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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.

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Dealing with discreteness

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  1. 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.

  2. 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?

  3. Continuous Example 4.2.1 • Graphite epoxy w • Design Laminate with Where on diagram?

  4. Different visualization • Fig. 4.1 (feasible domain)

  5. 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?

  6. 2.3 Bending deformation of isotropic layer –classical lamination theory • Bending response of a single layer • Bending stresses proportional to curvatures

  7. Hooke’s law • Moment resultants • D-matrix (EI per unit width)

  8. Bending of symmetrically laminated layers • As in in-plane case, we add contributions of all the layers. • We still get M=D, but

  9. 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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