Session: 02-04-02: Computer Methods and Reduced Order Modeling

Paper Number: 152001

152001 - Dynamic Stiffness Method for Free Vibration of Beams and Frameworks Using Higher Order Shear Deformation Theory 

One of the great advantages of using a higher order shear deformation theory in free vibration analysis of beams or frameworks is that it dispenses with the so-called shear correction factor generally adopted in the Timoshenko-Ehrenfest beam formulation to account for the non-uniform shear stress distribution through the thickness of the beam cross-section on an ad doc basis. A higher order shear deformation theory overcomes this limitation. Additionally, it should be noted that when carrying out the free vibration analysis of structures, the dynamic stiffness method (DSM) which is called an “exact” method is a powerful alternative to the conventional finite element method (FEM). With this pretext, the dynamic stiffness matrix of a beam using a higher order shear deformation theory together with the effect of rotatory inertia is developed in this paper. Starting with an appropriate choice of the displacement field, the potential and kinetic energies of the beam are formulated. Then, Hamilton’s principle is applied to derive the governing differential equations and associated natural boundary conditions, when the beam is undergoing free vibration. Next, the differential equations are solved in an exact sense to obtain the expressions for axial displacement, flexural displacement, bending rotation and the first derivative of the flexural displacement. The expressions for axial force, shear force, bending moment and the higher-order moment are obtained from the natural boundary conditions resulting from the Hamiltonian formulation. Finally, the force vector comprising the axial force, shear force, bending moment and the higher-order moment is related to the displacement vector  comprising the axial displacement, flexural displacement, bending rotation and the first derivative of the flexural displacement by using the frequency-dependent dynamic stiffness matrix relating the amplitudes of forces and displacements of the free vibratory motion.  Explicit expression for each element of the dynamic stiffness matrix is derived through rigorous application of symbolic computation.

A literature review about this research suggests that although higher order shear deformation theory has been used by some investigators to explore the free vibration characteristics of beams, publications relating to the application of the dynamic stiffness method to solve the problem more accurately, are scarce. Furthermore, most of the published literature deals with the free vibration behaviour of individual beams using higher order shear deformation theory, but the extension of the theory for applications to frameworks appears to be an open area of research, not undertaken by investigators. This paper is intended to fill this gap in the literature by developing the dynamic stiffness matrix of a beam using higher order shear deformation theory and then applying it to individual beams and frameworks.

One of the difficulties associated with the extension of the higher order shear deformation theory to a frame is that the higher order shear deformation theory introduces an additional degree of freedom at each node of the frame, making it somewhat difficult in the implementation of the theory to frameworks. This is perceived to be an additional complication. For a plane frame problem, the usually adopted three degrees of freedom at a node when Bernoulli-Euler or Timoshenko-Ehrenfest beam theories are used, will have an additional degree of freedom which is the first derivative of the flexural displacement of the beam at a node with respect to its length coordinate. Thus, when transforming the element mass and stiffness matrices in the FEM or the frequency-dependent element dynamic stiffness matrix in the DSM, from the local to global (or datum) coordinates, this additional degree of freedom which is different from the bending rotation, must be accounted for to deal with the free vibration problems of frameworks. Ultimately, the developed dynamic stiffness matrix is used with reference to the Wittrick-Williams algorithm as solution technique to compute the natural frequencies and mode shapes of some representative examples which include individual beams as well as frameworks. Results are discussed and significant conclusions are drawn.

Presenting Author: J Ranjan Banerjee City St George's, University of London

Presenting Author Biography: Professor Ranjan Banerjee received his Bachelor’s and Master’s degrees in mechanical engineering from the University of Calcutta (1969) and the Indian Institute of Technology, Kharagpur (1971) respectively. He joined the Structural Engineering Division of the Indian Space Research Organisation, Trivandrum and worked there for four years (1971-75) first as a Structural Engineer and then as a Senior Structural Engineer. He was involved in the research and development of multistage solid propellant rocket structures with particular emphasis on dynamic response. Later in the year 1975 he was awarded a Commonwealth Scholarship by the Association of Commonwealth Universities to study for a PhD degree at Cranfield University where he researched within the areas of structural dynamics and aeroelasticity. He completed his PhD in 1978. An important spin-off from his PhD work is the development of an aeroelastic package called CALFUN (CALculation of Flutter speed Using Normal modes) which has been extensively used as a teaching and research tool in aeroelastic studies.

Professor Banerjee joined the University of Cardiff as a Research Associate in 1979 and worked there for six years on vibration and buckling characteristics of space structures using the dynamic stiffness method. He worked in close collaboration with NASA, Langley Research Center, and was principally involved in the development of the well-established computer program BUNVIS (BUckling or Natural VIbration of Space Frames) which was later used by NASA and other organizations to analyse spacecraft structures. He was promoted to the position of Senior Research Associate in 1982.

Professor Banerjee joined City, University of London in 1985 as a Lecturer in Aircraft Structures. He was promoted to Senior Lectureship and Readership in 1994 and 1998 respectively. In March 2003 he was promoted to Professorship. He was elected to the EPSRC Peer Review College in 1996 and served until 1999, and was re-elected in 2002, and is currently serving in the College. He has been conducting research within the technical areas of structural dynamics, aeroelasticity and composite materials for well over40 years. To date he has published around two hundred and fifty papers in international journals and established conferences. In recognition of his research, he was awarded the degree of Doctor of Science (DSc) by City, University of London in 2017. Professor Banerjee is a Chartered Engineer and a Fellow of the Royal Aeronautical Society and the Institution of Structural Engineers and an Associate Fellow of the American Institute of Aeronautics and Astronautics.