Discontinuities　of　the　shear　force,　bending　moment,　slope,　and　deflection　can　be　caused　by　span　supports,　valves,　and　cracks　etc.　For　arbitrary　discontinuities　appearing　in　the　oil-conveying　pipe　resting　on　Pasternak　foundation,　the　general　analytical　solution　for　vibrations　of　this　structure　is　presented　in　this　paper.　To　deal　with　discontinuity　terms,　generalized　functions　are　employed　to　express　the　model　displacement　of　the　entire　pipe　as　a　single　function.　This　function　is　then　solved　by　Laplace　transform.　Continuity　conditions　at　discontinuity　points　are　also　given.　In　addition,　the　generalized　boundary　condition　(BC),　which　can　be　degenerated　to　many　classical　BCs,　is　considered.　Numerical　section　testifies　the　feasibility　and　the　validity　of　present　solutions,　and　discusses　the　effects　of　some　physical　parameters　on　natural　frequencies.　It　is　shown　that　the　closer　the　cracks　are　to　the　nearest　nodes　of　mode　shapes,　the　less　effects　they　have　on　the　corresponding　natural　frequencies;　locations　of　support　springs　change　effects　of　cracks　on　natural　frequencies　by　affecting　positions　of　structural　nodes;　foundation　parameters　have　different　effects　on　natural　frequencies.　The　present　method　can　be　applied　in　maintaining　safety　and　integrity　of　structures　of　pipes.　©　2021　Elsevier　Ltd