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Cantilevered laminated glass balustrades: the Conjugate Beam Effective Thickness method—part II: comparison and application

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Abstract

Cantilevered laminated glass installed in continuous U-profile base shoes are regularly constructed as structural glass guards, parapets, and windscreens. 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 lites. The most common simplified design approach consists of defining the effective thickness, i.e., the thickness of a monolithic section with equivalent properties. However, established effective thickness methods lack correlation with stress and deflection observed in numerical models simulating bearing support of cantilevered laminated glass in an ordinary U-profile. The analytical Conjugate Beam Effective Thickness (CBET) method proposed in Part I of the present work accounts for the influence of different boundary and loading conditions, and is readily applied to evaluate the flexural performance of two-ply cantilevered laminated glass beams. In this paper, results evaluated with the proposed CBET method are compared with existing analytical methods and numerical results for case study examples, demonstrating improved accuracy with respect to existing effective thickness methods for cantilevered laminated glass beams. The obtained closed-form formulas for evaluation of deflection- and stress-effective thickness are summarized in tables to facilitate the practical application of the CBET method in the design practice.

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Notes

  1. The contours shown in Fig. 2 have been evaluated by considering a LG element composed of 7.42 mm glass–1.52 mm interlayer–7.42 mm glass section. Considered mechanical properties of glass are \(E=71.7\) GPa, \(\nu =0.22\), and the interlayer are \(G=10\) MPa, \(\nu =0.49\).

  2. As in all ET models, here the secant stiffness approximation (Galuppi and Royer-Carfagni 2012a) used in engineering design practice for LG, is adopted. This consists of modeling the interlayer as an elastic material, with a shear secant elastic modulus G assumed equal to the relaxed modulus under constant strain at service temperature, for a duration appropriate for the characteristic design action.

  3. Discrete wedge blocks used in dry glazing result in gaps to continuous support. Stress concentrations may form around supports or at unsupported free edges. The simplified beam approach studied here may not be applicable for final design with discrete support conditions.

  4. Since the W–B method cannot account for different boundary conditions, it provides the same results for case (i) and (ii).

  5. Polymers typically used as an interlayer experience a nonlinear rubber-like response under large deformations, which can be analyzed with a geometric nonlinear analysis, together with a nonlinear viscoelastic interlayer material model (Hooper et al. 2012; Liu et al. 2012; Zhang et al. 2015), and can not accurately represent hyper-elastic interlayer material performance from concentrated transverse forces.

Abbreviations

ASTM:

ASTM International

CBET:

Conjugate Beam Effective Thickness

EET:

Enhanced Effective Thickness

ET:

Effective Thickness

FEA:

Finite Element Analysis

LG:

Laminated Glass

PVB:

Polyvinyl Butyral (a safety glass interlayer)

W-B:

Wölfel–Bennison effective thickness

L :

Length of the the cantilever span

a :

Length of simply-supported span

b :

Beam width

d :

Distance from free end of cantilevered span

i :

1, 2 Glass ply numbering

\(h_i\) :

Thickness of the ith glass ply

t :

Thickness of the interlayer

x, y, z :

Axial, through-the-thickness and lateral beam directions

E :

Young’s modulus of glass

G :

Secant shear modulus of the interlayer

\(A_i\) :

Cross sectional area of the ith glass ply

\(I_i\) :

Moment of inertia of the ith glass ply

H :

Distance between the midplane of two glass plies

\(I_L, I_M\) :

Moment of inertia of the beam at the layered and monolithic limits, respectively

p :

Transverse load per unit length acting on the laminated glass beam

F :

Concentrated transverse load acting on the laminated glass beam

\(I_{{ eff};w}\) :

Effective moment inertia

\(\mu \) :

Geometrical nondimensional parameter

\(\alpha \) :

Parameter dependent on glass and interlayer mechanical properties

\(\lambda \) :

Load and boundary condition pair hyperbolic coefficient

\(h_w\) :

Deflection-effective thickness

\(h_{\sigma ;i}\) :

Stress-effective thickness in the ith glass ply

\(h_{\sigma ;i;{ int}}\) :

Stress-effective thickness of the compressive surface of the ith glass ply

\(d_i\) :

Centroidal distance of the cross section to the ith glass ply

\(I_s\) :

Difference of moment of inertia between \(I_M\) and \(I_L\), divided by the beam width (W–B method)

\(\Gamma \) :

Shear coupling coefficient (W–B method)

\(h_{w;W{-}B}\) :

Deflection-effective thickness (W–B method)

\(h_{\sigma ;i;W{-}B}\) :

Stress-effective thickness in the ith glass ply (W–B method)

\(\eta \) :

Shear coupling coefficient (EET method)

\(\Psi \) :

Load and boundary condition pair coefficient (EET method)

g(x):

Shape function (EET method)

\(h_{w;{ EET}}\) :

Deflection-effective thickness (EET method)

\(h_{\sigma ;i;{ EET}}\) :

Stress-effective thickness in the ith glass ply (EET method)

\(\delta _{{ max}}\) :

Maximum out-of-plane deflection of effective beam (ET methods)

\(\delta _{\textit{max};W{-}B}\) :

Maximum out-of-plane deflection of effective beam (W–B methodl)

\(\delta _{\textit{max};{ EET}}\) :

Maximum out-of-plane deflection of effective beam (EET method)

\(\delta _{\textit{max};\textit{CBET}}\) :

Maximum out-of-plane deflection of effective beam (CBET method)

\(e_{W{-}B,w}\%\) :

Percent difference of out-of-plane deflection in the W–B method compared to the CBET method

\(e_{\textit{EET},w}\%\) :

Percent difference of out-of-plane deflection in the EET method compared to the CBET method

\(\sigma _{i;\textit{max}}\) :

Maximum tensile stress of effective beam (ET methods)

\(\sigma _{\textit{max};W{-}B}\) :

Maximum tensile stress of effective beam (W–B method)

\(\sigma _{\textit{max};{ EET}}\) :

Maximum tensile stress of effective beam (EET method)

\(\sigma _{\textit{max};\textit{CBET}}\) :

Maximum tensile stress of effective beam (CBET method)

\(e_{W{-}B,\sigma }\%\) :

Percent difference of tensile stress in the W–B method compared to the CBET method

\(e_{\textit{EET},\sigma }\%\) :

Percent difference of tensile stress in the EET method compared to the CBET method

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  1. Simpson Gumpertz and Heger Inc., 1625 Eye Street NW, Suite 900, Washington, DC, 20006, USA

    Adam J. Nizich

  2. Department of Engineering and Architecture, University of Parma, Parco Area delle Scienze 181/A, 43124, Parma, Italy

    Laura Galuppi

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Nizich, A.J., Galuppi, L. Cantilevered laminated glass balustrades: the Conjugate Beam Effective Thickness method—part II: comparison and application. Glass Struct Eng 7, 23–43 (2022). https://doi.org/10.1007/s40940-021-00165-7

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