2.3 Bending deformation of isotropic layer classical lamination theory

2.3 Bending deformation of isotropic layer classical lamination theoryBending 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 kinematicsNormals to mid-plane remain normal

Bending strains proportional to curvatures

Hookes lawMoment resultants

D-matrix (EI per unit width)

Bending of symmetrically laminated layers

The power of distance from mid-plane

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

B-matrixForce 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 MatrixA=0.2Qal+0.05Qbr

Checks:Ratios of diagonal terms.Ratios of diagonals to off diagonals.Diagonal terms approximately average moduli times total thickness (+10% correction due to Poissons ratio)

B-Matrix

11D-matrixFor all-aluminumFor all brass, 1.5 times larger.Calculated D

Is it reasonable? Other checks?

Strains