Calculation　of　probability　of　exceedance　for　nonstationary　non-Gaussian　responses　remains　a　great　challenge　to　researchers　in　the　field　of　structural　reliability.　In　this　paper,　an　analytical　solution　is　proposed　for　calculating　the　mean　upcrossing　rate　(MCR)　of　the　non-stationary　non-Gaussian　responses　by　approximating　the　displacement　and　velocity　responses　with　the　bivariate　vector　translation　process,　in　which　the　unified　Hermite　polynomial　model　(UHPM)　is　selected　as　the　mapping　function.　The　first　four　moments　(i.e.,　mean　value,　standard　deviation,　skewness,　and　kurtosis)　and　cross-correlation　function　of　the　displacement　and　velocity　responses　needed　in　UHPM　are　estimated　from　some　representative　samples　generated　by　random　function-spectral　representation　method　(RFSRM)　and　time-domain　analysis.　Under　the　Poisson　assumption　of　the　upcrossing　events,　the　calculation　of　extreme　value　distribution　or　probability　of　exceedance　for　structural　response　can　be　determined　with　the　proposed　method.　The　proposed　method　is　applicable　to　a　wide　range　of　structural　responses,　including　asymmetric　and　hardening　or　softening　responses.　Three　numerical　examples　are　provided　to　demonstrate　the　efficiency　and　accuracy　of　the　proposed　method.　It　can　be　concluded　that　the　proposed　method　provides　an　accurate　and　useful　tool　for　dynamic　reliability　assessment　in　engineering　applications.