In this paper, Hamilton's principle is used to derive the dynamic equilibrium equations of beams of generic section. The displacements are de"ned on an arbitrarily selected co-ordinate system. For Hamilton's principle, the dynamic behavior of non-prismatic beams is characterized by two energy functions: a kinetic energy and a potential energy. The formulation uses the procedure of variational operations. The dynamic equilibrium equations and natural boundary conditions obtained are strongly coupled. 2000 Academic Press C.-N. CHEN components of A can be expressed as ;(x, y, z, t)"u(z, t)!y (z, t), <(x, y, z, t)"v(z, t)#x (z, t), =(x, y, z, t)"w(z, t)!x *u(z, t) *z !y *v(z, t) *z # * (z, t) *z (x, y), (1)
in which (x, y) is the warping function de"ned on the cross-section. The warping function can be de"ned by using Saint Venant's torsion theory. Displacement components ; and < consist of lateral displacements on zN axis and the relative lateral displacements due to the rotation of the beam. The axial displacement = is composed of the average axial displacement, axial displacement due to #exural deformation and the warping displacement due to the warping of the beam. Using equation ( 1), the strain components can be obtained:
The boundary conditions on boundaries z"0 and l are
) !EI V *u *z !EI W *v *z #EA *w *z #EI S * *z "P or w"0, (9e) !EI SV *u *z !EI SW *v *z #EI S *w *z #EI SS * *z "M M S or * *z "0, (9f ) DYNAMIC EQUILIBRIUMS OF NON-PRISMATIC BEAMS # I VV *u *t *z # I VW *v *t *z ! I V *w *t ! I N x * *t *z "< M V #m W or u"0, (13b) EI VW *u *z #EI WW *v *z !EI W *w *z !EI N y * *z "!M M V or *v *z "0, (13c)