Probability　distributions　of　basic　random　variables　are　essential　for　the　accurate　evaluation　of　structural　reliability.　In　engineering　practice,　the　probability　distributions　of　some　random　variables　are　often　unknown　and　the　only　available　information　about　these　may　be　their　statistical　moments.　To　conduct　structural　reliability　analysis　without　the　exclusion　of　random　variables　with　unknown　probability　distributions,　the　fourth-moment　normal　transformation　(FMNT)　has　been　proposed.　However,　the　applicability　of　expression　of　the　FMNT　has　not　been　sufficiently　investigated.　Furthermore,　the　monotonic　regions　of　the　FMNT　are　not　defined　without　which　the　application　of　the　transformation　is　inconvenient,　or　even　unreliable　in　reliability　analysis.　In　the　present　paper,　a　complete　expression　of　the　FMNT　including　six　cases　with　different　combinations　of　skewness　and　kurtosis　is　derived,　and　the　monotonicity　of　each　case　of　the　FMNT　expression　is　confirmed.　Literature　suggests　that　the　complete　monotonic　expression　of　the　fourth-moment　normal　transformation　is　the　first　time　to　be　successfully　accomplished　up　to　date.　Through　the　numerical　examples,　the　FMNT　is　found　to　be　quite　efficient　for　normal　transformation　and　to　be　sufficiently　accurate　to　include　random　variables　with　unknown　probability　distributions　in　structural　reliability　analysis.　(C)　2017　Elsevier　Ltd.　All　rights　reserved.