Approximate Treatment of Transverse Shear Effects

The preceding subsection was devoted to a comparison of a special exact elasticity solution with classical lamination theory results. The importance of transverse shear effects was clearly demonstrated. However, that demonstration was for a special problem of rather narrow interest. The objective of this subsection is to display approaches and results for the approximate consideration of transverse shear effects for general laminated plates. 

The treatment of transverse shear stress effects in plates made of isotropic materials stems from the classical papers by Reissner [6-26] and Mindlin [6-27. Extension of Reissner s theory to plates made of orthotropic materials is due to Girkmann and Beer [6-28], Ambartsumyan [6-29] treated symmetrically laminated plates with orthotropic laminae having their principal material directions aligned with the plate axes. Whitney [6-30] extended Ambartsumyan s analysis to symmetrically laminated plates with orthotropic laminae of arbitrary orientation. 

The basic approaches as summarized by Ashton and Whitney [6-31] will now be discussed. First, a symmetric laminate with orthotropic laminae having principal material directions aligned with the plate axes will be treated. The transverse normal strain can be found from the orthotropic stress-strain relations, Equation (2.15), as 

The transverse shearing stress distribution is then approximated 

integration of the strain-displacement relation. Equation (2.2), with respect to z (with w assumed to be independent of z) yields 

---