The performance of a driving automation system (DAS) can influence the human drivers' trust in the system. This driving-simulator study examined how different types of DAS failures affected drivers' trust. The automation-failure type (no-failure, takeover-request, system-malfunction) was manipulated among 122 participants, when a critical hazard event occurred. The dependent measures included participants’ trust ratings after each of seven drives and their takeover performance following the hazard. Results showed that trust improved before any automation failure occurred, demonstrating proper trust calibration toward the errorless system. In the takeover-request and system-malfunction conditions, trust decreased similarly in response to the automation failures, although the takeover-request condition had better takeover performance. For the drives after the automation failure, trust was gradually repaired but did not recover to the original level. This study demonstrated how trust develops and responds to DAS failures, informing future research for trust repair interventions in designing DASs.