Coupled vibrations of beams—an exact dynamic element stiffness matrix
ABSTRACT A uniform linearly elastic beam element with non-coinciding centres of geometry, shear and mass is studied under stationary harmonic end excitation. The Euler-Bernoulli-Saint Venant theory is applied. Thus the effect of warping is not taken into account. The frequency-dependent 12 × 12 element stiffness matrix is established by use of an exact method. The translational and rotational displacement functions are represented as sums (real) of complex exponential terms where the complex exponents are numerically found. Built-up structures containing beam elements of the described type can be analysed with ease and certainty using existing library subroutines.
- SourceAvailable from: Mohammed Ali Hjaji[Show abstract] [Hide abstract]
ABSTRACT: The steady state response of thin-walled members with doubly symmetric cross-sections subjected to harmonic forces is investigated. Using the Hamiltonian functional, the governing differential equations and related boundary conditions are formulated based on the Vlasov thin walled beam theory. The formulation takes into account the effect of warping deformation and translational and rotary inertia. The resulting governing field equations are then exactly solved and closed form solutions for transverse and torsional responses are obtained for common boundary conditions. Numerical examples are then presented and comparisons are made against other established Abaqus beam and shell solutions to assess the accuracy of the present analytical solutions.CSCE 2013 - 3rd Specialty Conference on Engineering Mechanics and Materials in Montreal, Quebec, Montreal, Quebec, Canada; 05/2013
- [Show abstract] [Hide abstract]
ABSTRACT: The governing field equations and boundary conditions based on a generalized Vlasov-Timoshenko beam the-ory are formulated using the principle of the stationary total potential energy. The formulation incorporates shear deformation effects due flexure and warping, and captures the flexural-torsional coupling in monosym-metric cross-sections. General closed-form solutions are obtained for cantilever boundary conditions. Numeri-cal examples for the flexural-torsional coupled analysis are presented and compared with Abaqus finite ele-ment solutions. Additional comparisons are performed against non-shear deformable beam theories and the ef-fect of transverse shear deformation is illustrated.SEMC 2013- The Fifth International Conference on Structural Engineering, Mechanics and Computation, 2-4 September 2013, Cape Town, South Africa; 09/2013
- [Show abstract] [Hide abstract]
ABSTRACT: Spectral finite element methods are used to compute exact vibration solutions of structural models at specific frequencies. The applicability of these methods to certain areas of structural dynamics is limited by two major factors: the lack of separate structural operators (mass, damping, and stiffness matrices), and the subsequent difficulty in computing mode shapes via eigenvalue decomposition. In the work presented in this article, a method is investigated to accurately calculate spectral finite elements while overcoming these limitations. The approach incorporates a two-dimensional, discrete solution utilizing a wavenumber-based gridding technique to compute frequency-dependent local mass, damping, and stiffness matrices which can be assembled into the global structural operators. Computed models are able to be used for precise vibration analysis as well as modal analysis via eigenvalue decomposition of the structural operators.Journal of Sound and Vibration 12/2013; 332(25):6659-6664. · 1.61 Impact Factor