You have just completed your registration at OpenAire.
Before you can login to the site, you will need to activate your account.
An e-mail will be sent to you with the proper instructions.
Important!
Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version
of the site upon release.
Transcendental stiffness matrices for vibration (or buckling) analysis have long been available for a range of structural members. Such stiffness matrices are exact in the sense that they are obtained from an analytical solution of the governing differential equations of the member. Hence, assembly of the member stiffnesses to obtain the overall stiffness matrix of the structure results in a transcendental eigenproblem that yields exact solutions and which can be solved with certainty using the Wittrick-Williams algorithm. Convergence is commonly achieved by bisection, despite the fact that the method is known to be relatively slow. Quicker methods are available, but their implementation is hampered by the highly volatile nature of the determinant of the structure's transcendental stiffness matrix, particularly in the vicinity of the poles, which may or may not correspond to eigenvalues. However, when the exact solution exists, the member has a recently discovered property that can also be expressed analytically and is called its member stiffness determinant. The member stiffness determinant is a property of the member when fully clamped boundary conditions are imposed upon it. It is then defined as the determinant of the member stiffness matrix when the member is sub-divided into an infinite number of identical sub-members. Each sub-member is therefore of infinitely small length so that its clamped-ended natural frequencies are infinitely large. Hence the contribution from the member stiffness matrix to the Jq count of the W-W algorithm will be zero. In general, the member stiffness determinant is normalised by dividing by its value when the eigenparameter (i.e. the frequency or buckling load factor) is zero, as otherwise it would become infinite. Part A of this thesis develops the first two applications of member stiffness determinants to the calculation of natural frequencies or elastic buckling loads of prismatic assemblies of isotropic and orthotopic plates subject to in-plane axial and transverse loads. A major advantage of the member stiffness determinant is that, when its values for all members of a structure are multiplied together and are also multiplied by the determinant of the transcendental overall stiffness matrix of the structure, the result is a determinant which has no poles and is substantially less volatile when plotted against the eigenparameter. Such plots provide a significantly better platform for the development of efficient, computer-based routines for convergence on eigenvalues by curve prediction techniques. On the other hand, Part B presents the development of exact dynamic stiffness matrices for three models of sandwich beams. The simplest one is only able to model the flexural vibration of asymmetric sandwich beams. Extending the first model to include axial and rotary inertia makes it possible to predict the axial and shear thickness modes of vibration in addition to those corresponding to flexure. This process culminates in a unique model for a three layer Timoshenko beam. The crucial difference of including axial inertia in the second model, enables the resulting member dynamic stiffness matrix (exact finite element) to be included in a general model of two dimensional structures for the first time. Although the developed element is straight, it can also be used to model curved structures by using an appropriate number of straight elements to model the geometry of the curve. Finally, it has been shown that considering a homogeneous deep beam as an equivalent three-layer beam allows the beam to have additional shear modes, besides the flexural, axial and fundamental shear thickness modes. Also for every combination of layer thickness, the frequencies of the three-layer beam are less than the corresponding frequencies calculated for the equivalent beam model with only one layer, since it is equivalent to providing additional flexibility to the system. However, a suitable combination of layer thicknesses for any mode may be found that yields the minimum frequency. It is anticipated that these frequencies would probably be generated by a single layer model of the homogeneous beam if at least a third order shear deformation theory was incorporated. Numerous examples have been given to validate the theories and to indicate their range of application. The results presented in these examples are identical to those that are available from alternative exact theories and otherwise show good correlation with a selection of comparable approximate results that are available in the literature. In the latter case, the differences in the results are attributable to many factors that vary widely from different solution techniques to differences in basic assumptions.
Abramovich, H., Eisenberger, M., and Shulepov, O. (1995). "Vibrations of multispan nonsymmetrical composite beams." Composites Engineering, 5(4), 397-404.
Ahmed, K. M. (1971). "Free vibration of curved sandwich beams by the method of finite element." Journal o fSound and vibration, 18, 61-74.
Ahmed, K. M. (1972). "Dynamic analysis of sandwich beams." Journal o f Sound and vibration, 21(3), 263-276.
Allen, H. G. (1969). Analysis and design o fstructural sandwich panels, Pergamon press, Oxford.
Ammar, S., Dhatt, G., and Fafard, M. (1996). "Exact stability model of space frames." Computers & Structures, 60(1), 59-71.
Anderson, M. S., and Williams, F. W. (1986). "BUNVIS-RG: An exact buckling and vibration program for latice structures, with repetitive geometry and substructuring options." 27th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conference, San Antonio, Texas, 211-220.
Armstrong, I. D. (1969). "The natural frequencies of multi-story frames." Structural Engineering, 47, 299-308.
Baber, T. T., Maddox, R. A., and Orozco, C. E. (1998). "A finite element model for harmonically excited viscoelastic sandwich beams." Computers & Structures, 66(1), 105-113.
Banerjee, J. R. (1989). "Coupled bending-torsional dynamic stiffiiess matrix for beam elements." International Journalfor Numerical Methods in Engineering, 28(6), 1283-98.
Banerjee, J. R. (1997). "Dynamic stiffiiess formulation for structural elements: A general approach." Computers & Structures, 63(1), 101-103.
Banerjee, J. R. (1998). "Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method." Computers & Structures, 69(2), 197-208.
Banerjee, J. R. (2001). "Free vibration analysis of a twisted beam using the dynamic stiffness method." International Journal o fSolids and Structures, 38(38-39), 6703-6722.
Banerjee, J. R. (2003). "Free vibration of sandwich beams using the dynamic stiffness method." Computers & Structures, 81(18-19), 1915-1922.
Banerjee, J. R. (2004). "Development of an exact dynamic stiffiiess matrix for free vibration analysis of a twisted Timoshenko beam." Journal o f Sound and Vibration, 270(1-2), 379- 401.
Banerjee, J. R., Guo, S., and Howson, W. P. (1996). "Exact dynamic stiffiiess matrix o f a bending-torsion coupled beam including warping." Computers & Structures, 59(4), 613- 621.
Banerjee, J. R., and Williams, F. W. (1985). "Exact Bemoulli-Euler Dynamic Stiffiiess Matrix for a Range o f Tapered Beams." International Journal for Numerical Methods in Engineering,, 21(12), 2289-2302.
Banerjee, J. R., and Williams, F. W. (1992). "Coupled Bending-Torsional Dynamic Stiffiiess Matrix for Timoshenko Beam Elements." Computers & Structures, 42(3), 301-310.
Banerjee, J. R., and Williams, F. W. (1994a). "Coupled Bending-Torsional Dynamic Stiffiiess Matrix o f an Axially Loaded Timoshenko Beam Element." International Journal o f Solids and Structures, 31(6), 749-762.
Banerjee, J. R., and Williams, F. W. (1994b). "An Exact Dynamic Stiffiiess Matrix for Coupled Extensional Torsional Vibration o f Structural Members." Computers & Structures, 50(2), 161-166.
Banerjee, J. R., and Williams, F. W. (1995). "Free-Vibration o f Composite Beams - an Exact Method Using Symbolic Computation." Journal o fAircraft, 32(3), 636-642.
Banerjee, J. R., and Williams, F. W. (1996). "Exact dynamic stiffiiess matrix for composite Timoshenko beams with applications." Journal o fSound and Vibration, 194(4), 573-585.
Bercin, A. N. (1995). "Analysis o f orthotropic plate structures by the direct-dynamic stiffiiess method." Mechanics Research Communications, 22(5), 461-466.
Bercin, A. N. (1997). "Eigenffequencies o f rectangular plate assemblies." Computers & Structures, 65(5), 703-711.
Bercin, A. N., and Langley, R. S. (1996). "Application o f the dynamic stiffiiess technique to the in-plane vibrations o f plate structures." Computers & Structures, 59(5), 869-875.
Bitzer, T. N. (1992). "Honeycomb material and application." Sandwich construction 2, UK, 681 - 691.
Bozhevolnaya, E., and Sun, J. Q. (2004). "Free vibration analysis of curved sandwich beams." Journal o f Sandwich Structures <£Materials, 6(1), 47-73.
Busool, W., and Eisenberger, M. (2002). "Free vibration o f helicoidal beams of arbitrary shape and variable cross section." Journal o f Vibration and Acoustics-Transactions o f the ASME, 124(3), 397-409.
Cabanska-Placzkiewicz, K. (2000). "Dynamic analysis of viscoelastic sandwich beams." International Applied Mechanics, 36(5), 673-681.
Eisenberger, M., and Effaim, E. (2001). "In-plane vibrations of shear deformable curved beams.” International Journalfor Numerical Methods in Engineering, 52(11), 1221-1234.
Farghaly, S. H., and Shebl, M. G. (1992). "Vibration characteristics o f end mass loaded undamped sandwich beams with elastically constrained ends." Journal o f Sound and Vibration, 159(2), 237-249.
Fasana, A., and Marchesiello, S. (2001). "Rayleigh-Ritz analysis of sandwich beams." Journal o f Sound and Vibration, 241(4), 643-652.
Fontgalland, G., Baudrand, H., and Guglielmi, M. (1998). "Application o f a monotonous function to the analysis o f ridged waveguides." Journal o f Microwaves and Optoelectronics, 1(3), 25-34.
Fotiu, P. (1987). "Dynamic analysis o f viscoplastic sandwich beams using Green functions." Zeitschrift Fur Angewandte Mathematik UndMechanik, 67(4), T75-T77.
Friberg, P. O. (1983). "Coupled Vibrations o f Beams - an Exact Dynamic Element Stiffiiess Matrix." International Journalfor Numerical Methods in Engineering, 19(4), 479-493.
Frid, A. (1989). "Fluid vibration in piping systems-a structural mechanics approach, I. Theory." Journal o fSound and Vibration, 133(3), 423-38.
Frid, A. (1990). "A modified Wittrick-Williams algorithm for eigenvalue analysis o f fluid vibration in piping systems." Journal o fSound and Vibration, 141(2), 355-358.
Frostig, Y. (1992). "Behaviour o f delaminated sandwich beam with transversely flexible core - High-order theory." Composite Structures, 20(1), 1-16.
Frostig, Y. (1993). "High-order behaviour o f sandwich beams with flexible core and transverse diaphragms." Journal ofEngineering Mechanics-ASCE, 119(5), 955-972.
Frostig, Y., and Baruch, M. (1994). "Free vibrations o f sandwich beams with a transversely flexible core: A high order approach." Journal o fSound and Vibration, 176(2), 195-208.
Frostig, Y., and Thomsen, O. T. (2004). "High-order free vibration o f sandwich panels with a flexible core." International Journal o fSolids and Structures, 41(5-6), 1697-1724.
Ganapathi, M., Patel, B. P., and Kumar, T. S. (1999a). "Torsional vibration and damping analysis o f sandwich beams." Journal o fReinforced Plastics and Composites, 18(2), 96-117.
Ganapathi, M., Patel, B. P., Polit, O., and Touratier, M. (1999b). "A C-l finite element including transverse shear and torsion warping for rectangular sandwich beams." International Journalfor Numerical Methods in Engineering, 45(1), 47-75.
Gorman, D. J. (1982). Free vibration analysis o f rectangularplates, Elsevier, New York.
Grove, E. A., and Ladas, G. (1974). Introduction to complex variables, Houghton Mifflin, Boston.
Hallauer, W. L. J., and Liu, R. Y. L. (1982). "Beam bending-torsion dynamic stiffiiess method for calculation o f exact vibration modes." Journal o fSound and Vibration, 85(1), 105-13.
Hashemi, S. M., and Richard, M. J. (2000a). "A Dynamic Finite Element (DFE) method for free vibrations o f bending-torsion coupled beams." Aerospace Science and Technology, 4(1), 41-55.
Hashemi, S. M., and Richard, M. J. (2000b). "Free vibrational analysis o f axially loaded bendingtorsion coupled beams: a dynamic finite element." Computers & Structures, 77(6), 711- 724.
Hashemi, S. M., and Richard, M. J. (2001). "Natural frequencies o f rotating uniform beams with Coriolis effects." Journal o f Vibration and Acoustics-Transactions o f the ASME, 123(4), 444-455.
He, S., and Rao, M. D. (1993). "Vibration and damping analysis of multi-span sandwich beams with arbitrary boundary conditions." Journal o fSound and Vibration, 164(1), 125-142.
Heinisuo, M. (1988). "An exact finite element technique for layered beams." Computers & Structures, 30(3), 615-22.
Henshell, and Warburton. (1969). "Transmission o f vibration in beam systems." International Journalfor Numerical Methods in Engineering, 1, 47-66.
Hjelmgren, J. P., Lunden, R., and Akesson, B. (1993). "Some comments on damped 2nd-order Rayleigh Timoshenko beam vibration in space - An exact complex dynamic member stiffiiess matrix." International Journal for Numerical Methods in Engineering, 36(24), 4267-4268.
Howson, W. P. (1979). "A compact method for computing the eigenvalues and eigenvectors o f plane frames." Advances in Engineering Software, 1(4), 181-190.
Howson, W. P., and Jemah, A. K. (1999a). "Exact dynamic stiffiiess method for planar natural frequencies o f curved Timoshenko beams." Proceedings o f the Institution o f Mechanical Engineers Part C- Journal o fMechanical Engineering Science, 213(7), 687-696.
Howson, W. P., and Jemah, A. K. (1999b). "Exact out-of-plane natural frequencies of curved Timoshenko beams." Journal o fEngineering Mechanics-ASCE, 125(1), 19-25.
Howson, W. P., and Williams, F. W. (1973). "Natural frequencies o f frames with axially loaded Timoshenko members." Journal o fSound and Vibration, 26(4), 503-515.
Howson, W. P., Williams, F. W., and Watson, A. (2001). "A methodology for finding interdisciplinary applications of the Wittrick-Williams algorithm." 7th EPMESC Conference, Shanghai, China, 143-55.
Howson, W. P., and Zare, A. (2004). "Exact dynamic stiffiiess matrix for flexural vibration o f three-layered sandwich beams." Journal o f Sound and vibration, In Press, Corrected Proof, Available online 5 November 2004.
Huang, C. S., Tseng, Y. P., Chang, S. H., and Hung, C. L. (2000). "Out-of-plane dynamic analysis o f beams with arbitrarily varying curvature and cross-section by dynamic stiffiiess matrix method." International Journal o fSolids and Structures, 37(3), 495-513.
Huang, C. S., Tseng, Y. P., Leissa, A. W., and Nieh, K. Y. (1998). "An exact solution for in-plane vibrations o f an arch having variable curvature and cross section." International Journal o fMechanical Sciences, 40( 11), 1159-1173.
Jabareen, M., and Eisenberger, M. (2001). "Free vibrations of non-homogeneous circular and annular membranes." Journal o fSound and Vibration, 240(3), 409-29.
Kant, T., and Gupta, A. (1988). "A finite element model for a higher order shear deformable beam theory." Journal o fSound and Vibration, 125(1), 193-202.
Kant, T., Marur, S. R., and Rao, G. S. (1998). "Analytical solution to the dynamic analysis o f laminated beams using higher order refined theory." Composite Structures, 40(1), 1-9.
Kennedy, D., Watkins, W. J., and Williams, F. W. (1995). "Hybrid Parallel Computation o f Transcendental Structural Eigenvalues." AIAA Journal, 33(11), 2194-2198.
Kennedy, D., and Williams, F. W. (1991). "More efficient use o f determinants to solve transcendental structural eigenvalue problems reliably." Computers <£ Structures, 41(5), 973-979.
Kennedy, D., and Williams, F. W. (1997). "Parallel computation for transcendental structural eigenproblems." Structural Engineering and Mechanics, 5(5), 635-644.
Kerwin, E. M. (1959). "Damping o f flexural waves by a constrained visco-elastic layer." Journal o f the Acoustical Society o fAmerica, 31, 952-962.
Kim, M. Y., Yun, H. T., and Kim, N. I. (2003). "Exact dynamic and static element stiffiiess matrices o f nonsymmetric thin-walled beam-columns." Computers & Structures, 81(14), 1425-1448.
Kolousek, V. (1941). "Anwendung des Gesetzes der virtuellen Verschiebungen und des Reziprozitatssatzes in der Stabwerksdynamic." Ing.-Archiv, 12, 363-370.
Kolousek, V. (1943). "Berechnung der schwingenden Stockwerahmen nach der Deformationmethode." Der Stahlbau, 16, 5-6 and 11-13.
Kolousek, V. (1973). Dynamics in Engineering Structures, Butterworths, London.
Kumpinsky, E. (1992). "The solution o f transcendental trigonometric characteristic equations." Industrial & Engineering Chemistry Research, 31(1), 440-445.
Laursen, H. I., Shubinski, R. P., and Clough, R. W. (1962). "Dynamic matrix analysis o f framed structures." Thefourth U.S. National Congress on Applied Mechanics.
Lee, J., and Thompson, D. J. (2001). "Dynamic stiffiiess formulation, free vibration and wave motion o f helical springs." Journal o fSound and Vibration, 239(2), 297-320.
Lekhnitskii, S. J. (1968). Anisotropic plates, Gordon and Breach, New York.
Leung, A. Y. T. (1992). "Dynamic stiffiiess analysis o f thin-walled structures." Thin-Walled Structures, 14(3), 209-222.
Leung, A. Y. T. (1993). Dynamic stiffiiess and substructures, Springer, London.
Szilard, R. (1974). Theory and analysis o fplates, Printice-Hall, New Jersy.
Tanghe-Camer, F., and Gay, D. (2000). "Nonuniform warping torsion o f orthotropic composite beams.”Archive o fApplied Mechanics, 70(8-9), 635-648.
Timoshenko, S. P. (1921). "On the correction for shear o f the differential equation for transverse vibration o f prismatic bars." Philosophical Magazine, 41.
Veletsos, A. S., and Newmark, N. M. (1956). "Determination o f natural frequencies o f continous plates hinged along two opposite edges." Journal o fApplied Mechanics-Transactions o f the ASME, 23, 97-102.
Vinson, J. R. (1999). The behaviour o f sandwich structures o f isotropic and composite materials, Technomic, Lancaster PA,.
Vinson, J. R. (2001). "Sandwich structures." Applied Mechanics Review, 54(3), 201-214.
Vuo-Quoc, L., and Ebcioglu, I. K. (1995). "Dynamic formulation for geometrically exact sandwich beams and one-dimensional plates." Journal o f Applied MechanicsTransactions o f the ASME, 62(3), 756-763.
Vu-Quoc, L., and Deng, H. (1995). "Dynamics o f geometrically exact sandwich beams/l-D plates: computational aspects." American Society o f Mechanical Engineers, Design Engineering Division (Publication) DE, 84(3), 703-713.
VuQuoc, L., Deng, H., and Ebcioglu, I. K. (1996). "Multilayer beams: A geometrically exact formulation." Journal o fNonlinear Science, 6(3), 239-270.
Wang, T. M., and Kinsman, T. A. (1971). "Vibration o f frame structures according to the Timoshenko theory." Journal o fSound and Vibration, 14, 215-227.
Williams, F. W., Howson, W. P., and Watson, A. (2004a). "Application o f the Wittrick-Williams algorithm to the Sturm- Liouville problem on homogeneous trees: A structural mechanics analogy." Proceedings o f the Royal Society o f London Series A- Mathematical Physical and Engineering Sciences, 460(2045), 1243-1268.
Williams, F. W., and Kennedy, D. (1988). "Reliable use o f determinant to solve non-linear structural eigenvalue analysis." International Journal o f Numerical Methods for Engineering, 26, 1825-41.
Williams, F. W., and Kennedy, D. (1996). "Accelerated solutions for transcendental stiffiiess matrix eigenproblems." Shock and Vibration, 3(4), 287-292.
Williams, F. W., and Kennedy, D. (2003). "Derivation o f new transcendental member stiffiiess determinant for vibrating frames." International Journal o f Structural Stability and Dynamics, 3(2), 299-305.
Williams, F. W., Kennedy, D., and Djoudi, M. S. (2002a). "The member stiffiiess determinant and its uses for the transcendental eigenproblems o f structural engineering and other disciplines." Proceedings o f the Royal Society o fLondon Series A- Mathematical Physical and Engineering Sciences, 459(2032), 1001-1019.
Zare, A., Howson, W. P., and Kennedy, D. (2003a). "An overview o f member stiffiiess determinants for prismatic plate assemblies." ICCES'03: International Conference on Computational and Experimental Engineering and Sciences, Corfu, Greece, 1-6.
Zare, A., Howson, W. P., Kennedy, D., and Williams, F. W. (2003b). "The exact member stiffiiess determinant o f orthotropic prismatic plates and a proposed approximation for anisotropic plates." International Journal o f Mechanical Sciences, Submitted for publication.
Zare, A., Howson, W. P., Kennedy, D., and Williams, F. W. (2003c). "Member stiffiiess determinants for use with axially loaded isotropic prismatic plate assemblies." Journal o f Sound and Vibration, Submitted for publication.
Zenkert, D. (1997). The handbook o fsandwich construction, Chameleon press, London.
Zhang, Q. J., and Sainsbury, M. G. (2000). "The Galerkin element method applied to the vibration of rectangular damped sandwich plates." Computers & Structures, 74(6), 717-730.