By　Laplace　transform,　the　interaction　between　a　dynamically　loaded　fix　pile-tip　and　a　soil　layer　is　theoretically　investigated　first.　The　displacement　and　velocity　response　at　pile　head　in　frequency　domain　and　the　dynamic　stiffness　of　a　pile　are　obtained　in　a　closed　form.　A　semi-analytical　solution　for　the　velocity　response　of　a　pile　subjected　to　a　semi-sine　wave　exciting　torque　is　also　achieved.　The　comparison　between　theories　based　on　plane　strain　assumption　and　solutions　developed　in　this　paper　is　made,　and　some　important　conclusions　are　drawn.