The geometrically nonlinear vibration of 3D beams with different end conditions (such as fixed, free and pinned etc.,) is studied. Rectangular cross sectional beams with different materials are considered for the analysis. The beams, which experience bending, torsional and longitudinal deformations in space, are studied using H- Version, P-Version and HP- Version finite element (FE) method. Among the available beam models, such as Timoshenko's theory for flexure[1] that assumes that the entire cross section of the beam rotates as a rigid body under torsion and it is warping free in the longitudinal direction and Bernoulli - Euler theory [2-5] for bending are considered in the present study. Both models assume St.Venant's theory for torsion.
The geometrical nonlinearity is considered based on Green's strain tensor [6]. The equations of motion for beams with different end conditions are derived by using principle of virtual work. General Hook's law is used. Bending linear natural frequencies of beams with different materials and different end conditions are calculated using different version of FE Model. The significance of warping is evaluated for different cross sectional dimensions of the rectangular beams [7]. Nonlinear static analysis is performed for different loads such as point loads and moment with different shape functions. Twist angle for different dimensions of rectangular beams is calculated for moment with and without including the linearization of the trigonometric terms in strain - displacement relations. A comparative study is performed by including and neglecting the higher order terms that appears in the direct strain equation for two different beams models. The consequences of quadratic terms of the on longitudinal displacement are studied. For the harmonic excitation, the dynamic response of the beam is calculated by considering the bending torsion coupling.

Keywords: Nonlinear vibration, 3D beam, Warping, HP method.