As demonstrated by recent events in Italy, New-Zealand, Haiti, and Nepal, earthquakes continue to pose threats to civil infrastructure, including buildings. For a long time seismic ultimate limit states have not been considered in design codes for regions with a low to moderate seismicity (for instance, earthquake hazard has been explicitly introduced into Swiss building codes in 1989). Therefore, earthquakes in such regions are likely to result in important post-earthquake assessment needs. Current post-earthquake assessment methodologies heavily rely on expert-conducted visual inspections, which are potentially subjective and often slow. By implementing model-based data-interpretation methodologies, improvements with regard to both main shortcomings are possible: speeding up integrity assessment and providing an objective complement to visual inspection. The main objective is to improve accuracy and precision of residual-capacity predictions for damaged concrete and masonry buildings. A reduction of the time between an earthquake event and the clearance for occupancy of buildings increases the resilience of cities by limiting the needs for provisional housing and the loss of business opportunities. Ambient-vibration measurements are an interesting data source for structural identification because they are inexpensive and fast to acquire. However, ambient vibrations are characterized by low acceleration amplitudes. Thus, modal properties are representative of the linear and elastic range of structural behavior. True behavior under earthquake-like loading may differ from the behavior under ambient vibrations. Civil-engineering behavior models are approximate and safe and therefore, they provide biased representations. Even sophisticated high-fidelity models omit several parameters such as nonlinear soil-structure interactions, environmental factors, as well as non-structural elements (windows, separation walls, heavy furniture). Therefore, a data-interpretation technique that is accurate in a context of biased and correlated uncertainties is included in the framework. In addition, post-earthquake situations are characterized by uncertain and unforeseen circumstances and gradual knowledge acquisition. Thus, a data-interpretation framework that allows transparent implementation of gradual knowledge acquisition is proposed, based on error-domain model falsification. Using data recorded on full-scale concrete and masonry structures, it is concluded that ambient-vibration measurements are practicable for structural identification. However, highly nonlinear boundary conditions need to be considered in the estimation of discrepancies between model predictions and measurements. A main challenge related to nonlinear behavior predictions based on linearly elastic measurements relies on model extrapolation. Model extrapolation is complicated by modeling uncertainties. The uncertainties that arise in predicting residual load-bearing capacities are reviewed and integrated into this framework in a transparent manner and are applied to illustrative case studies. The proposed framework for data interpretation provides accurate predictions of nonlinear load-bearing behavior, even based on scarce linear measurement data and in the absence of measurements and information related to the initial building condition. In addition, simplified models can provide accurate results by employing conservative estimates for systematic modelling uncertainties.