Let us consider a Kirchhoff axisymmetric annular plate with internal and external radii \(\,R_i\,\) and \(\,R_e\,\), respectively, and a uniform thickness \(\,h\,\). A cylindrical coordinate system \(\,r, \theta , z\,\) is conveniently introduced and the two-dimensional domain of plate cross-section in the \(\,r\,\theta \,\)-plane is defined by \(\,\Omega =[\,R_i,\, R_e] \times [0,\, 2 \pi ]\,\). Boundary of domain is denoted by \(\,\partial \Omega _k\) for \(\,r = R_k\,\) with \(\,k = \{i, e\}\,\). The base vectors in the \(\,r\,\theta \,\) coordinate plane are radial \(\,\mathbf{e} _r\,\) and circumferential \(\,\mathbf{e} _\theta \,\) unit vectors.
According to linearized Kirchhoff theory, the geometric curvature tensor is defined as
\(\,\varvec{\chi }:=\nabla \nabla u\,\), being
\(\,u:[\,R_i,\, R_e] \rightarrow \Re \,\) the transverse displacement field and
\(\,\nabla \,\) the gradient operator. Denoting by
\(\,\partial _r\,\) derivative along the radial axis and by
\(\,\otimes \,\) tensor product, the curvature can be explicitly expressed as follows:
$$\begin{aligned} \varvec{\chi }= \partial _r^2 u\,\mathbf{e} _r\otimes \mathbf{e} _r+ \frac{\partial _ru}{r}\,\mathbf{e} _\theta \otimes \mathbf{e} _\theta , \end{aligned}$$
(1)
where the eigenvalues
\(\,\partial _r^2 u\,\) and
\(\,\displaystyle \frac{\partial _ru}{r}\,\) are radial
\(\,\chi _r\,\) and circumferential
\(\,\chi _{\theta }\,\) bending curvatures, respectively. By duality, stress tensor field is the bending interaction
\(\,\mathbf{M } = M_r\,\mathbf{e} _r\otimes \mathbf{e} _r+ M_\theta \,\mathbf{e} _\theta \otimes \mathbf{e} _\theta \,\), being
\(\,M_r\,\) and
\(\,M_\theta \,\) the radial and circumferential moments, respectively. Denoting by
\(\,:\,\) the scalar product, equilibrium is expressed by the variational condition that the external virtual power of the loading system is equal to the internal virtual power
$$\begin{aligned} \int _{\Omega }\,\mathbf{M }:\varvec{\chi }_{\delta u}\, \mathrm{d}A, \end{aligned}$$
(2)
for all virtual displacement fields
\(\,\delta u\,\) fulfilling homogeneous kinematic boundary conditions, being
\(\,\varvec{\chi }_{\delta u}\,\) the bending curvature kinematically compatible with
\(\,\delta u\,\). Integration by parts and localization procedures lead to the following differential equation of equilibrium:
$$\begin{aligned} \frac{1}{r}\bigg {(}\partial _r^2(M_r\,r)-\partial _rM_\theta \bigg {)} = q, \quad \, r \,\in \Omega , \end{aligned}$$
(3)
equipped with boundary conditions
$$\begin{aligned} \left\{ \begin{aligned}&M_r\,\partial _r\delta u = -\,{\bar{M}}_i \, \partial _r\delta u, \quad r \in \partial \Omega _i\\&M_r\,\partial _r\delta u = \,{\bar{M}}_e \, \partial _r\delta u, \quad r \in \partial \Omega _e\\&\bigg {(}M_\theta - \partial _r(M_r\,r)\bigg {)}\, \delta u = - {\bar{Q}}_i \,r\,\delta u, \quad r \in \partial \Omega _i\\&\bigg {(}M_\theta - \partial _r(M_r\,r)\bigg {)}\, \delta u = {\bar{Q}}_e \,r\,\delta u, \quad r \in \partial \Omega _e, \end{aligned} \right. \end{aligned}$$
(4)
with
\(\,q\,\) transverse distributed loading,
\(\,\{{\bar{M}}_i, \,\bar{M}_e\}\,\) distributed edge bending couples and
\(\{\,{\bar{Q}}_i,\, \bar{Q}_e\}\,\) distributed edge transverse forces. From Eq. (
4), it can be observed that the shear force is consequently defined as
\(\,Q_r := \displaystyle \frac{M_\theta - \partial _r(M_r\,r)}{r}\,\).
Finally, it is worth noting that circular plates can be derived as a particular case of annular plates free on internal boundary for vanishing internal radius \(\,R_i\rightarrow 0\,\).