2.3 Bending deformation of isotropic layer –classical lamination theory Bending response of a single layer Assumption of linear variation is far from reality, but gives reasonable results. Kirchoff-Love plate theory corresponds to Euler Bernoulli beam theory.

Basic kinematics Normals to mid-plane remain normal Bending strains proportional to curvatures

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

Bending of symmetrically laminated layers

The power of distance from mid-plane

Bending-extension coupling of unsymmetrical laminates With unsymmetrical laminates, mid-plane is not neutral surface when only moment is applied. Conversely pure bending deformation require both force and moment.

B-matrix Force resultants needed to produce pure bending How can we see that is B zero for symmetrical laminate? Under both in-plane strains and curvatures

Under in-plane strains

Example 2.3.1

A Matrix A=0.2Q al +0.05Q br Checks: – Ratios of diagonal terms. – Ratios of diagonals to off diagonals. – Diagonal terms approximately average moduli times total thickness (+10% correction due to Poisson’s ratio)

B-Matrix

D-matrix For all-aluminum For all brass, 1.5 times larger. Calculated D Is it reasonable? Other checks?

Strains