In　this　paper,　a　new　method　for　normal　transformation　is　proposed　to　transform　correlated　non-normal　random　variables　into　independent　standard　normal　ones　based　on　their　first　four　linear　moments　(L-moments),　standard　deviations　and　correlation　matrix.　The　complete　monotonic　expressions　of　the　equivalent　correlation　coefficient　are　proposed　and　the　applicable　range　of　the　original　correlation　coefficient　to　ensure　the　transformation＇s　executability　is　identified.　Numerical　studies　demonstrate　that　the　admissible　range　of　the　normal　transformation　for　correlated　random　variables　based　on　L-moments　is　larger　in　scope　than　that　based　on　ordinary　central　moments　(C-moments),　and　the　proposed　method　is　effective　for　normal　transformations　and　sufficiently　accurate　for　reliability　engineering　practices.