Coupled vibration of beams—An exact dynamic element stiffness matrix

Division of Solid Mechanics, Chalmers University of Technology, Gothenburg, Sweden
International Journal for Numerical Methods in Engineering (Impact Factor: 2.06). 04/1983; 19(4):479 - 493. DOI: 10.1002/nme.1620190403


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.

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    • "Several studies investigated the coupled bendingtorsional buckling of thin-walled beams. Based on the Vlasov beam theory, Friberg (1983), Leung (1991, 1992), Banerjee et al (1996), Kim et al (2007) developed the dynamic stiffness matrix in which shear deformation effects are neglected. Kollar and Pluzsik (2002) formulated the stiffness matrix for composite beams of open thin-walled sections. "
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    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
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    • "The theory is based on two kinematic assumptions; (i) the beam cross-section does not deform in its own plane, and (ii) the transverse shear strains at the mid-surface are negligible. Friberg (1983), Leung (1991,1992), Chen and Tamma (1994), Li et al. (2004) and Kim et al. (2007) developed the dynamic stiffness matrix of Vlasov beam in which the shear deformation is ignored. Using the normal mode method, Eslimy-Isfahany et al. (1996) developed a solution for the response of coupled bending-torsion vibration of thin-walled beams under deterministic and random excitations. "
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    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
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    • "Based on Vlasov theory, Friberg (1983), Leung (1991) and (1992), Chen and Tamma (1994), Li et al. (2004a) and Kim et al. (2007) developed the dynamic stiffness matrix of Vlasov beam in which shear deformation is entirely ignored. Many researchers modified the Vlasov theory for the analysis of elastic thin-walled members to capture the transverse shear deformations. "
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    ABSTRACT: This paper investigates the behavior of doubly symmetric thin-walled members of open sections under harmonic excitation. The governing equilibrium equations and associated boundary conditions based on a generalized Timoshenko-Vlasov theory are derived by applying Hamilton's variational principle. Shear deformation effects due to bending and non-uniform warping as well as rotary inertia effects are incorporated in the formulation. The steady state response of the system is obtained by exactly solving the coupled differential equations obtained. The applicability of the analytical solution is demonstrated via examples with various harmonic loads. In order to assess the accuracy of the analytical solutions, comparisons are made against established shell finite element solutions. Additional comparisons are made against non-shear deformable theories and the effect of shear deformation is established.
    2nd International Engineering Mechanics and Materials Specialty Conference, Ottawa, Ontario, Canada; 06/2011
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