  We have systematically studied the nematic susceptibility in
non-superconducting Ba(Fe$_{1-x}$TM$_{x}$)$_2$As$_2$ (TM = Cr, Mn, V and Cu) by
measuring the uniaxial pressure dependence of the resistivity along the
Fe-As-Fe direction. The nematic susceptibilities in all samples show the
Curie-Weiss-like behavior at high temperatures, where the nematic Curie
constant $A_n$ can be derived, similar to the Curie constant in a
paramagnetism. While all these dopants do not introduce superconductivity in
BaFe$_2$As$_2$, their effects on nematic fluctuations are different. In Mn, Cr
and V doped samples, $|A_n|$ decreases significantly with the increasing doping
level. On the other hand, $|A_n|$ increases dramatically with Cu doping,
similar to the superconducting Ni-doped BaFe$_2$As$_2$. However, the nematic
susceptibility is suppressed at low temperatures for $x$ larger than $0.04$,
which may be related to the short-range antiferromagnetic order that survives
up to very high doping level. Doping Mn, Cr and Cu into the optimally-doped
superconducting BaFe$_2$(As$_{0.69}$P$_{0.31}$)$_2$ also strongly reduces
$|A_n|$. Compared with those systems that clearly exhibit superconductivity,
such as Ni, K or P doped samples, our results suggest a strong connection
between the nematic and spin degrees of freedom. Moreover, the reason of the
suppression of superconductivity by dopants such as Cr, Mn, V and Cu may be
correlated with the suppression of nematic fluctuations.
