We study the asymptotic behavior of the solution of an anisotropic, heterogeneous, linearized elasticity problem in a cylinder whose diameter $\varepsilon$ tends to zero. The cylinder is assumed to be fixed (homogeneous Dirichlet boundary condition) on the whole of one of its extremities, but only on a small part (of size $\varepsilon r^\varepsilon$) of the second one; the Neumann boundary condition is assumed on the remainder of the boundary. We show that the result depends on $r^\varepsilon$, and that there are 3 critical sizes, namely $r^\varepsilon=\varepsilon^3$, $r^\varepsilon=\varepsilon$, and $r^\varepsilon=\varepsilon^{1/3}$, and in total 7 different regimes. We also prove a corrector result for each behavior of $r^\varepsilon$.