In　this　article,　an　analytical　moment-based　procedure　is　developed　for　estimating　the　first　passage　probability　of　stationary　non-Gaussian　structural　responses　for　practical　applications.　In　the　procedure,　an　improved　explicit　third-order　polynomial　transformation　(fourth-moment　Gaussian　transformation)　is　proposed,　and　the　coefficients　of　the　third-order　polynomial　transformation　are　first　determined　by　the　first　four　moments　(i.e.　mean,　standard　deviation,　skewness,　and　kurtosis)　of　the　structural　response.　The　inverse　transformation　(the　equivalent　Gaussian　fractile)　of　the　third-order　polynomial　transformation　is　then　used　to　map　the　marginal　distributions　of　a　non-Gaussian　response　into　the　standard　Gaussian　distributions.　Finally,　the　first　passage　probabilities　can　be　calculated　with　the　consideration　of　the　effects　of　clumping　crossings　and　initial　conditions.　The　accuracy　and　efficiency　of　the　proposed　transformation　are　demonstrated　through　several　numerical　examples　for　both　the　＂softening＂　responses　(with　wider　tails　than　Gaussian　distribution;　for　example,　kurtosis　＞　3)　and　＂hardening＂　responses　(with　narrower　tails;　for　example,　kurtosis　＜　3).　It　is　found　that　the　proposed　method　has　better　accuracy　for　estimating　the　first　passage　probabilities　than　the　existing　methods,　which　provides　an　efficient　and　rational　tool　for　the　first　passage　probability　assessment　of　stationary　non-Gaussian　process.