 sin 2 𝛼 cos 2 𝛼+ 𝑆 66 ( sin 4 𝛼+ cos 4 𝛼) α = 0° 𝑆 11 ´ = 𝑆 11 , 𝑆 12 ´ = 𝑆 21 ´ = 𝑆 12 =𝑆 21 , 𝑆 22 ´ = 𝑆 22 , 𝑆 66 ´ = 𝑆 66 𝜀 𝑥 = 𝑆 11 ´ 𝜎 𝑥 + 𝑆 12 ´ 𝜎 𝑦 +0 𝜎 𝑥𝑦 = 𝑆 11 𝜎 𝑥 + 𝑆 12 0+0∙0= 𝜎 𝑥 𝐸 𝐿 𝜀 𝑦 = 𝑆 21 ´ 𝜎 𝑥 + 𝑆 22 ´ 𝜎 𝑦 +0 𝜎 𝑥𝑦 = 𝑆 21 𝜎 𝑥 + 𝑆 22 0+0∙0=− 𝜈 𝐿𝑇 𝐸 𝐿 𝜎 𝑥 𝛾 𝑥𝑦 =0 𝜎 𝑥 +0 𝜎 𝑦 + 𝑆 66 ´ 𝜎 𝑥𝑦 =0

Stress transformation 𝑆 11 ´ = 𝑆 11 cos 4 𝛼+ 𝑆 22 sin 4 𝛼+ 2 𝑆 12 + 𝑆 66 sin 2 𝛼 cos 2 𝛼 𝑆 12 ´ = 𝑆 21 ´ = 𝑆 11 + 𝑆 22 − 𝑆 66 sin 2 𝛼 cos 2 𝛼+ 𝑆 12 (cos 4 𝛼+ sin 4 𝛼) 𝑆 22 ´ = 𝑆 11 sin 4 𝛼+ 𝑆 22 cos 4 𝛼+ 2 𝑆 12 + 𝑆 66 sin 2 𝛼 cos 2 𝛼 𝑆 66 ´ =2 2 𝑆 11 + 𝑆 22 − 2𝑆 12 − 𝑆 66 sin 2 𝛼 cos 2 𝛼+ 𝑆 66 ( sin 4 𝛼+ cos 4 𝛼) α = 90° 𝑆 11 ´ = 𝑆 22 , 𝑆 12 ´ = 𝑆 21 ´ = 𝑆 12 =𝑆 21 , 𝑆 22 ´ = 𝑆 11 , 𝑆 66 ´ = 𝑆 66 𝜀 𝑥 = 𝑆 11 ´ 𝜎 𝑥 + 𝑆 12 ´ 𝜎 𝑦 +0 𝜎 𝑥𝑦 = 𝑆 22 𝜎 𝑥 + 𝑆 12 0+0∙0= 𝜎 𝑥 𝐸 𝑇 𝜀 𝑦 = 𝑆 21 ´ 𝜎 𝑥 + 𝑆 22 ´ 𝜎 𝑦 +0 𝜎 𝑥𝑦 = 𝑆 21 𝜎 𝑥 + 𝑆 11 0+0∙0=− 𝜈 𝐿𝑇 𝐸 𝐿 𝜎 𝑥 𝛾 𝑥𝑦 =0 𝜎 𝑥 +0 𝜎 𝑦 + 𝑆 66 ´ 𝜎 𝑥𝑦 =0

Stress transformation 𝑆 11 ´ = 𝑆 11 cos 4 𝛼+ 𝑆 22 sin 4 𝛼+ 2 𝑆 12 + 𝑆 66 sin 2 𝛼 cos 2 𝛼 𝑆 12 ´ = 𝑆 21 ´ = 𝑆 11 + 𝑆 22 − 𝑆 66 sin 2 𝛼 cos 2 𝛼+ 𝑆 12 (cos 4 𝛼+ sin 4 𝛼) 𝑆 22 ´ = 𝑆 11 sin 4 𝛼+ 𝑆 22 cos 4 𝛼+ 2 𝑆 12 + 𝑆 66 sin 2 𝛼 cos 2 𝛼 𝑆 66 ´ =2 2 𝑆 11 + 𝑆 22 − 2𝑆 12 − 𝑆 66 sin 2 𝛼 cos 2 𝛼+ 𝑆 66 ( sin 4 𝛼+ cos 4 𝛼) α = 15° 𝑆 11 ´ = 0,87𝑆 11 +0,0045 𝑆 22 +0,0625(2 𝑆 12 + 𝑆 66 ) 𝑆 12 ´ = 𝑆 21 ´ =0,0625 𝑆 11 + 𝑆 22 − 𝑆 66 +0,875 𝑆 12 𝑆 22 ´ = 0,0045𝑆 11 +0,87 𝑆 22 +0,0625 2 𝑆 12 + 𝑆 66 𝑆 66 ´ =2 2 𝑆 11 + 𝑆 22 − 2𝑆 12 − 𝑆 66 0,0625+ 0,875𝑆 66

Stress transformation α = 15° 𝑆 11 ´ = 0,87𝑆 11 +0,0045 𝑆 22 +0,0625(2 𝑆 12 + 𝑆 66 ) 𝑆 12 ´ = 𝑆 21 ´ =0,0625 𝑆 11 + 𝑆 22 − 𝑆 66 +0,875 𝑆 12 𝑆 22 ´ = 0,0045𝑆 11 +0,87 𝑆 22 +0,0625 2 𝑆 12 + 𝑆 66 𝑆 66 ´ =2 2 𝑆 11 + 𝑆 22 − 2𝑆 12 − 𝑆 66 0,0625+ 0,875𝑆 66 𝜀 𝑥 = 𝑆 11 ´ 𝜎 𝑥 + 𝑆 12 ´ 𝜎 𝑦 +0 𝜎 𝑥𝑦 = 0,87𝑆 11 +0,0045 𝑆 22 +0,0625(2 𝑆 12 + 𝑆 66 ) 𝜎 𝑥 𝜀 𝑦 = 𝑆 21 ´ 𝜎 𝑥 + 𝑆 22 ´ 𝜎 𝑦 +0 𝜎 𝑥𝑦 = 0,0625 𝑆 11 + 𝑆 22 − 𝑆 66 +0,875 𝑆 12 𝜎 𝑥 𝛾 𝑥𝑦 =0 𝜎 𝑥 +0 𝜎 𝑦 + 𝑆 66 ´ 𝜎 𝑥𝑦 =0