shear deformation effects classical plate theory (cpt), of which classical lamination theory (clt)...
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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).
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