A STRAIN-DIFFERENCE BASED NONLOCAL ELASTICITY THEORY FOR SMALL-SCALE SHEAR-DEFORMABLE BEAMS WITH PARAMETRIC WARPING

The strain-difference based nonlocal elasticity theory devised by the authors is applied to homogeneous isotropic beams subjected to static loads. Shear deformation is taken into account and a warping parameter ω is used to fix the warping shape of the cross sections. On letting ω vary from zero to infinity, a continuous family of beam models is generated, which spans from the Euler-Bernoulli beam (ω = 0) to the Timoshenko beam (ω → ∞), and identifies itself with the Reddy beam for ω = 2. Taking as basic unknowns the axial stretching e, the Euler-Bernoulli curvature χEB, and the shear curvature η, the boundary-value problem proves to be governed by three uncoupled integral equations whose input terms contain, besides the load data, eight arbitrary constants. These equations are solved by addressing a set of eight uncoupled auxiliary integral equations independent of the boundary conditions, each of which is either a Fredholm integral equation of the second kind, or is more complex but has strong similarities with the latter type of equation. This makes it possible to express (e, χEB, η), the axial and transverse displacements (u, w), and the shear angle γ to within the mentioned constants, which is useful to enforce the eight boundary conditions. The numerical solutions for simple beam problems are reported and graphically illustrated with particular concern for size effects and for their sensitivity to shear deformation.