This paper investigates the existence of strong global and exponential attractors and their robustness on the perturbed parameter for an extensible beam equation with nonlocal energy damping in Omega subset of R-N: u(tt) + Delta(2)u-kappa phi(parallel to del u parallel to(2))Delta u - M(parallel to Delta u parallel to(2) + parallel to u(t)parallel to(2)) Delta u(t) + f(u) = h, where kappa epsilon Lambda (index set) is an extensibility parameter, and where the "strong" means that the compactness, the attractiveness and the finiteness of the fractal dimension of the attractors are all in the topology of the stronger space H-2 where the attractors lie in. Under the assumptions that either the nonlinearity f(u) is of optimal subcritical growth or even f(u) is a true source term, we show that (i) the semi-flow originating from any point in the natural energy space 7t lies in the stronger strong solution space H-2 when t > 0; (ii) the related solution semi group S-kappa(t) has a strong (H, H-2)-global attractor A(kappa) for each kappa and the family of A(kappa), kappa epsilon Lambda is upper semicontinuous on kappa in the topology of stronger space H-2; (iii) S-kappa(t) has a strong (H, H-2)-exponential attractor A(exp)(kappa) for each kappa and it is Holder continuous on kappa in the topology of H-2. These results break through long-standing existed restriction for the attractors of the extensible beam models in energy space and show the optimal topology properties of them in the stronger phase space.