Shear deformation effects

• Classical plate theory (CPT), of which classical lamination theory (CLT) assume that there is no shear deformation.

• Strains vary linearly through the thickness and normal remain normal (Kirchoff-Love assumptions, 1888).

• Gustav Kirchoff (1824-1887, German),Augustus Love (1863-1940, British)

• There are shear deformation theories that remove the second assumption, and theories that remove both.

• We will go over the Timoshenko beam theory that removes the second assumption for beams.

• Then we will look at some results for plates.

Timoshenko beam theory (Wikipedia)

• Stephen Timoshenko (1878-1972, Ukraine, US) proposed in 1921.

• Strain is still linear function of z, but normal do not remain normal

Basic equations

• Displacements • Governing equations

• Combined

2

2( )

1

d dEI q x

dx dx

dw d dEI

dx kAG dx dx

4 2

4 2( )

d w EI d qEI q xdx kAG dx

Beam under end load P

• Tip displacement• Ratio of shear to bending deformation

• k is 1 for an ideal I beam, 5/6 for rectangular section.• For metals, 3E/kG is close to 10, so shear

deformation is negligible except for stubby beams with radius of inertia over L less than 10.

• For composites G is much smaller, hence shear deformation more important

3

3tip

PL PLw

EI kGA

23

/shear

bending

w EI A

w kG L

Representative results

• Source: Whitney’s Structural Analysis of laminated Anisotropic Plates, Chapter 10, Technomic, 1987.

• Material characteristics • Weaker in shear than the materials we have

used.

Displacements

Buckling

Frequency

Shear stresses

• Shear loading leads to shear stresses, which are important for delamination failure.

• Shear stresses can be approximated as they are done in beam theory.

• Good paper: Simplified shear solution for determination of shear stress distribution in a composite panel from the applied shear resultant, by Bednarcyk, Aboudi, Yarrington and Collier (see link in schedule).