  An extra-stabilised Morley finite element method (FEM) directly computes
guaranteed lower eigenvalue bounds with optimal a priori convergence rates for
the bi-Laplace Dirichlet eigenvalues. The smallness assumption
$\min\{\lambda_h,\lambda\}h_{\max}^{4}$ $\le 184.9570$ in $2$D (resp. $\le
21.2912$ in $3$D) on the maximal mesh-size $h_{\max}$ makes the computed $k$-th
discrete eigenvalue $\lambda_h\le \lambda$ a lower eigenvalue bound for the
$k$-th Dirichlet eigenvalue $\lambda$. This holds for multiple and clusters of
eigenvalues and serves for the localisation of the bi-Laplacian Dirichlet
eigenvalues in particular for coarse meshes. The analysis requires
interpolation error estimates for the Morley FEM with explicit constants in any
space dimension $n\ge 2$, which are of independent interest. The convergence
analysis in $3$D follows the Babu\v{s}ka-Osborn theory and relies on a
companion operator for the Morley finite element method. This is based on the
Worsey-Farin $3$D version of the Hsieh-Clough-Tocher macro element with a
careful selection of center points in a further decomposition of each
tetrahedron into $12$ sub-tetrahedra. Numerical experiments in $2$D support the
optimal convergence rates of the extra-stabilised Morley FEM and suggest an
adaptive algorithm with optimal empirical convergence rates.
