Within the framework of performance‐based earthquake engineering, the intensity‐measure approach (IMA) has become the standard option for the characterization of engineering‐demand parameters (EDPs) for systems undergoing significant inelastic behavior. Within this approach, the rates of exceedance of the EDPs are computed from a hazard curve corresponding to an intensity measure (IM; usually the spectral acceleration at the first mode period SA(T1)) and the conditional probability density function (CPDF) of the EDP given the IM. In view of the lack of strong ground motion records associated with large values of currently used IMs, the parameters required to establish a CPDF are obtained from an incremental dynamic analysis that considers the SA(T1) linear scaling of motions recorded during seismic events of moderate intensity. However, from a seismological perspective, the linearly scaling method is too simple and may lead to unrealistic ground‐motion records that may affect the accuracy of the IMA. The dynamic response of single‐degree‐of‐freedom systems subjected to simulated ground motions is analyzed in order to assess the limitations of the SA(T1) linearly scaling method. These simulated ground motions were obtained via a stochastic simulation technique that has a solid seismological basis. The results presented herein are useful to numerically understand the limitations of the SA(T1) linearly scaling method and to identify situations in which a more sophisticated analysis is warranted.