The author studies a variational inequality connected with a beam of the form

where $\epsilon ,r_{\epsilon },t_{\epsilon }$ are positive parameters such that $\frac{\epsilon r_{\epsilon }}{t_{\epsilon }}\to 0$. The variational inequality has the form: Find $U_{\epsilon }\in M_{\epsilon }$ such that

with $A_{\epsilon },B_{\epsilon },{\text{Φ}}_{\epsilon }$ given functions and $M_{\epsilon }$ a closed, convex, nonempty cone in ${𝒱}_{\epsilon }=\{V\in H^1\left({\text{Ω}}_{\epsilon }\right)$, $V=0$ on ${\text{Γ}}_{\epsilon }\}$, where ${\text{Γ}}_{\epsilon }={\text{Γ}}_{\epsilon }^{-}\cup {\text{Γ}}_{\epsilon }^{+}$ and ${\text{Γ}}_{\epsilon }^{-}=\{-1\}\times \left(\epsilon S\right),{\text{Γ}}_{\epsilon }^{+}=\left\{1\right\}\times \left(\epsilon S\right)$ are two bases of the beam. The limit variational inequality is obtained.