t Baseline Over upper E Subscript normal upper T Baseline EndFraction plus StartFraction nu Subscript normal a Baseline nu Subscript normal upper A Baseline Over upper E Subscript normal upper A Baseline EndFraction right-parenthesis equals nu overTilde Subscript normal t Baseline comma"/>(2.224)
so that (2.223) may now be written as
Integration with respect to x3 then leads to
As the shear stresses σ13 and σ23 are everywhere zero, it follows from (2.143), (2.209) and (2.211) that
Integration then yields
These relations must be consistent with (2.226) so that
where the displacement component has been selected to be zero at the origin.
The through-thickness displacement of the top surface of the beam, at x3=0, can be defined in terms of two lengths, R1 and R2, which are the radii of curvature of this surface in the x1–x3 plane and the x2–x3 plane, respectively. The exact relationships are given by the well-known formulae
For small deflections
Thus, it follows from (2.229) that
providing a useful physical interpretation of the strain parameters ε^A and ε^T.
The final requirement is to determine the loading state that is consistent with the various strain parameter values. It is assumed that stresses within the beam can arise from an applied in-plane loading that is equivalent to an applied axial force FA and a transverse force FT acting in the mid-plane between the upper and lower surfaces of the beam, and an axial applied bending moment per unit area of cross section MA and a transverse applied bending moment per unit area of cross section MT. From mechanical equilibrium
