Coupled bending-torsional dynamic stiffness matrix of an axially loaded timoshenko beam element

Division of Structural Engineering, School of Engineering, University of Wales College of Cardiff, Newport Road, Cardiff CF2 1YF, U.K.
International Journal of Solids and Structures (Impact Factor: 2.04). 03/1994; 31(6):749-762. DOI: 10.1016/0020-7683(94)90075-2

ABSTRACT Analytical expressions for the coupled bending-torsional dynamic stiffness matrix terms of an axially loaded uniform Timoshenko beam element are derived in an exact sense by solving the governing differential equations of motion of the element. The symbolic computing package REDUCE has been used to generate an analytical expression for each of the dynamic stiffness terms in a concise form. For check purposes, numerical values of the dynamic stiffness matrix terms were obtained using the derived explicit expressions as well as by an alternative nonanalytical method based on matrix inversions and matrix multiplications. Stiffnesses obtained from both methods agreed with each other to machine accuracy. Application of the developed theory is discussed with particular reference to an established algorithm. The influence of axial force, shear deformation and rotatory inertia on the natural frequencies of a bending-torsion coupled beam with cantilever end-conditions is demonstrated by numerical results. Such results are not generally available in the literature. Therefore, results obtained by partially restricting the present theory are compared with the existing literature wherever possible. The results indicate that the method is accurate and efficient.

  • [Show abstract] [Hide abstract]
    ABSTRACT: A simple but efficient method to obtain the exact static stiffness matrices is developed in order to perform the spatially coupled elastic and buckling analyses of shear deformable uniform beam-columns having non-symmetric thin-walled sections. First this numerical technique is accomplished via a generalized eigenvalue problem associated with 14 displacement parameters which produces both complex eigenvalues and multiple zero eigenvalues. Next polynomial expressions are assumed as trial solutions for displacement parameters and eigenmodes containing undetermined parameters equal to the number of zero eigenvalues are determined by invoking the identity condition to governing equations. And then the exact displacement functions are constructed by combining eigenvectors and polynomial solutions corresponding to non-zero and zero eigenvalues, respectively. Consequently exact static stiffness matrices are evaluated by applying member force–displacement relationships to these displacement functions. The lateral-torsional deflections and buckling loads of thin-walled beam-columns are evaluated and compared with analytic solutions and the results by straight beam element and ABAQUS’s shell element.
    Thin-Walled Structures 05/2004; · 1.43 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Vibration characteristics of columns are influenced by their axial loads. Numerous methods have been developed to quantify axial load and deformation in individual columns based on their natural frequencies. However, these methods cannot be applied to columns in a structural framing system as the natural frequency is a global parameter of the entire framing system. This paper presents an innovative method to quantify axial deformations of columns in a structural framing system using its vibration characteristics, incorporating the influence of load tributary areas, boundary conditions and load migration among the columns.
    Structural Engineering & Mechanics 04/2014; 50(1). · 0.80 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The free vibration of beams, subjected to a constant axial load and end moment and various boundary conditions, is examined. Based on the Euler-Bernoulli bending and St. Venant torsion beam theories, the differential equations governing coupled flexural-torsional vibrations and stability of a uniform, slender, isotropic, homogeneous, and linearly elastic beam, undergoing linear harmonic vibration, are first reviewed. The existing formulations are then briefly discussed and a conventional finite element method (FEM) is developed. Exploiting the MATLAB-based code, the resulting linear Eigenvalue problem is then solved to determine the Eigensolutions (i.e., natural frequencies and modes) of illustrative examples, exhibiting geometric bending-torsion coupling. Various classical boundary conditions are considered and the FEM frequency results are validated against those obtained from a commercial software (ANSYS) and the data available in the literature. Tensile axial force is found to increase natural frequencies, indicating beam stiffening. However, when a force and an end moment are acting in combination, the moment reduces the stiffness of the beam and the stiffness of the beam is found to be more sensitive to the changes in the magnitude of the axial force compared to the moment. A buckling analysis of the beam is also carried out to determine the critical buckling end moment and axial compressive force.
    Shock and Vibration 10/2014; 2014. · 0.61 Impact Factor