The paper is devoted to the dynamics of the model for a beam with strong damping ( P ε ) ε 2 u t t + ε δ u t + α u x x x x + u t x x x x − [ g ( ∫ 0 l u ξ 2 d ξ ) + ε σ ∫ 0 l u t ξ u ξ d ξ ] u x x = 0 , ( x , t ) ∈ ] 0 , l [ × ] 0 , ∞ [ , where g : R → R is continuously differentiable, δ , σ ∈ R and α , l , ε > 0 , subject to boundary conditions corresponding to hinged or clamped ends. We show that for ε → 0 + the dynamics of the equation is close to the dynamics of equation ( P 0) u t = − α u − g ( ∫ 0 l u ξ 2 d ξ ) A − 1 / 2 u , where A u : = u x x x x with the domain determined by one of the above boundary conditions. Specifically, we show that isolated invariant sets of ( P 0) continue to isolated invariant sets of ( P ε ) , ε > 0 small, having the same Conley index. Moreover, isolated Morse decompositions with respect to ( P 0) continue to isolated Morse decompositions of ( P ε ) , ε > 0 small, having isomorphic homology index braids. Under some additional assumptions we establish existence and upper semicontinuity results for attractors of ( P 0) and ( P ε ) , ε > 0 small, extending previous results by Ševčovič.