We study the asymptotic behavior of a three-dimensional elastic material reinforced with highly contrasted thin vertical strips constructed on horizontal iterated Sierpinski gasket curves. We use $\mathrm{\Gamma }$-convergence methods in order to study the asymptotic behavior of the composite as the thickness of the strips vanishes, their Lamé constants tend to infinity, and the sequence of the iterated curves converges to the Sierpinski gasket in the Hausdorff metric. We derive the effective energy of the composite. This energy contains new degrees of freedom implying a nonlocal effect associated with thin boundary layer phenomena taking place near the fractal strips and a singular energy term supported on the Sierpinski gasket.