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
where σA and σT are the effective axial and transverse applied stresses. On substituting (2.220) and (2.221) into (2.233), the following effective axial and transverse stresses are obtained
The relations (2.234) are now expressed in the form
On substituting (2.220) and (2.221) into (2.237) the following relations, enabling the determination of the effective axial and transverse bending moments per unit area of cross section, are obtained