Let　F　be　a　family　of　digraphs.　A　digraph　D　is　F-free　if　it　has　no　isomorphic　copy　of　any　member　of　F.　The　Tur　&　aacute;n　number　ex(n,F)　is　the　largest　number　of　arcs　of　F-free　digraphs　on　n　vertices.　Bermond,　Germa,　Heydemann　and　Sotteau　in　1980　[Girth　in　digraphs,　J.　Graph　Theory,　4　(1980),　337-341]　determined　the　Tur　&　aacute;n　number　of　C-k-free　strong　digraphs　on　n　vertices　for　k　＞=　2,　where　C-k　=　{C-2,C-3,...　,　C-k}　and　Ci　is　a　directed　cycle　of　length　i　is　an　element　of　{2,　3,　...　,　k}.　In　this　paper,　we　determine　all　Tur　&　aacute;n　number　of　strong　digraphs　without　t　＞=　2　triangles,　extending　the　previous　result　for　the　case　k　=　3.