Shear deformation effects

1. 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.

2. 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

3. Basic equations • Displacements • Governing equations • Combined

4. 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/kGis 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

5. 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.

6. Displacements

7. Buckling

8. Frequency

9. 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).