  We consider a thin multi-domain of $\mathbb R^N$, with $N\geq 2$, consisting
of a vertical rod upon a horizontal disk. In this thin multi-domain, we
introduce a bulk energy density of the kind $W(D^2U)$, where $W$ is a
continuous function with linear growth at $\infty$ and $D^2U$ denotes the
Hessian tensor of a vector-valued function $U$ that represents a deformation of
the multi-domain. Considering suitable boundary conditions on the admissible
deformations and assuming that the two volumes tend to zero with same rate, we
prove that the limit model is well posed in the union of the limit domains,
with dimensions $1$ and $N-1$, respectively. Moreover, we show that the limit
problem is uncoupled if $N\geq 3$, and ``partially" coupled if $N=2$.
