1. STRESS TRANSFORMATION
Zdeněk Padovec

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

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

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

5. 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 ´ 𝑆 66 ´ 𝜎 𝑥 𝜎 𝑦 𝜎 𝑥𝑦

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

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

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

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