Taking into account the tensile forces (Fig. 2.3) and considering the equilibrium of a segment of dimension dr, rdQ and thickness t, the sum of the projection of theses forces onto the radial direction, after division by rdrdQ, is
After calculation of the Nr derivative with respect to z, u, and t, substitution in (2.27) and division by Etr/(1 - v2), we obtain the first equation of equilibrium d2u dr2
The second equation is given by the equilibrium of the bending moments. This is (2.13) in which Qr stands for the total shearing force Q* that takes also into account r z z
the axial component of the radial force Nr in the middle surface. The total shearing force Q* is expressed by
where the shearing force Qr is defined by the external loading cases in (2.15), (2.18), or (2.20). From the expression of Nr in (2.26a), we obtain
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