In order to investigate the effect of load variation on the quasi-zero-stiffness(QZS) vibration isolator, firstly, the statics of a QZS vibration isolator with piecewise nonlinear characteristic is analyzed. Then, the smoothing function is introduced here and the periodic solutions of the QZS vibration isolator are obtained using incremental harmonic balance(IHB) method. The effects of load and excitation amplitude on the dynamic characteristics are studied. Finally, the force transmissibility of the QZS vibration isolator is defined. And the effects of load and excitation amplitude on the isolation performance are investigated. The results show that the cross and folding phenomenon occurs in the constant term of the amplitude-frequency curves near the resonance region when the excitation amplitude is relatively large; The variations of load and excitation amplitude lead to the changes of resonance frequency, peak value of transmissibility and vibration isolation frequency band. However, the QZS vibration isolator still outperforms the corresponding linear counterpart.