Let
$u(\varepsilon)$
be the rescaled three‐dimensional displacement field solution of the linear elastic model for a clamped prismatic straight rod
${\varOmega}^\varepsilon$
having cross section with diameter of order
$\varepsilon$
, and let
$u^0$
be the corresponding Bernoulli–Navier displacement. In this article we establish that the error
$\|u(\varepsilon)-u^0\|_{1,{\varOmega}}$
in the reference space
$[H^1({\varOmega})]^3$
is of order
$\varepsilon^{1/2}$
. We mainly use an auxiliar corrector function and we prove that this estimation cannot be improved using other corrector functions of the same family.
