The model of single degree of freedom perfectly elastoplastic structures under external loading is treated in this paper. It is analyzed and pointed out that for the perfectly elastoplastic structure the equation of motion is nothing but a two-phase linear system with an on-off switch. Then the exact solutions of the dynamic responses are derived for sinusoidal loadings. For such loading we prove that no matter how large the amplitude is, the structure is impossible undergoing a permanent plastic motion; conversely, the long term behavior exhibits either stable hysteresis loop and limit cycle, or elastic shakedown. A phase plane estimation method of the steady response is developed, which includes not only the formulae for calculating the amplitude of displacement and two time lags, but also the identification of the parameters’ values for elastic shakedown and the maximum size of dissipation loop. The accuracy of the formulae is confirmed by comparing with the exact solutions. The mean displacement however depends on the initial conditions, a transient response, not being determined by the steady state estimation.

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