We prove that the von Kármán model for vibrating beams can be obtained as a singular limit of a modified Mindlin-Timoshenko system when the modulus of elasticity in shear k tends to infinity, provided a regularizing term through a fourth-order dispersive operator is added. We also show that the energy of solutions for this modified Mindlin-Timoshenko system decays exponentially, uniformly with respect to the parameter k, when suitable damping terms are added. As k → ∞, one deduces the uniform exponential decay of the energy of the von Kármán model.