The Conjugate Beam Effective Thickness Method Laminated Glass, Effective Thickness, Structural Analysis

The structural performance of laminated glass is strongly dependent on the shear coupling offered by the interlayer between the bounding layered and monolithic limits of the glass plies. The most common simplified design approach consists of defining the effective thickness, i.e., the thickness of a monolithic section with equivalent flexural section properties. The Enhanced Effective Thickness (EET) method has been verified to estimate deflection in laminated glass for a range of load and boundary conditions for two-, three-, and multi-ply beams; however, for some static schemes, the EET method is less accurate for predicting stress. The recently proposed Conjugate Beam Effective Thickness (CBET) method, initially developed for cantilevered laminated glass balustrade applications, accounts for the relative displacement of glass plies across the interlayer for a range of loads and statically determinate boundary conditions. In this paper, the CBET method is extended to the evaluation of two-ply simply supported beams under concentrated, uniformly, and tapered distributed out-of-plane loads. Predicted deflection- and stress-effective thickness obtained from effective thickness methods is compared with finite element model results in illustrative examples, demonstrating improved accuracy. Closed-form formulas are summarized in tables to facilitate the practical application of the CBET method in the design practice.


Introduction
Laminated Glass (LG) is a composite material formed by two or more glass plies bonded with a polymeric interlayer(s) to achieve the desired stiffness, strength, and reliability. LG stiffness and strength are governed by the shear coupling of the polymer between the glass plies, as first mentioned by Hooper (1973), and a function of the section properties and external load and boundary conditions, as highlighted by Galuppi and Royer-Carfagni (2012).
LG beams in flexure have a varying distribution of internal flexural and axial forces, depending on the interlayer shear modulus, G. As observed in Figure 1, for a simply supported LG beam with a concentrated load, the stress distribution on the tensile surface of the beam from axial and flexural forces is hyperbolic between the bounding layered (where → 0) and monolithic (where → ∞) limits. The conjugate beam analogy proposed by Galuppi and Royer-Carfagni (2020) allows for the determination of ply surface stresses based on the precise evaluation of axial forces in two glass plies, due to the shear stress transmitted by the interlayer, as observed in Figure 2.  The complex structural mechanics of an LG beam can be simplified into a monolithic section with an Effective Thickness (ET) for evaluation with classical beam analysis equations corresponding to the associated loading and boundary conditions. Various effective thickness formulations have been proposed for the design of LG in flexure. The most well-known effective thickness methods are: (a) the 1 Assumed structural parameters for the beam are L = 3150 mm, b = 1000 mm, h 1 = h 2 = 10 mm, t = 0.76 mm, E = 70.0 GPa, with a P = 1 kN concentrated load distributed across the width and through the lite thickness. Wölfel-Bennison effective thickness method recorded in ASTM E1300 (2016), based on formulas derived by Bennison and others (Calderone et al. 2009) following the seminal work by Wölfel (1987), and (b) the Enhanced Effective Thickness (EET) method for beams proposed by Galuppi and Royer-Carfagni (2012) and Galuppi et al. (2013). These methods have limitations for evaluation of stress-ET for concentrated loads, and for evaluation of deflection-and stress-ET where slip occurs between plies in cantilever support conditions (Nizich and Galuppi 2019).
The Conjugate Beam Effective Thickness (CBET) method for cantilevered LG balustrades was recently proposed by Galuppi and Nizich (2021) and compared to existing effective thickness models by Nizich and Galuppi (2022). For the application of cantilevered LG supported in a conventional U-profile, particular attention was given to constrained and unconstrained slip between two glass plies at the support. Here, the method is extended to simply supported LG beams under concentrated, uniformly, and tapered distributed loads.

Effective Thickness Models
The method consists of defining the LG effective thickness (i.e., the thickness of a monolithic section with equivalent flexural section properties) between the bounding layered and monolithic limits. The reference geometry, shown in Figure 3, is a LG beam of width b, composed of two glass plies with a Young's modulus E and thickness ℎ 1 and ℎ 2 , respectively, bonded by an interlayer of thickness ≪ ℎ 1 , ℎ 2 , with a shear modulus G. With reference to Figure 3, the section properties are defined as: where and are the moment of inertia of the beam at the layered and monolithic limits, respectively.

Enhanced Effective Thickness Method
The Enhanced Effective Thickness (EET) method assumes that the deflection of a LG beam has the same shape as that of a monolithic beam subjected to the same boundary and loading conditions, with an effective inertia I ef , being the weighted harmonic mean of the moment of inertia at the layered I L and monolithic I M limits. Under this assumption, the optimal response of the LG beam is evaluated, in a variational framework, by minimizing the total energy. This method is described in the Italian Glass Code CNR-DT 210 (2013), appears in the draft of the upcoming Eurocode on structural glass CEN/TS 19100-2 (2021), and has been extended to three-and multi-ply LG beams (Galuppi and Royer-Carfagni 2014) with equivalent accuracy. According to the EET model, the deflection-and stress-ET, for a twoply LG beam may be evaluated as: where ∈ [0,1] is the nondimensional shear coupling coefficient, which corresponds to the beam's geometry, loading and boundary conditions, and glass and interlayer mechanical properties, evaluated as: (3) The coefficient accounting for the boundary and load conditions of the beam Ψ is tabulated in Table 1 for select applications, as evaluated by Galuppi et al. (2013) for traditional combinations of loading and boundary conditions, and by Nizich and Galuppi (2022) for cantilevered LG balustrades, e.g., for cantilevered beams with a fixed end support as well as overhanging one support under different loading conditions.
The EET method has demonstrated adaptability to variations of load and boundary conditions for beams in flexure. However, as discussed by Nizich and Galuppi (2022), the traditional EET approach cannot account for the possibility of rigid axial displacement of one glass ply with respect to the other, affecting the interlayer shear strain and, consequently, the shear coupling and the LG beam flexural response. Its accuracy decreases in applications with asymmetric boundary conditions, point loads, and/or where relative displacement between glass plies does not equal zero at a known position (e.g., at a beam-end, midspan, etc.).

Conjugate Beam Effective Thickness Method
The CBET method recently proposed by Galuppi and Nizich (2021) is based on the conjugate beam analogy proposed by Galuppi and Royer-Carfagni (2020). According to the conjugate beam analogy, the axial force (equal and opposite in the two glass plies) is related to the external bending moment M(x) by: As defined, is a geometrical nondimensional square root of the ratio of monolithic and layered limits, dependent on ℎ 1 , ℎ 2 , and , while α also depends on the mechanical properties of both glass and the interlayer. The shear stress transmitted by the interlayer is related to the axial force by: At the layered limit, the axial force is null, corresponding to null shear stress transmitted by the interlayer.
Galuppi and Royer-Carfagni (2020) demonstrated that the axial force ( ) corresponds to the deflection of a conjugate beam subjected to a transverse load, indicated as � in Figure 4, similar in shape to that load acting on the LG beam, and to a tensile axial force � , both dependent on the shear coupling. The boundary conditions for the conjugate beam depends not only on the support conditions for the laminate, but also on the axial constraints for slip between the two glass plies. This analogy allows evaluation of the axial force and shear coupling of the LG beam, simply by studying the deflection of the conjugate beam ( Figure 4). Indeed, the axial force ( ) corresponds to the deflection of the conjugate beam at , while the shear stress ( ) is the slope of the elastic curve of the conjugate beam at the same point. Once the axial load is evaluated by means of the conjugate beam analogy (Eq. 4), this may be used to evaluate the effective thickness of the LG beam.

Deflection-Effective Thickness
The deflection of the LG beam corresponds to the deflection of a beam with moment of inertia , subjected to a fictitious bending moment (accounting for the shear stress transmitted by the interlayer) equal to ( ) + ( ) . Hence, the deflection ( ) may be calculated by solving the differential equation: with appropriate boundary conditions. To find the deflection-ET (h w ), ( ) must be compared with the maximum deflection of an equivalent monolithic beam, of thickness h w and inertia = ℎ 3 12 ⁄ , i.e., with the solution of: Since the deflection of the two beams are different in shape, because they are subjected to qualitatively different bending moments, the comparison is made in terms of their maximum values. It may be verified that this provides a deflection-ET formula in the form of: where geometric ( ) and coupling ( ) coefficients are dependent on the loading and boundary conditions, recorded in Table 2. The effective laminate thickness approaches the equivalent monolithic limit for stiff interlayers where → 0.

Stress-Effective Thickness
The i-th glass ply ( = 1, 2) is subjected to a part of the fictitious bending moment, proportional to its inertia, and to the axial force ( ). Its surface stress could be hence evaluated as: To evaluate the corresponding stress-effective thickness, this value is compared with the maximum stress acting in a monolithic effective beam of thickness ℎ ; , subjected to the same bending moment ( ) of the LG element, which is: Again, due to the different shape of bending moment and normal force, the maximum stresses ( ) and ; ( ) could be located at different locations along the beam, and the comparison should be made in terms of their maximum value. Therefore, the coupling coefficient for stress can be different from , when the maximum LG stress and effective-stress occur at different locations, as recorded in Table 2. Nizich and Galuppi 2022 verified that this provides stress-ET formula in the form: The CBET method was originally developed to address the structural performance of cantilevered LG balustrades. Here, the CBET method is applied to the analysis of simply supported LG beams under concentrated, uniformly distributed, and tapered distributed loads.

Evaluation and Comparison
To compare the analytical effective thickness models, numerical analyses have been performed with Strand7 (2021), by modeling the LG beam as a 2D sandwich section, using four-noded plane-stress finite elements. The mesh was scaled proportional to the interlayer thickness and elements maintained a maximum aspect ratio of 2:1, with three subdivisions through the interlayer thickness. Loads and boundary conditions were applied in accordance with the modeling and analysis criteria discussed by Nizich and Galuppi (2022).
To perform accurate comparison of detailed 2D numerical models with the analytical beam models, concentrated forces (representative of a quasi-concentrated load) are distributed through the thickness of glass section, thus avoiding a nodal stress concentration. Evaluated models are representative of more complex 3D applications; marginal differences are expected and are in general agreement with engineering theory.

Simply Supported LG Beam Example
The beam section, length, and loads used for original comparison of the EET method with a numerical model (Galuppi and Royer-Carfagni 2012) are adopted for the four simply supported beam load cases represented in Figure  In Figures 6 -9 below, deflection-and stress-effective thickness for each beam are compared, as bounded by the layered and monolithic limits. We observe good correlation of the EET, CBET, and numerical methods for the distributed load cases in Figures 6 and 7. Indeed, further accuracy in comparison to the numerical model is obtained with the CBET method. For point load cases in Figures 8 and 9, we observe good correlation of the EET, CBET, and numerical methods for deflection-ET. Significant deviation of stress-ET is observed between the EET and CBET methods. Remarkably, the numerical results coincide with the CBET method.

Cantilevered LG Beam Example
Many cantilevered laminated glass balustrades with an approximate L = 1100 mm cantilever free length have been constructed with a pair of fully tempered 6 mm plies (nominal thickness) around a 1.52 mm interlayer to resist live loads. We compare the effective thickness for this geometry for two cantilevered beam support conditions in Table 3 and evaluate deflection and stress using classical formulas for the corresponding loading and boundary conditions. Assumed structural parameters for all cases are: L = 1100 mm, a = 50 mm, b = 1500 mm, h 1 = h 2 = 5.56 mm, t = 1.52 mm, and E = 71.7 GPa as per the minimum thickness and mechanical properties in ASTM E1300. A 0.73 kN/m service level design load (unfactored) is applied to the free end of the beam as a P = 1.095 kN point load. In Figure 10 below, deflection-and stress-effective thickness for each support condition is compared. We observe good correlation of the EET, CBET, and numerical methods of deflection-ET for a fixed end support (case i), however, only the CBET method predicts the response of for a beam overhanging one support (case ii). In Figure 11, we observe a significant difference in the predicted deflection and stress between the two idealized support cases, corresponding to the classical beam equations in Table 3. A fixed-end support (Case i) implies that the plies are constrained from relative slip at the support. Given that many balustrades are installed with nonadhesive grouts, or discrete wedge blocks, we assume that a slip-critical connection is generally not developed in conventional installations. If a fixedend support boundary condition is assumed, and not realized in construction, the predicted deflection is at risk of increasing by 118% and maximum surface stress by 82% to perform as a cantilevered beam overhanging one support (Case ii) for the evaluated conditions. Variations in deflection, glass stress, and interlayer shear strain for different supports are observable in a 3D-layered finite element model ( Figure 12) with an interlayer shear stiffness of G = 10 MPa. LG balustrade with support that constraints slip between plies at the support (left), and support that allows slip (right). Interlayer shear strain and glass maximum principal surface stress 11 are plotted on the deformed shape.

Current Work
Task groups within the ASTM E06.52 subcommittee and the European Committee for Standardization (CEN/TS 2021) are currently collaborating on new structural glass standards. In support of this work, a new ASTM standard for the effective thickness evaluation of laminated glass is under preparation to encompass methods for assessment of flexural and torsional section properties in two-, three-, and multi-ply LG sections.

Conclusion
The Conjugate Beam Effective Thickness (CBET) method has been verified for both uniformly distributed and concentrated loads for a range of statically determinate beam boundary conditions, applicable to the design of one-and two-edge supported glass. The method is simple to use and allows for evaluation of deflection and stress with classical engineering beam equations. Recent advances in effective thickness methods can inform the discussion on design strength for laminated glass and inform the development of structural glass building codes.