The structural behavior of a tensile fabric structure, known as hypar, is investigated. Seismic-induced stresses in the fabric and axial forces in masts and cables are obtained using accelerograms recorded at different regions of the world. Time-history analysis using each recording are performed for the hypar by using finite element simulation. It is found that while the seismic stresses in the fabric are not critical for design, the seismic tensile forces in cables and the seismic compressive forces in masts should not be disregarded by designers. This is important, because the seismic design is usually not considered so relevant, as compared for instance with wind design, for these types of structures. The most relevant findings of this study are: 1) dynamic axial forces can have an increase of up to twice the static loading when the TFS is subjected to seismic demands, 2) large peak ground accelerations seem to be the key parameter for significant seismic-induced axial forces, but not clear trend is found to relate such forces with earthquakes and site characteristics and, 3) the inclusion or exclusion of the form-finding in the analysis procedure importantly affects results of seismic stresses in the fabric, but not in the frame.