  The article, presents the study of the regularity of two thermoelastic beam
systems defined by the Timoshenko beam model coupled with the heat conduction
of Green-Naghdiy theory of type III, both mathematical models are
differentiated by their coupling terms that arise as a consequence of the
constitutive laws initially considered. The systems presented in this work have
3 fractional dampings: $\mu_1(-\Delta)^\tau \phi_t$, $\mu_2(-\Delta)^\sigma
\psi_t$ and $K(-\Delta)^\xi \theta_t$, where $\phi,\psi$ and $\theta$ are
transverse displacement, rotation angle and empirical temperature of the bean
respectively and the parameters $(\tau,\sigma,\xi)\in [0,1]^3$. It is noted
that for values 0 and 1 of the parameter $\tau$, the so-called frictional or
viscous damping will be faced, respectively. The main contribution of this
article is to show that the corresponding semigroup
$S_i(t)=e^{\mathcal{B}_it}$, with $i=1,2$, is of Gevrey class
$s>\frac{r+1}{2r}$ for $r=\min \{\tau,\sigma,\xi\}$ for all
  $(\tau,\sigma,\xi )\in R_{CG}:= (0, 1)^3$. It is also showed that
$S_1(t)=e^{\mathcal{B}_1t}$ is analytic in the region $R_{A_1}:=\{(\tau,\sigma,
\xi )\in [\frac{1}{2},1]^3\}$ and $S_2(t)=e^{\mathcal{B}_2t}$ is analytic in
the region $R_{A_2}:=\{(\tau,\sigma, \xi )\in [\frac{1}{2},1]^3/ \tau=\xi\}$.
