Structural　reliability　analysis　without　the　exclusion　of　random　variables　with　unknown　distributions　has　attracted　increasing　attention,　and　many　endeavors　have　been　devoted　to　this　aspect.　Recently,　linear　moments　(L-moments)　have　been　found　increasing　use　for　characterizing　random　variables　with　unknown　distributions,　in　part　since　they　show　less　sensitivity　to　distribution　tails　and　are　more　stable　than　the　ordinary　central　moments　(C-moments).　In　this　paper,　the　clear　definition　of　the　complete　monotonic　expressions　of　the　third-order　polynomial　normal　transformation　(TPNT)　under　different　combinations　of　the　first　four　L-moments　of　random　variables　is　first　proposed.　Additionally,　the　applicable　boundaries　and　monotonic　regions　of　TPNT　based　on　L-moments　are　also　determined.　The　advantages　of　the　TPNT　based　on　L-moments　are　that　it　has　a　wider　applicable　range　and　is　much　more　stable　under　a　small　sample　size　compared　with　the　TPNT　based　on　C-moments,　which　are　demonstrated　through　numerical　example　studies.　Through　the　numerical　examples,　the　proposed　TPNT　based　on　the　first　four　L-moments　is　found　to　be　quite　effective　for　normal　transformation.