Transcript Shear deformation effects
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 𝑢 = − 𝑧𝜑 𝑥 , 𝑤 = 𝑤 𝑥 , 𝑣 = 0 Governing equations
d
2
dx
2
EI d
dx
dw dx
Combined 1
d kAG dx EI d
dx EI
4
d w
dx
4 2
EI d q kAG dx
2
• • • • •
Beam under end load P
Tip displacement
w tip
PL
3 3
EI
PL kGA
Ratio of shear to bending deformation 2
w shear
3
E
w bending
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
Representative results
• • • Source: Whitney’s Structural Analysis of laminated Anisotropic Plates, Chapter 10, Technomic, 1987.
Material characteristics 25, 𝐺 𝐿𝑇 𝐸 𝑇 = 0.5, 𝐺 𝑇𝑇 𝐸 𝑇 = 0.2, 𝜈 𝐿𝑇 = 0.25
𝐸 𝐿 𝐸 𝑇 = 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).