Elsevier

Composites Part B: Engineering

Volume 151, 15 October 2018, Pages 92-105
Composites Part B: Engineering

On modal analysis of laminated glass: Usability of simplified methods and Enhanced Effective Thickness

https://doi.org/10.1016/j.compositesb.2018.05.032Get rights and content

Abstract

This paper focuses on the modal analysis of laminated glass beams. In these multilayer elements, the stiff glass plates are connected by compliant interlayers with frequency and temperature-dependent behavior. The aim of our study is (i) to assess whether approximate techniques can accurately predict the behavior of laminated glass structures and (ii) to propose a simple tool for modal analysis based on the Enhanced Effective Thickness concept.

For this purpose, we consider four approaches to the solution of the related nonlinear eigenvalue problem: a complex-eigenvalue solver based on the Newton method and three simplified approaches. In particular, we limit our attention to the modal strain energy method, the dynamic effective thickness method, and the Enhanced Effective Thickness method. A comparative study of free vibrating laminated glass beams is performed considering different geometries of cross-sections, boundary conditions, and material parameters for interlayers under two ambient temperatures. The viscoelastic response of polymer foils is represented by the generalized Maxwell model.

We show that the simplified approaches predict natural frequencies with an acceptable accuracy for most of the examples. However, there is considerable scatter in predicted loss factors. The Enhanced Effective Thickness approach adjusted to modal analysis results in lower errors in both quantities compared to the other two simplified procedures. It reduces the extreme error in loss factors by half compared to the modal strain energy method and to one quarter compared to the original dynamic effective thickness method.

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)

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