Dealing with discreteness
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Dealing with discreteness. Laminate thickness must be integer multiple of basic ply thickness. Ply orientations often need to be selected from a small set of angles, e.g. In terms of optimization algorithms we transition from algorithms that use derivatives to algorithms that do not.

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Dealing with discreteness

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Dealing with discreteness

Dealing with discreteness

  • Laminate thickness must be integer multiple of basic ply thickness.

  • Ply orientations often need to be selected from a small set of angles, e.g.

  • In terms of optimization algorithms we transition from algorithms that use derivatives to algorithms that do not.

  • Integer programming is usually NP hard.


Miki s diagram for

Miki’s diagram for

  • Finite number of points and excluded regions

  • Which points do we lose with balance condition?

  • Diagram is for 8-ply laminate. What will change and what will remain the same for 12 plies?


Continuous example 4 2 1

Continuous Example 4.2.1

  • Graphite epoxy w

  • Design Laminate with

Where on diagram?


Different visualization

Different visualization

  • Fig. 4.1 (feasible domain)


Example 4 3 1

Example 4.3.1

  • Solve 4.2.1 for 16-ply balanced symmetric laminate of plies.

  • What is common for the first five designs besides the shear modulus?


2 3 bending deformation of isotropic layer classical lamination theory

2.3 Bending deformation of isotropic layer –classical lamination theory

  • Bending response of a single layer

  • Bending stresses proportional to curvatures


Hooke s law

Hooke’s law

  • Moment resultants

  • D-matrix (EI per unit width)


Bending of symmetrically laminated layers

Bending of symmetrically laminated layers

  • As in in-plane case, we add contributions of all the layers.

  • We still get M=D, but


The power of distance from mid plane

The power of distance from mid-plane

  • In Example 2.21 we had a laminate made of brass and aluminum

  • For in-plane loads laminate was twice as close to aluminum than brass.

  • For bending, brass contribution proportional to . Aluminum contribution


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