  In this work we introduce novel stress-only formulations of linear elasticity
with special attention to their approximate solution using weighted residual
methods. We present four sets of boundary value problems for a pure stress
formulation of three-dimensional solids, and in two dimensions for plane stress
and plane strain. The associated governing equations are derived by
modifications and combinations of the Beltrami-Michell equations and the
Navier-Cauchy equations. The corresponding variational forms of dimension $d
\in \{2,3\}$ allow to approximate the stress tensor directly, without any
presupposed potential stress functions, and are shown to be well-posed in $H^1
\otimes \mathrm{Sym}(d)$ in the framework of functional analysis via the
Lax-Milgram theorem, making their finite element implementation using
$C^0$-continuous elements straightforward. Further, in the finite element
setting we provide a treatment for constant and piece-wise constant body forces
via distributions. The operators and differential identities in this work are
provided in modern tensor notation and rely on exact sequences, making the
resulting equations and differential relations directly comprehensible.
Finally, numerical benchmarks for convergence as well as spectral analysis are
used to test the limits and identify viable use-cases of the equations.
