  Bresse-Timoshenko beam model with thermal, mass diffusion and theormoelastic
effects is studied. We state and prove the well-posedness of problem. The
global existence and uniqueness of the solution is proved by using the
classical Faedo-Galerkin approximations along with two a priori estimates. We
prove an exponential stability estimate for problem under an unusual
assumption, and by using a multiplier technique in two different cases, with
frictional damping in the angular rotation and with frictional damping in the
vertical displacement. In numerical parts, we first obtained a numerical scheme
for problem by $P_1$-finite element method for space discretization and
implicit Euler scheme for time discretization. Then, we showed that the
discrete energy decays, later a priori error estimates are established. Finally
, some numerical simulations are presented.
