On modal analysis of laminated glass: Usability of simplified methods and Enhanced Effective Thickness
Introduction
Laminated glass is a multilayer composite made of glass layers and plastic interlayers, typically polymers. These foils improve the post-fracture behavior of the originally brittle glass elements [3,18,26], increase their damping [16,31], and therefore allow for applications unsuitable for traditional glass, such as load-bearing and fail-safe transparent structures. Avoiding resonance and reducing the noise and vibrations of laminated glass components is important not only in the context of building structures but also for car or ship design processes and other applications. Thus, reliable prediction of natural frequencies and the damping characteristics associated with each vibration mode is an essential issue for the design of dynamically loaded structures [2,14,19]. Due to the viscoelastic behavior of polymer foils [1], free vibration analysis leads to an eigenvalue problem with complex eigenvalues and eigenvectors which correspond to natural angular frequencies and mode shapes. In addition, nonlinearity due to the frequency/temperature-sensitive response of polymer interlayers adds complexity to the analysis.
Several approaches to analyzing the vibrations of viscoelastically damped layered composites can be found in the literature, e.g., [9,16,38]. In this paper, we broadly divide these methods into three groups: (i) numerical approaches solving the complex eigenvalue problem directly [13], (ii) simplified numerical approximations dealing only with a real eigenmode problem [5], and (iii) analytical methods and effective thickness methods derived from analytical models [20].
The comparison of selected non-linear solvers for complex-valued problems in [13] shows that while most of them converge towards the same eigenvalues, their computational times and the numbers of iterations differ. Computational costs can be reduced using simplified numerical methods, which deal only with a real eigenvalue problem corresponding to delayed elasticity or which take into account only the real part of the complex stiffness of the core [5]. Then, the damping parameters are obtained by post-processing the real-valued eigenvalues and eigenmodes using, e.g., the modal strain energy method [17] which will be discussed later in the paper.
For three-layer structures with simple boundary conditions and geometries, analytical solutions can be derived [23,24,34]. Because of the frequency-dependent behavior of the polymer foil, they provide natural frequencies and loss factors using an iterative algorithm. Recently, the dynamic effective thickness approach for laminated glass beams was proposed by López-Aenlle and Pelayo [20], using the complex flexural stiffness introduced in [34] and assuming constant wavenumbers for an Euler-Bernoulli beam. This concept can be extended towards plates [21] and multilayer laminated glass beams [27]. The validation of this dynamic effective thickness method against results gained during experimental testing in Refs. [20,21,27] shows that, using this approach, natural frequencies can be predicted with good accuracy but there is a high scatter in loss factors.
Therefore, in this paper we analyze the accuracy of the response of effective thickness and other simplified approaches in order to investigate their usability in the modal analysis of laminated glass elements and to propose some improvements. More specifically, we:
- •
perform a comparative study for free vibrating beams using selected solvers representing the three groups, as introduced above, and
- •
propose an easy tool for modal analysis based on the Enhanced Effective Thickness concept [12].
To our best knowledge, no such comparison of complex and approximate models has yet been performed for laminated glass.
All methods are compared for simply-supported, clamped-clamped, and free-free beams with symmetric and asymmetric cross-sections under different ambient temperatures. The viscoelastic behavior of polymer foil is described using the generalized Maxwell model. Several sets of parameters of the chain were taken from the literature [1,25,36] and used in our case study in order to evaluate and discuss the effect of various materials used in laminated glass structures and also of different Maxwell chain parameters describing the same type of interlayer.
The structure of this paper is as follows. The geometry of a three-layer laminated glass beam and material characterization of the glass and polymer layers is outlined in Section 2. The approaches based on the finite element methods, i.e., the Newton method and the modal strain energy method, are introduced in Section 3. The closed-form formula for the complex-valued natural frequencies presented in Section 4 is combined with the effective thickness concept, using the dynamic effective thickness from [21] and the Enhanced Effective Thickness [12] adjusted for modal analysis. The results of our case study are presented and analyzed in Section 5. Finally, we summarize our findings in Section 6.
Section snippets
Configuration of laminated glass beams
In this paper, the most common three-layer configuration (with two face glass plies and one polymer interlayer, see Fig. 1) is used for simplicity. However, an extension towards multilayer elements is possible for all approaches discussed. No slipping on the interface of the glass ply and the polymer foil is assumed.
Materials
The constitutive behavior of glass and polymer layers remains the same for all methods presented. Glass is treated as an elastic material, whereas the behavior of polymer is assumed
Refined beam element for three-layer laminated glass
Because of the relatively small thickness of the interlayer, we assume that shear deformation in the viscoelastic foil is responsible for all the damping and the transverse compressive strain is negligible. Thus, we treat each layer as a one-dimensional beam element in our numerical analysis. We assume the deformed cross-sections of individual beam elements to be planar. This, however, does not hold for the laminate. The three layers of a laminated glass beam are constrained together with
Closed-form expression for natural angular frequencies of beams
A few effective thickness formulations can be found in the literature for laminated glass beams and plates under static loading, whereas, to the best of our knowledge, only one effective thickness approach is available for dynamic problems [20,21]. In general, the effective thickness methods are based on calculating a constant thickness of a monolithic element with the same width and length which yields the same response as for a laminated glass beam under identical loading and boundary
Case study
In this section, the usability of the three simplified methods introduced in Sections 3.6 and 4 is assessed for laminated glass beams. The section is divided into two parts: the first introducing the selected test examples and the second discussing the results and effect of input data on the usability of the modal strain energy method and the two effective thickness approaches.
Conclusions
Four methods for modal analysis of laminated glass structures were introduced in this paper, i.e., the numerical complex-valued eigensolver based on the Newton method, the real-valued eigensolver complemented with the modal strain energy method, and two dynamic effective thickness methods. The aim of this paper was to assess the usability of the last three practical methods by comparing their predictions to those generated by the complex-valued eigensolver. For the Enhanced Effective Thickness
Acknowledgments
This publication was supported by the Czech Science Foundation under project No. 16-14770S. We would also like to thank Stephanie Krueger and Kaitlyn Haines (National Library of Technology) for their helpful comments regarding the manuscript.
References (41)
- et al.
Dynamic torsion tests to characterize the thermo-viscoelastic properties of polymeric interlayers for laminated glass
Construct Build Mater
(2014) - et al.
Viscoelastic vibration damping identification methods. Application to laminated glass
Procedia Eng
(2011) - et al.
Low velocity impact performance investigation on square hollow glass columns via full-scale experiments and finite element analyses
Compos Struct
(2017) - et al.
Linear and nonlinear vibrations analysis of viscoelastic sandwich beams
J Sound Vib
(2010) - et al.
Long term response of glass–PVB double-lap joints
Compos B Eng
(2014) - et al.
A numerical method for nonlinear eigenvalue problems application to vibrations of viscoelastic structures
Comput Struct
(2001) - et al.
Effective thickness of laminated glass beams: new expression via a variational approach
Eng Struct
(2012) - et al.
Comparison of non-linear eigensolvers for modal analysis of frequency dependent laminated visco-elastic sandwich plates
Finite Elem Anal Des
(2016) - et al.
Damping mechanism of elastic-viscoelastic-elastic sandwich structures
Compos Struct
(2016) - et al.
Experimental and numerical investigations of laminated glass subjected to blast loading
Int J Impact Eng
(2012)
Dynamic effective thickness in laminated-glass beams and plates
Compos B Eng
The measurement of the loss factors of beams and plates with constrained and unconstrained damping layers: a critical assessment
J Sound Vib
The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions
J Sound Vib
Loss factors and resonant frequencies of encastré damped sandwich beams
J Sound Vib
Quasi-static bending and low velocity impact performance of monolithic and laminated glass windows employing chemically strengthened glass
Eur J Mech Solid
Deformation and damage mechanisms of laminated glass windows subjected to high velocity soft impact
Int J Solid Struct
Recent applications of viscoelastic damping for noise control in automobiles and commercial airplanes
J Sound Vib
A new identification method of viscoelastic behavior: application to the generalized maxwell model
Mech Syst Signal Process
Finite element analysis of damping the vibrations of laminated composites
Comput Struct
A practical, nondestructive method to determine the shear relaxation modulus behavior of polymeric interlayers for laminated glass
Polym Test
Cited by (21)
Time-domain numerical analysis of single pedestrian random walks on laminated glass slabs in pre- or post-breakage regime
2022, Engineering StructuresCitation Excerpt :More precisely, the study aims at addressing the potential or limits of time-domain numerical approaches and deterministic models to reproduce walking paths when specifically applied to structural LG components which are characterized by typically high slenderness and flexibility, compared to other slab solutions of typical use in buildings, as well as limited mass, compared to occupants [22–25]. The vibration response assessment of glass structures, in the same way of post-breakage considerations, is a relatively recent research topic [26–30], and research in support of design is still needed. As a basic rule, occupied glass slabs should be verified against vibrations in the same way of pedestrian structures composed of other constructional materials [22].
Structural assessment of glass used in façade industry
2021, StructuresCitation Excerpt :Extreme deflection can lead to distortion; therefore, codes recommend the limitation of deflection as a function of the glass pane span. The deflection limits of well-known building codes for glass systems are shown in Table 1 [27–29]. The determination of glass resistance to withstand external loads and the allowable stresses depends on both the glass type and the load duration.
Experimental investigation on vibration sensitivity of an indoor glass footbridge to walking conditions
2020, Journal of Building EngineeringCitation Excerpt :While several calculation methods and design recommendations are available for the design and assessment of glass members/structures with respect to expected deformations and stress peaks/distributions, the dynamic vibration response is a relatively recent research topic. Most of the existing studies, see Refs. [30–35], include experimental, analytical and numerical analyses devoted to exploring the dynamic behaviour of simple glass members, even in presence of flexible restraints/bonding layers with viscoelastic phenomena. Even more attention should be spent for pedestrian glass structures, given the common use of glass in wide surfaces for floors and roofs [36–40].
Novel treatment methods for improving fatigue behavior of laminated glass
2019, Composites Part B: EngineeringCitation Excerpt :The second goal of the study was to develop finite element (FE) model that can predict the effect of treatment on the fatigue life and complement with the experimental data for providing stress-life and strain-life relations. The non-uniform and different types of flaws having different dimensions [13–15], rarely investigated interfacial properties of the glass and interlayer required for accurate modeling [16,17], heterogeneity, possible changes in viscoelastic behavior and elastic modulus of interlayer due to treatment [18], geometric dependency and non linearity of LG [19,20] are among the few challenges to overcome for the accurate predictions for LG by any analytical or FE based model. Recently published review articles on FE modeling of LG [19,21] provide sufficient background of modeling approaches used for LG.
Diagnostic analysis and dynamic identification of a glass suspension footbridge via on-site vibration experiments and FE numerical modelling
2019, Composite StructuresCitation Excerpt :The same issue was further examined in [37], based on OMA investigations, FE analyses and video-tracking based dynamic estimates. Dynamic effective analytical models were proposed in [38,39], for the reliable estimation of natural frequencies and damping ratios of multi-layered LG beams, by accounting for the viscous effect of PVB, SentryGlas® (SG) or Ethylene Vinyl Acetate (EVA) interlayers (see for example Fig. 2(b)). In general terms, glass floors should be checked – under ordinary service loads – towards maximum deformations (Service Limit State – SLS) and stresses (Ultimate Limit State – ULS) due to permanent and accidental (i.e., human induced) live loads, see for example [2,3].