1. CLASSIC LAMINATION THEORY Zdeněk Padovec

2. Classic lamination theory
layered laminates
each lamina is orthotropic and quasi homogeneous
thickness << width, length plane stress
displacement of particular points are very small
ideal, infinitely thin joint between laminas – continuous displacements, linear in through thickness directions
Kirchhoff ´s hypothesis, εz  0
linear relationship between stress and strain

3. Classic lamination theory
forces and moments working on lamina
𝑁 𝑥 = ℎ 𝑘−1 ℎ 𝑘 𝜎 𝑥𝑥 𝑑𝑧 , 𝑁 𝑦 = ℎ 𝑘−1 ℎ 𝑘 𝜎 𝑦𝑦 𝑑𝑧, 𝑁 𝑥𝑦 = ℎ 𝑘−1 ℎ 𝑘 𝜎 𝑥𝑦 𝑑𝑧
𝑀 𝑥 = ℎ 𝑘−1 ℎ 𝑘 𝜎 𝑥𝑥 𝑧𝑑𝑧 , 𝑀 𝑦 = ℎ 𝑘−1 ℎ 𝑘 𝜎 𝑦𝑦 𝑧𝑑𝑧, 𝑀 𝑥𝑦 = ℎ 𝑘−1 ℎ 𝑘 𝜎 𝑥𝑦 𝑧𝑑𝑧

4. Classic lamination theory
σ is substituted according to Hookes´s law
𝑁 𝑥 𝑁 𝑦 𝑁 𝑥𝑦 = 𝑘=1 𝑛 ℎ 𝑘−1 ℎ 𝑘 𝑄 11 𝑄 12 𝑄 16 𝑄 21 𝑄 22 𝑄 26 𝑄 61 𝑄 62 𝑄 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝛾 𝑥𝑦 𝑑𝑧+ ℎ 𝑘−1 ℎ 𝑘 𝑄 11 𝑄 12 𝑄 16 𝑄 21 𝑄 22 𝑄 26 𝑄 61 𝑄 62 𝑄 𝑘 𝑥 𝑘 𝑦 𝑘 𝑥𝑦 𝑧𝑑𝑧
𝑀 𝑥 𝑀 𝑦 𝑀 𝑥𝑦 = 𝑘=1 𝑛 ℎ 𝑘−1 ℎ 𝑘 𝑄 11 𝑄 12 𝑄 16 𝑄 21 𝑄 22 𝑄 26 𝑄 61 𝑄 62 𝑄 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝛾 𝑥𝑦 𝑧𝑑𝑧+ ℎ 𝑘−1 ℎ 𝑘 𝑄 11 𝑄 12 𝑄 16 𝑄 21 𝑄 22 𝑄 26 𝑄 61 𝑄 62 𝑄 𝑘 𝑥 𝑘 𝑦 𝑘 𝑥𝑦 𝑧 2 𝑑𝑧
𝑁 𝑥 𝑁 𝑦 𝑁 𝑥𝑦 𝑀 𝑥 𝑀 𝑦 𝑀 𝑥𝑦 = 𝐴 11 𝐴 𝐴 𝐵 11 𝐵 21 𝐵 𝐴 12 𝐴 𝐴 𝐵 12 𝐵 22 𝐵 𝐴 16 𝐴 𝐴 𝐵 16 𝐵 26 𝐵 𝐵 11 𝐵 𝐵 𝐷 11 𝐷 21 𝐷 𝐵 12 𝐵 𝐵 𝐷 12 𝐷 22 𝐷 𝐵 16 𝐵 𝐵 𝐷 16 𝐷 26 𝐷 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝛾 𝑥𝑦 𝑘 𝑥 𝑘 𝑦 𝑘 𝑥𝑦 .
𝐴 𝑖𝑗 = 𝑘=1 𝑛 𝑄 𝑖𝑗 𝑘 ℎ 𝑘 − ℎ 𝑘−1
𝐵 𝑖𝑗 = 1 2 𝑘=1 𝑛 𝑄 𝑖𝑗 𝑘 ℎ 𝑘 2 − ℎ 𝑘−1 2
𝐷 𝑖𝑗 = 1 3 𝑘=1 𝑛 𝑄 𝑖𝑗 𝑘 ℎ 𝑘 3 − ℎ 𝑘−1 3

5. Classic lamination theory
A…membrane stiffness of a laminate
B...bending – extension coupling stiffness of a laminate
D…bending stiffness of a laminate
𝐍 𝐌 = 𝐀 𝐁 𝐁 𝐃 𝛆 𝐦 𝟎 𝐤
𝛆 𝐦 𝟎 𝐤 = 𝐀 𝐁 𝐁 𝐃 𝐍 𝐌

6. Laminate analysis and design
general laminates, e.g. [40|-30|0|60]
𝑁 𝑥 𝑁 𝑦 𝑁 𝑥𝑦 𝑀 𝑥 𝑀 𝑦 𝑀 𝑥𝑦 = 𝐴 11 𝐴 𝐴 𝐵 11 𝐵 21 𝐵 𝐴 12 𝐴 𝐴 𝐵 12 𝐵 22 𝐵 𝐴 16 𝐴 𝐴 𝐵 16 𝐵 26 𝐵 𝐵 11 𝐵 𝐵 𝐷 11 𝐷 21 𝐷 𝐵 12 𝐵 𝐵 𝐷 12 𝐷 22 𝐷 𝐵 16 𝐵 𝐵 𝐷 16 𝐷 26 𝐷 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝛾 𝑥𝑦 𝑘 𝑥 𝑘 𝑦 𝑘 𝑥𝑦 .
extension,
shear, bending,
torsion
extension, shear, bending, torsion

7. Laminate analysis and design
balanced laminate (no coupling between normal and shear forces), e.g. [30|-60|0|60|-30], same thickness
𝑁 𝑥 𝑁 𝑦 𝑁 𝑥𝑦 𝑀 𝑥 𝑀 𝑦 𝑀 𝑥𝑦 = 𝐴 11 𝐴 𝐵 11 𝐵 21 𝐵 𝐴 12 𝐴 𝐵 12 𝐵 22 𝐵 𝐴 𝐵 16 𝐵 26 𝐵 𝐵 11 𝐵 𝐵 𝐷 11 𝐷 21 𝐷 𝐵 12 𝐵 𝐵 𝐷 12 𝐷 22 𝐷 𝐵 16 𝐵 𝐵 𝐷 16 𝐷 26 𝐷 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝛾 𝑥𝑦 𝑘 𝑥 𝑘 𝑦 𝑘 𝑥𝑦 .
extension, bending, torsion
bending, torsion

8. Laminate analysis and design
symmetric laminates (no coupling between forces and moments), e.g. [40|90|0|60|-30]s, different thickness
𝑁 𝑥 𝑁 𝑦 𝑁 𝑥𝑦 𝑀 𝑥 𝑀 𝑦 𝑀 𝑥𝑦 = 𝐴 11 𝐴 𝐴 𝐴 12 𝐴 𝐴 𝐴 16 𝐴 𝐴 𝐷 11 𝐷 21 𝐷 𝐷 12 𝐷 22 𝐷 𝐷 16 𝐷 26 𝐷 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝛾 𝑥𝑦 𝑘 𝑥 𝑘 𝑦 𝑘 𝑥𝑦 .
extension, shear
bending, torsion

9. Laminate analysis and design
symmetric balanced laminates (no coupling between forces and moments and also between normal and shear stresses), e.g. [40|90|0|60|-30]s, same thickness of symmetric layers
𝑁 𝑥 𝑁 𝑦 𝑁 𝑥𝑦 𝑀 𝑥 𝑀 𝑦 𝑀 𝑥𝑦 = 𝐴 11 𝐴 𝐴 12 𝐴 𝐴 𝐷 11 𝐷 21 𝐷 𝐷 12 𝐷 22 𝐷 𝐷 16 𝐷 26 𝐷 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝛾 𝑥𝑦 𝑘 𝑥 𝑘 𝑦 𝑘 𝑥𝑦 .
extension
bending,
torsion

10. Laminate analysis and design
symmetric cross ply laminate (just 0° and 90° symmetric)
𝑁 𝑥 𝑁 𝑦 𝑁 𝑥𝑦 𝑀 𝑥 𝑀 𝑦 𝑀 𝑥𝑦 = 𝐴 11 𝐴 𝐴 12 𝐴 𝐴 𝐷 11 𝐷 𝐷 12 𝐷 𝐷 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝛾 𝑥𝑦 𝑘 𝑥 𝑘 𝑦 𝑘 𝑥𝑦 .
extension
bending