Stability and natural frequencies of axial-loaded shear-deformable beams using an improved two-node finite element

The linear flexural stiffness, incremental stiffness, mass, and consistent force matrices for a simple two-node Tirnoshenko beam element are developed based upon Hamilton's principle, where interdependent cubic and quadratic polynomials are used for the transverse and rotational displacements, respectively. The resulting linear flexural stiffness matrix is in agreement with the exact 2-node Timoshenko beam stiffness matrix. Numerical results are presented to show that the current element can accurately predict the buckling load and natural frequencies of axially-loaded isotropic and composite beams for a wide variety of beam-lengths and boundary conditions. The current element consistently outperforms the existing finite element approaches in studies involving the static, buckling, and vibration behavior of axially-loaded short beams. Axially-loaded beam-type structural components can be found in almost all aerospace, civil, and mechanical systems. The presence of a tensile axial load is well known to increase all of the bending natural frequencies of the beam member, whereas a compressive axial load will decrease all of the bending natural frequencies. The magnitude of the compressive axial load can only be increased up to the point where the first bending frequency goes to zero, then the motion of the beam member becomes unstable (i.e. buckling occurs). This vibration and stability behavior has been studied analytically using a Bernoulli-Euler theory for long slender uniform beam columns [1,2] and using the Timoshenko beam theory [3,4] for short uniform beam columns 151, where the latter approach can be used to study beams composed of advanced composite materials because the effects of shear deformation and rotatory inertia are included. For complex structural systems (i.e. aircraft fuselages with stringers and skins), the above simple analytical solutions are generally not applicable and thus one uses the finite element method to study the vibration and stability behavior. Most researchers and almost all developers of commercial finite element computer programs (MSCINASTRAN, ANSYS, etc) use a 2-node (4 DOF) finite element. For long slender beams, the linear flexural stiffness, the mass, and the incremental t Associate Professor, member AlAA (or geometric) stiffness matrices are developed in a consistent manner using a cubic Hemitian polynomial for the transverse displacement and the Bernoulli-Euler form of the beam strain energy, the kinetic energy, and the potential energy associated with the axial loads, respectively. For short thick or composite beams, a different approach is used. The linear flexural stiffness matrix is developed using force equilibrium relations (not a displacement formulation) so that the resulting matrix form is in agreement with the exact 2-node Tirnoshenko stiffness matrix developed by Przemieniecki [6j. But the mass matrix and incremental stiffness matrix are not altered. Accuracy and convergence problems are known to occur when one attempts to study the vibration and/or stability of axially-loaded short or composite beams using finite elements because of the inconsistencies in the development of the flexural stiffness matrix (force equilibrium) with the mass and incremental stiffness matrices (displacement formulation). Recently, Friedman and Kosmatka [q developed a set of interdependent cubic and quadratic shape functions for the transverse and rotational displacements of the beam, respectively, which exactly satisfy the homogeneous differential equations associated with Timoshenko's beam theory. The stiffness, mass, and consistent force matrices for a two-node beam element with (4 DOF) were developed based upon Hamilton's principle, where the resulting linear flexural stiffness matrix is in agreement with the aforementioned exact 2node Timoshenko stiffness matrix developed by Przemieniecki [6]. Published results showed that the element exactly predicts the transverse displacements of short beams subjected to complex distributed badings using only one element, and predicts shear and moment resultants and natural frequencies better than any existing 2-node Timoshenko beam-type finite element. In the current paper, these interdependent shape functions are used to develop an incremental stiffness matrix, which is fully compatible with the exact linear flexural stiffness matrix and thus accuracy and convergence problems which occur in stability and vibration calculations of axially-loaded short or composite beams can be eliminated. Numerical results are presented to show that the current element can accurately predict the buckling load and natural frequencies of axially-loaded isotropic and composite beams for a variety of beam-lengths and boundary conditions. Review of Timoshenko Beam EauCopyright 1994 by J. B. Kosmatka. Published by the We begin by considering a prismatic isotropic American Institute of Aeronautics and Astronautics, Inc. with beam, of length L, having a general homogeneous permission. cross-section of area A. A Cartesian coordinate system 19 3 (x,y,z) is defined on the beam where the x axis is coincident with the centroidal axis and y and z are coincident with the principal axes of the root crosssection. It is assumed that the centroidal axis is coincident with the elastic axis so that bending-torsion coupling is negligible. Limiting our discussion to transverse displacements in the x-y plane only, the kinematic displacement distributions can be written as: where, v is the time-dependent transverse displacement of the centroidal axis and 8 is the time-dependent rotation of the cross-section about the positive z-axis. See Fig. 1. The nonzero strain components of the beam are determined using Eqns. (1 .a-c) as The equations of motion are derived via Hamilton's principle where SU, SVA, ST, and 6We are the variations of the strain energy, the potential energy associated with initial stress, the kinetic energy, and the work of external forces, respectively. The strain energy is given as and can be rewritten, by making use of Eqns. (2.a,b), the beam material constitutive relations, and integrating over the cross-section, as where El and kGA are flexural rigidity and the shear rigidity, respectively, with k being a shear coefficient that is dependent upon the material definition and crosssection geometry [8]. The potential energy of the beam associated with an initial axial tension load (P) is given as The kinetic energy of the beam is given as: and is rewritten using Eqns. (2.a.b) and integrating over the cross-section, as where p is the mass density. Finally, the work of external forces is given as where q and rn are the distributed forces and moments along the length of the beam. The two differential equations of motion and associated boundary conditions are obtained by Substituting Eqns. (4.b), (4.c), (5.b), and (6) into (3) and integrating by parts: where the two geometric and natural boundary conditions that must be specified at the beam ends (x=O,L) include