Acta Polytechnica CTU Proceedings doi:10.14311/APP.2017.13.0109 Acta Polytechnica CTU Proceedings 13:109–114, 2017 © Czech Technical University in Prague, 2017 available online at http://ojs.cvut.cz/ojs/index.php/app VARIATIONALLY-BASED EFFECTIVE DYNAMIC THICKNESS FOR LAMINATED GLASS BEAMS Jaroslav Schmidt∗, Alena Zemanová, Tomáš Janda, Jan Zeman, Michal Šejnoha Czech Technical University in Prague, Faculty of Civil Engineering, Thákurova 7, 166 29 Prague 6, Czech Republic ∗ corresponding author: jarasit@gmail.com Abstract. Laminated glass, consisting of glass layers connected with transparent foils, has found its applications in civil, automotive, or marine engineering. Due to a high contrast in layer properties, mechanical response of laminated glass structures cannot be predicted using classical laminate theories. On the other hand, engineering applications demand easy-to-use formulas of acceptable accuracy. This contribution addresses such simplified models for free vibrations of laminated glass beams, with the goal to determine their natural frequencies and modal damping properties. Our strategy is to approximate the complex behavior of a laminated structure with that of an equivalent monolithic beam. Its effective thickness is determined by the variational method proposed by Galuppi and Royer-Carfagni for static problems, which we extended for modal analysis. We show that this new approach overcomes inaccuracies of the currently used dynamic effective thickness model by López-Aenlle and Pelayo. Keywords: Laminated glass, dynamic effective thickness, modal analysis. 1. Motivation In the past, glass was seen as an infill material for transparent areas with no load-bearing function in a structural system. The invention of laminated glass became one of the milestones that allowed glass to attain a genuine structural role [1]. Starting from windscreens in the car industry, its use has expanded into other fields of industry, including civil engineering. Today, layered glass is used, for example, for floor and roof systems, columns, staircases etc. Laminated glass is a layered composite material con- sisting of several layers of glass and polymer. Apart from its primary function in enhancing the post-peak load-bearing capacity, the polymer interlayer also pro- vides damping of vibrations [2]. Therefore, it can also reduce the possible noise and vibration problems, which can occur in structures under dynamic loading. On the other hand, the nature of the polymer response is strongly frequency-dependent and also temperature-sensitive, e.g., [3], [4], or [5]. Its behavior is often described by a complex-valued dynamic shear modulus, see Figure 1. This complex-valued formu- lation allows us to transmit the information about the resonance behavior and damping of the structural element in a single complex number. For these rea- sons, the eigenvalue problem which describes the free vibrations of laminated glass becomes complex and nonlinear; see Section 2.1 for more details. To the best of our knowledge, such type of problems cannot be currently addressed with standard finite element systems used in civil engineering. As engineering practice demands easy-to-use ap- proaches of acceptable accuracy, this contribution addresses simplified models for free vibrations of lam- inated glass beams, with the goal to determine their natural frequencies and modal damping properties. Our focus is on the effective thickness approaches – one group of the simplified methods popular in the design of laminated glass structures, e.g., [6], [7], [8]. In particular, we demonstrate on selected examples the (in)accuracy of the state-of-the-art effective thick- ness approaches from literature, and we discuss our proposal how to increase their accuracy by employ- ing principles of variationally-based effective thickness methods. The structure of the paper is as follows. In Sec- tion 2, we briefly describe all approaches used in this study. The examples are introduced and the results of our comparison are discussed in Section 3. Finally, the main findings of our study are summarized in Sec- tion 4. 2. Methods Three methods for vibration analysis of laminated glass beams are used in this section. The first method is based on solving the nonlinear and complex eigen- value problem using a finite element discretization and the Newton method. This method provides us with a reference solution to assess the accuracy of effective thickness approaches. The second method is the dy- namic effective thickness method (DET) proposed by López-Aenlle and Pelayo [6]. Finally, the last method is derived from the enhanced effective thickness ap- proach (EET) by Galuppi and Royer-Carfagni for deflections under static loading [7], which we adjusted for modal analysis. 109 http://dx.doi.org/10.14311/APP.2017.13.0109 http://ojs.cvut.cz/ojs/index.php/app J. Schmidt, A. Zemanová, T. Janda et al. Acta Polytechnica CTU Proceedings 10−1 100 101 102 103 FREQUENCY [Hz] 10−3 10−2 10−1 100 101 102 103 S H E A R M O D U L U S [M P a ] 25◦C 50◦C STORAGE MODULUS LOSS MODULUS Figure 1. The frequency dependence of the real and imaginary part of the complex shear modulus for polyvinyl butyral (PVB) according to [3]. For the sake of simplicity, we focus on the most com- mon case of laminated glass configuration – a three- layered beam. However, the generalization of all meth- ods for multi-layered beams can be also found in the literature [9]. In our analysis, we consider a three- layered beam with the same material parameters for both glass layers and with constant properties along the length according to Figure 2. Then, the cross- section areas Ai and the second moments of area Ii are defined by the standard relations Ii = 1 12 bh3i , Ai = bhi, i = {1, 2, 3}, (1) where b and hi stand for the width and the thicknesses of the layers; see Figure 2. 2.1. Reference method (RM) Our reference method is based on the finite element method (FEM) together with the Newton method due to the nonlinearity of the solved eigenvalue prob- lem. The system of equations of natural vibrations is written in the FEM matrix form as follows( K(ω) −ω2M ) U = 0, (2) where ω is the natural angular frequency, U is the corresponding vector of the mode shape, K(ω) is the complex frequency-dependent stiffness matrix, and M is the real constant mass matrix. The eigenvalue problem described by Eq. (2) specifies the mode shape up to an arbitrary constant. Therefore, we use a regularization condition U 0 T(U − U 0) = 0, (3) in order to ensure the uniqueness of the solution. Here, U 0 is the initial mode shape solving the real eigenvalue problem with the constant stiffness matrix accounting only for the initial shear modulus of the polymer interlayer. For each initial eigen-pair ω0 and U 0, we solve itera- tively a linearized system of equations, derived by the Newton method [10] from Eq. (2) and Eq. (3), until the convergence is achieved. Then, the natural frequencies fHz and the loss factors η (representing damping) cor- respond to the real part and to the ratio of imaginary and real part of the complex eigen-frequencies fHz = √ Re[ω2] 2π , (4a) η = Im[ω2] Re[ω2] . (4b) This numerical solution converges for a small stop- ping tolerance to the exact solution. That is why this method is taken as the reference one, when assess- ing the accuracy of the simplified methods considered next. 2.2. Dynamic effective thickness (DET) To our best knowledge, only one effective thickness approach for modal analysis of laminated glass beams can be found in literature. This approach by López- Aenlle and Pelayo [6] is derived from a closed-form solution of free vibration for a three-layer beam with simply-supported ends. For the other boundary con- ditions, the authors adjust the formulation using the wavenumbers of an Euler-Bernoulli beam, see ahead Eq. (7) and Table 2. The analytical expression for this dynamic effective thickness has the following form hef(ω) = 3 √√√√(h31 + h33) ( 1 + Y 1 + h1 q(ω)(h1+h3) ) , (5) where Y is a geometric parameter, which depends only on the thicknesses of layers and is given by Y = 12h1h3(0.5h1 + h2 + 0.5h3) (h31 + h33)(h1 + h3) , (6) and q(ω) is a material parameter given by the identity q(ω) = G2(ω) E1h3h2β2n . (7) In this equation, G2(ω) is the frequency-dependent complex shear modulus of the interlayer, E1 is the Young modulus of glass layers, and βn is the wavenum- ber corresponding to the given boundary conditions and to the n-th mode, see ahead Table 2. Because the shear modulus G2(ω) is complex-valued, the effective thickness from Eq. (5) is also complex. Then, the complex-valued natural angular frequen- cies can be expressed analytically, similarly as for the monolithic beam ω2 = β4nE1h 3 ef(ω) 12m̄ , (8) 110 vol. 13/2017 Variationally-based effective dynamic thickness h1 h2 h3 l b Figure 2. The layout of a three-layered laminated glass beam. where m̄ = ρ1h1 + ρ2h2 + ρ3h3 (9) is the effective mass per unit of length and width. The natural frequencies fHz and loss factors η are obtained from this complex-valued angular frequency ω according to Eq. (4). 2.3. Enhanced effective thickness (EET) Firstly, we introduce the formula for the deflection- effective thickness according to Galuppi and Royer- Carfagni [7], and subsequently, we propose its exten- sion to the free vibration analysis. To shorten the notation, H is defined as a distance of centerlines of glass layers, so that H = h2 + 0.5(h1 + h3). (10) Further, Itot is the second moment of area of glass layers for the monolithic case Itot = I1 + I3 + A1A3 A1 + A3 H2, (11) and Is is the last term from (11) divided by the width of the beam b, thus Is = h1h3 h1 + h3 H2. (12) Using these quantities, the enhanced effective thick- ness for the deflection under static loading from [7] has the following form hef = 3 √√√√√ 1ζ h31 + h33 + 12Is + 1 − ζ h31 + h33 , (13) where ζ is the coefficient of shear cohesion ζ = 1 1 + I1 + I3 µItot A1A3 A1 + A3 ψ , (14) and µ is the non-dimensional ratio of the glass and interlayer stiffnesses µ = G2b E1h2 . (15) Finally, ψ is a coefficient which depends on the shape of the deflection curve and is described by the following equation ψ = ∫ l 0 (g ′′(x))2dx∫ l 0 (g ′(x))2dx , (16) Geometry length l = 1 m width b = 0.1 m thicknesses of glass h1 = h3 = 10 mm thickness of interlayer h2 = 1.52 mm Glass density ρ1 = ρ3 = 2500 kg/m3 Poisson’s ratio ν1 = ν3 = 0.22 Young’s modulus E1 = E3 = 72 GPa Interlayer density ρ2 = 1100 kg/m3 Poisson’s ratio ν2 = 0.49 Prony series for GMM taken from [3] Table 1. The properties of laminated glass beams (GMM – Generalized Maxwell Model). where the deflection curve is represented by the func- tion g(x), which describes the shape of the curve regardless of the stiffness of the beam. This deflection-effective thickness was derived by minimization of the strain energy functional for a laminated glass beam in bending under static load- ing. For its use in modal analysis, we made just two intuitive adjustments of this method: (1.) We used the complex-valued interlayer shear mod- ulus G2(ω) in Eq. (15). (2.) We replaced the shape function of the deflection under static loading in Eq. (16) with the one corre- sponding to the n-th mode shape of a monolithic beam under given boundary conditions. This adjustment leads to a complex effective thick- ness in Eq. (13), similarly to the DET approach. For the evaluation of the natural frequencies and the damping, we again use Eq. (4). 3. Comparison of methods 3.1. Examples In this section, we discuss the applicability of effec- tive thickness approaches in modal analysis and how our adjustment changes the errors of the method. The comparison is provided for three types of bound- ary conditions: simply-supported, fixed-fixed, and free-free. 111 J. Schmidt, A. Zemanová, T. Janda et al. Acta Polytechnica CTU Proceedings Mode shape functions simply-supported beam g(x) = sin nπx l fixed-fixed beam g(x) = cosh βnx− cos βnx− sinh βnl + sin βnl cosh βnl− cos βnl (sinh βnx− sin βnx) free-free beam g(x) = sinh βnx + sin βnx− sinh βnl− sin βnl cosh βnl− cos βnl (cosh βnx + cos βnx) Table 2. The overview of the n-th mode shape functions for the coordinate x ∈ 〈0, l〉 for a simply-supported, fixed-fixed, and free-free beam. (The products of the wavenumbers and the beam length βnl for the first three modes are β1l = 4.7300, β2l = 7.8532, and β3l = 10.996.) The geometry and the material properties of glass and interlayer appear in Table 1. Whereas glass is treated as an elastic material, the viscoelastic behav- ior of polymer, which is in our case polyvinyl bu- tyral (PVB), is described using a generalized Maxwell model, whose parameters are taken from [3]. Due to the temperature-sensitiveness of PVB, two different ambient temperatures were assumed – the room tem- perature 25◦C and the temperature 50◦C correspond- ing to an external panel under summer sunlight. 3.2. Evaluation of shape coefficients Recall that our extension of the enhanced effective thickness approach [7] to the modal analysis of lami- nated glass beams requires the evaluation of the shape coefficients ψ according Eq. (16). For these coeffi- cients, we need mode shape functions corresponding to the three given boundary conditions. Their overview taken from [11] is shown in Table 2. For a simply- supported beam, one closed-form expression can be written for all mode shape functions; however, the formulas for the other boundary conditions are more involved and contain a product of the wavenumbers βn and the beam length l that needs to be evaluated numerically from a characteristic equation for free vibrations [11]. Then, we used the integral formula Eq. (16) and evaluated, for all boundary conditions, the shape co- efficients ψ summarized in Table 3 using the mode shape functions for a monolithic Euler-Bernoulli beam from Table 2. With these results at hand, we can calculate the complex enhanced effective thickness and subsequently the natural frequencies and the loss factors for our comparison. 3.3. Results and discussion In this section, the results obtained by the refer- ence method (RM), the effective thickness approaches from [6] (DET) or from [7] (EET), their errors, and their comparison are shown. The first three natural frequencies fHz and loss factors η are summarized in Table 4. The errors of the effective thickness ap- proaches are evaluated against the reference method based on the finite element complex-eigenvalue solver. Shape coefficients ψ beam mode 1 2 3 simply-supported π2 l2 (2π)2 l2 (3π)2 l2 fixed-fixed 40.7 l2 82.6 l2 148 l2 free-free 10.1 l2 34.9 l2 78.2 l2 Table 3. Summary of the shape coefficients ψ for three different boundary conditions and the first three mode shapes. It is evident from the first two subtables in Table 4 that the adjusted EET approach gives the same re- sults as the DET method for the simply-supported beam for both temperatures. Both effective thickness methods give the errors in natural frequencies less than 1% and in loss factors less than 10%. For the room temperature, even the errors in loss factors are under 1%. For the two other boundary condition, the DET approach provides quite good predictions for natural frequencies with the errors under 13%. However, the errors in loss factors can be very high (up to 80%). Therefore, this method is mostly unable to provide a good estimation of damping. The adjusted EET approach helps to overcome these inaccuracies. For the fixed-fixed beam, this approach gives the largest error in natural frequencies 3% (against the 8% error by the DET) and in loss fac- tors 9% (against the 44% error by the DET). For the free-free beam, the largest value of errors in natural frequencies is 3% (against the 13% error by the DET) and in loss factors 19% (against the 78% error by the DET). This demonstrates that our intuitive extension of the variationally-based enhanced effective thickness concept from [7] provides better (or for the simply- supported beam the same) estimates of both - the natural frequencies and loss factors for laminated glass beams. 112 vol. 13/2017 Variationally-based effective dynamic thickness Simply-supported beam at 25◦C Mode Natural frequency [Hz] Error [%] Mode Loss factor [%] Error [%] RM DET EET DET EET RM DET EET DET EET 1 52.03 52.07 52.07 0.09 0.09 1 1.93 1.93 1.93 -0.25 -0.25 2 197.9 198.5 198.5 0.30 0.30 2 4.24 4.23 4.23 -0.32 -0.32 3 419.1 421.4 421.4 0.56 0.56 3 6.15 6.14 6.14 -0.20 -0.20 Simply-supported beam at 50◦C Mode Natural frequency [Hz] Error [%] Mode Loss factor [%] Error [%] RM DET EET DET EET RM DET EET DET EET 1 30.81 30.97 30.97 0.53 0.53 1 24.41 22.17 22.17 -9.18 -9.18 2 108.1 108.5 108.5 0.42 0.42 2 16.89 16.25 16.25 -3.81 -3.81 3 232.0 233.1 233.1 0.47 0.47 3 14.46 13.90 13.90 -3.84 -3.84 Fixed-fixed beam at 25◦C Mode Natural frequency [Hz] Error [%] Mode Loss factor [%] Error [%] RM DET EET DET EET RM DET EET DET EET 1 110.2 115.4 111.3 4.78 1.09 1 5.42 3.09 5.08 -43.05 -6.27 2 285.3 301.4 292.7 5.65 2.59 2 7.15 5.29 6.50 -25.94 -9.02 3 527.2 556.3 541.1 5.53 2.65 3 7.86 6.74 7.56 -14.28 -3.82 Fixed-fixed beam at 50◦C Mode Natural frequency [Hz] Error [%] Mode Loss factor [%] Error [%] RM DET EET DET EET RM DET EET DET EET 1 60.06 64.63 60.21 7.62 0.25 1 12.99 18.74 12.72 44.19 -2.14 2 160.2 164.6 160.9 2.79 0.46 2 11.51 14.91 11.80 29.52 2.49 3 308.8 314.0 310.2 1.66 0.43 3 10.98 12.89 10.87 17.39 -1.03 Free-free beam at 25◦C Mode Natural frequency [Hz] Error [%] Mode Loss factor [%] Error [%] RM DET EET DET EET RM DET EET DET EET 1 117.8 115.4 118.4 -2.00 0.56 1 1.73 3.09 1.50 78.05 -13.64 2 307.9 301.4 314.1 -2.12 2.00 2 4.13 5.29 3.37 28.20 -18.41 3 568.4 556.3 584.5 -2.12 2.84 3 5.77 6.74 5.04 16.74 -12.64 Free-free beam at 50◦C Mode Natural frequency [Hz] Error [%] Mode Loss factor [%] Error [%] RM DET EET DET EET RM DET EET DET EET 1 73.95 64.63 74.62 -12.60 0.90 1 26.44 18.74 25.39 -29.13 -3.96 2 170.5 164.6 175.6 -3.45 2.97 2 20.85 14.91 22.20 -28.51 6.48 3 322.5 314.0 325.3 -2.66 0.86 3 17.57 12.89 18.20 -26.61 3.62 Table 4. Natural frequencies and loss factors for the three-layer laminated glass beam for the first three modes determined by the reference method (RM), the dynamic effective thickness method (DET), and the adjusted enhanced effective thickness approach (EET) with their errors against the reference method (RM). 113 J. Schmidt, A. Zemanová, T. Janda et al. Acta Polytechnica CTU Proceedings 4. Conclusions Finally, we would like to summarize that in this paper • we verified the effective thickness approaches against the finite element complex-eigenvalue solver, and • we proposed a new dynamic effective thickness con- cept derived from [7]. It follows from our comparison that this extended enhanced effective thickness method • provides the same results for the simply-supported beam as the dynamic effective thickness from [6] with the errors less than 1% in natural frequencies and 10% in loss factors, and • improves the estimates of natural frequencies and loss factors for the other two boundary conditions with the errors less than 3% in natural frequencies and 19% in loss factors. In our future work, we would like to justify this intuitive approach by rigorously extending the variationally-based derivation of the effective thickness from [7] to the dynamic problems. Acknowledgements This publication was supported by the Czech Science Foundation, the grant No. 16-14770S and by the Grant Agency of the Czech Technical University in Prague, grant No. SGS17/043/OHK1/1T/11. References [1] M. Haldimann, A. Luible, M. Overend. Structural Use of Glass, vol. 10 of Structural Engineering Documents. IABSE, Zürich, Switzerland, 2008. 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Dynamic effective thickness in laminated-glass beams and plates. Composites: Part B 67:332–347, 2014. [7] L. Galuppi, G. F. Royer-Carfagni. Effective thickness of laminated glass beams: New expression via a variational approach. Engineering Structures 38:53 – 67, 2012. doi:10.1016/j.engstruct.2011.12.039. [8] L. Galuppi, G. Royer-Carfagni. The effective thickness of laminated glass: Inconsistency of the formulation in a proposal of EN-standards. Composites Part B: Engineering 55:109–118, 2013. [9] F. Pelayo, M. López-Aenlle. Natural frequencies and damping ratios of multi-layered laminated glass beams using a dynamic effective thickness. Journal of Sandwich Structures and Materials 0:1–25, 2017. doi:10.1177/1099636217695479. [10] K. Schreiber. Nonlinear Eigenvalue Problems: Newton-type Methods and Nonlinear Rayleigh Functionals. Ph.D. thesis, Technischen Universitat Berlin, 2008. [11] R. W. Clough, J. Penzien. Dynamics of structures. Computers & Structures, Inc, 2003. 114 http://dx.doi.org/10.1016/j.compstruct.2016.05.105 http://dx.doi.org/10.1016/j.polymertesting.2014.04.011 http://dx.doi.org/10.1016/j.euromechsol.2017.01.006 http://dx.doi.org/10.1016/j.conbuildmat.2014.04.003 http://dx.doi.org/10.1016/j.engstruct.2011.12.039 http://dx.doi.org/10.1177/1099636217695479 Acta Polytechnica CTU Proceedings 13:110–115, 2017 1 Motivation 2 Methods 2.1 Reference method (RM) 2.2 Dynamic effective thickness (DET) 2.3 Enhanced effective thickness (EET) 3 Comparison of methods 3.1 Examples 3.2 Evaluation of shape coefficients 3.3 Results and discussion 4 Conclusions Acknowledgements References