Review Articles

Frequency Response of Laminated Glass Elements: Analytical Modeling and Effective Thickness

[+] Author and Article Information
M. L. Aenlle

e-mail: aenlle@uniovi.es

F. Pelayo

Department of Construction and
Manufacturing Engineering,
University of Oviedo,
Campus de Gijón,
Zona Oeste, Edificio 7,
Gijón 33203, Spain

1Corresponding author.

Manuscript received August 9, 2012; final manuscript received February 21, 2013; published online April 11, 2013. Editor: Harry Dankowicz.

Appl. Mech. Rev 65(2), 020802 (Apr 11, 2013) (13 pages) Paper No: AMR-12-1040; doi: 10.1115/1.4023929 History: Received August 09, 2012; Revised February 21, 2013

Laminated glass elements are sandwich structures where the glass presents linear-elastic behavior, whereas the polymer interlayer is, in general, a linear-viscoelastic material. Several analytical models have been proposed since the 1950s to determine the response of laminated glass elements to both frequency and thermal conditions. In this paper, it is proved that Ross, Kerwin, and Ungar's model can be considered as a particular case of the Mead and Markus model when the exponential decay rate per unit length is neglected. The predictions of these models are compared with those obtained from operational modal tests carried out on a laminated glass beam at different temperatures. Finally, a new effective thickness for the dynamic behavior of laminated glass beams, which allows the determination of the dynamic response using a simple monolithic elastic model, is proposed.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Jones, D. I. G., 1996, “Reflections on Damping Technology at the End of the Twentieth Century,” J. Sound Vib., 190(3), pp. 449–462. [CrossRef]
Koutsawa, Y., and Daya, E. M., 2007, “Static and Free Vibration Analysis of Laminated Glass Beam on Viscoelastic Supports,” Int. J. Solids Struct., 44, pp. 8735–8750. [CrossRef]
Hooper, J. A., 1973, “On the Bending of Architectural Laminated Glass,” Int. J. Mech. Sci., 15, pp. 309–323. [CrossRef]
Behr, R. A., Minor, J. E., and Norville, H. S., 1993, “Structural Behavior of Architectural Laminated Glass,” J. Struct. Eng., 119(1), pp. 202–222. [CrossRef]
Edel, M. T., 1997, “The Effect of Temperature on the Bending of Laminated Glass Units,” M.S. thesis, Department of Civil Engineering, Texas A&M University, College Station, TX.
Norville, H. S., King, K. W., and Swoord, J. L., 1998, “Behavior and Strength of Laminated Glass,” J. Eng. Mech., 124(1), pp. 46–53. [CrossRef]
Asik, M. Z., and Tezcan, S., 2005, “A Mathematical Model for the Behavior of Laminated Glass Beams,” Comput. Struct., 83, pp. 1742–1753. [CrossRef]
Ivanov, I. V., 2006, “Analysis, Modeling and Optimization of Laminated Glasses as Plane Beam,” Int. J. Solids Struct., 43(22-23), pp. 6887–6907. [CrossRef]
Galuppi, L., and Royer-Carfagni, G. F., 2012, “Laminated Beams With Viscoelastic Interlayer,” J. Solids Struct., 49(18), pp. 2637–2645. [CrossRef]
Calderone, I., Davies, P. S., and Benninson, S. J., 2009, “Effective Laminate Thickness for the Design of Laminated Glass,” Glass Processing Days, Tampere, Finland.
Wölfel, E., 1987, “Nachgiebiger Verbund Eine Näherungslösung und Deren Anwendungsmöglichkeiten,” Stahlbau, 6, pp. 173–180.
Galuppi, L., and Royer-Carfagni, G. F., 2012, “Effective Thickness of Laminated Glass Beams: New Expression via a Variational Approach,” Eng. Struct., 38, pp. 53–67. [CrossRef]
Galuppi, L., and Royer-Carfagni, G. F., 2012, “The Effective Thickness of Laminated Glass Plates,” J. Mech. Mater. Struct., 7(4), pp. 375–400. [CrossRef]
Kerwin, E. M., 1959, “Damping of Flexural Waves by a Constrained Viscoelastic Layer,” J. Acoust. Soc. Am., 31(7), pp. 952–962. [CrossRef]
Ross, D., Ungar, E. E., and Kerwin, E. M., 1959, “Damping of Plate Flexural Vibrations by Means of Viscoelastic Laminate,” Structural Damping, American Society of Mechanical Engineers (ASME), New York, pp. 49–88.
Lu, Y. P., and Douglas, B. E., 1974, “On the Forced Vibrations of Three Layer Damped Sandwich Beams,” J. Sound Vib., 32(4), pp. 513–516. [CrossRef]
Sadasiva Rao, Y. V. K., and Nakra, B. C., 1974, “Vibrations of Unsymmetrical Sandwich Beams and Plates With Viscoelastic Cores,” J. Sound Vib., 34(3), pp. 309–326. [CrossRef]
DiTaranto, R. A., and McGraw, J. R., Jr., 1969, “Vibratory Bending of Damped Laminated Plates,” ASME J. Eng. Industry, 91(4), pp. 1081–1090. [CrossRef]
DiTaranto, R. A., 1965, “Theory of Vibratory Bending for Elastic and Viscoelastic Layered Finite-Length Beams,” ASME J. Appl. Mech., 32(4), pp. 881–886. [CrossRef]
Mead, D. J., and Markus, S., 1969, “The Forced Vibration of a Three-Layer, Damped Sandwich Beam With Arbitrary Boundary Conditions,” J. Sound Vib., 10(2), pp. 163–175. [CrossRef]
Mead, D. J., and Markus, S., 1970, “Loss Factors and Resonant Frequencies of Encastré Damped Sandwich Beam,” J. Sound Vib., 12(1), pp. 99–112. [CrossRef]
Rao, D. K., 1978, “Frequency and Loss Factors of Sandwich Beams Under Various Boundary Conditions,” J. Mech. Eng. Sci., 20(5), pp. 271–282. [CrossRef]
Mead, D. J., 2007, “The Measurements of the Loss Factors of Beams and Plates With Constrained and Unconstrained Layers: A Critical Comparison,” J. Sound Vib., 300, pp. 744–762. [CrossRef]
Bennison, S. J., Jagota, A., and Smith, C. A., 1999, “Fracture of Glass/PVB Laminates in Biaxial Flexure,” J. Am. Ceram. Soc., 82(7), pp. 1761–1770. [CrossRef]
Lee, E. H., 1955, “Stress Analysis in Viscoelastic Bodies,” Q. J. Mech. Appl. Math., 13, pp. 183–190.
Read, W. T., 1950, “Stress Analysis for Compressible Viscoelastic Materials,” J. Appl. Phys., 21, pp. 671–674. [CrossRef]
Ferry, J. D., 1980, Viscoelastic Properties of Polymers, 3rd ed., John Wiley and Sons, New York.
Williams, M. L., Landel, R. F., and Ferry, J., 1955, “The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-Forming Liquids,” J. Am. Chem. Soc., 77, pp. 3701–3707. [CrossRef]
Van Duser, A., Jagota, A., and Bennison, S. J., 1999, “Analysis of Glass/Polyvinyl Butyral Laminates Subjected to Uniform Pressure,” J. Eng. Mech., 125(4), pp. 435–442. [CrossRef]
Park, S. W., and Schapery, R. A., 1999, “Methods of Interconversion Between Linear Viscoelastic Material Functions: Part I: A Numerical Method Based on Prony Series,” Int. J. Solids Struct., 36(11), pp. 1653–1675. [CrossRef]
Fernández, P., Rodríguez, D., Lamela, M. J., and Fernández-Canteli, A., 2010, “Study of the Interconversion Between Viscoelastic Behaviour Functions of PMMA,” Mech. Time-Depend. Mater., 15(2), pp. 169–180. [CrossRef]
Garcia-Barruetabeña, J., Cortés, F., Abete, J. M., Fernández, P., Lamela, M. J., and Fernández-Canteli, A., “Relaxation Modulus—Complex Modulus Interconversion for Linear Viscoelastic Materials,” Mech. Time-Depend. Mater. (in press). [CrossRef]
Tschoegl, N. W., 1989, The Phenomenological Theory of Linear Viscoelastic Behavior, Springer-Verlag, Berlin.
Tzikang, C., 2000, “Determining a Prony Series for a Viscoelastic Material From Time Varying Strain Data,” Report No. NASA /TM–2000–210123, ARL–TR–2206.
Jones, D. I. G., 2001, Handbook of Viscoelastic Vibration Damping, John Wiley and Sons, New York.
Povolo, F., and Hermida, E., 1988, “Scaling Concept and the Williams-Landel-Ferry Relationship,” J. Mater. Sci., 23, pp. 1255–1259. [CrossRef]
Enelund, M., 1995, “Vibration and Damping of a Plate on a Viscous Fluid Layer,” Proceedings of the 13th IMAC, pp. 261–267.
Brincker, R., Zhang, L.-M., and Anderson, P., 2000, “Modal Identification From Ambient Response Using Frequency Domain Decomposition,” Proceedings of the 18th IMAC, pp. 625–630.
Van Overschee, P., and De Moor, B., 1996, Subspace Identification for Linear Systems: Theory, Implementation and Applications, Kluwer Academic, Dordrecht, The Netherlands.
Wanbo, L., 2008, “Experimental and Analytical Estimation of Damping in Beams and Plates With Damping Treatments,” Ph.D. thesis, University of Kansas, Lawrence, KS.


Grahic Jump Location
Fig. 1

Laminated-glass beam

Grahic Jump Location
Fig. 2

Tensile and shear moduli of PVB

Grahic Jump Location
Fig. 3

Exponential decay rate and wavenumber for mode 1

Grahic Jump Location
Fig. 4

Exponential decay rate and wavenumber for mode 2

Grahic Jump Location
Fig. 5

Exponential decay rate and wavenumber for mode 3

Grahic Jump Location
Fig. 6

Exponential decay rate and wavenumber for mode 4

Grahic Jump Location
Fig. 7

Natural frequencies and loss factors for mode 1

Grahic Jump Location
Fig. 8

Natural frequencies and loss factors for mode 2

Grahic Jump Location
Fig. 9

Natural frequencies and loss factors for mode 3

Grahic Jump Location
Fig. 10

Natural frequencies and loss factors for mode 4

Grahic Jump Location
Fig. 11

Complex effective stiffness of the laminated glass beam

Grahic Jump Location
Fig. 12

Test setup with the location of the accelerometers

Grahic Jump Location
Fig. 13

Singular value decomposition at 20 and 40 °C



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In