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Cantilevered laminated glass balustrades: the Conjugate Beam Effective Thickness method—part I: the analytical model


The proper assessment of shear coupling is necessary for the evaluation of laminated glass performance between the bounding layered and monolithic limits. The most common simplified design approach consists in defining the effective thickness, i.e., the thickness of a monolithic section with equivalent flexural properties. Cantilevered laminated glass balustrades are common applications of structural glass. However, the use of existing effective thickness methods presents strong limitations for their design. Here, the conjugate beam effective thickness (CBET) method is presented, based on the conjugate beam analogy recently proposed to evaluate the response of inflected laminates formed by external elastic beams bonded by an adhesive ply. The conjugate beam analogy, applied to laminated glass beams, allows accurate evaluation of the shear stress transmitted by the interlayer, based on the response of a monolithic conjugate beam, with the option to constrain relative sliding of plies at a beam end. Once the shear coupling is known, the effective thickness may be evaluated with the proposed CBET model by comparing the maximum stress and deflection of the laminated beam with a monolithic Euler–Bernoulli beam. The CBET method’s formulas can be readily applied to evaluate the maximum stress and cantilever free-end deflection for different load and boundary conditions, representative of cantilevered laminated glass balustrade supported in a U-profile.

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  1. The contours shown in Figure 1 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. It can be verified that \(q''(x)=-[M(x)+N(x)H]\), and hence Eqs. (2.10) formally coincides with (2.4)

  3. Notice that, hereafter, the term equivalent is used for the beam represented in Fig. 3a subject to a load per unit length q(x) accounting for the shear coupling. On the other hand, the term effective refers to the beam of Fig. 3b, which is a monolithic beam with effective thickness, subject to the same load p(x) of the real laminate.

  4. The axial force could be, equivalently, evaluated by solving Eq. (2.5), with appropriate boundary conditions on N(x) and \(N'(x)\) at the beam’s ends. However, the conjugate beam analysis is prefered, since it provides a synthetic view of the shear-coupling effect.

  5. Again, \(\langle \rangle \) denotes the Macaulay brackets. Hence, \(\langle x-a \rangle ^0\) represents the heaviside unit step function acting at \(x=a\), while \(\langle x-a \rangle ^1\) is the ramp function (Yavari et al. 2000; Falsone 2002).

  6. The viscoelastic interlayer response is time- and temperature-dependent. Depending upon the polymer type, service temperature and characteristic load-duration, the secant shear modulus of the interlayer may vary approximately from 0.01 MPa (for PVB interlayers at 50 \(^{\circ }\)C, under permanent load) up to 420 MPa (for stiff PVB and ionoplast interlayers at 0 \(^{\circ }\)C, under 1 s loads).

  7. More precisely, when the laminate is supported at an interior point, say \(x={\bar{x}}\), there is a jump in shear \(V({\bar{x}}^-)\ne V({\bar{x}}^+)\), where \(V'(x)=p(x)\). This results in a jump discontinuity of the shear in the conjugate beam \( {\widetilde{V}}(x)=EI\ \alpha ^2 (\mu ^2-1)\frac{V(x)}{H} \).

  8. Note the maximum deflection, in absolute values, is attained between the two supports where a is large relative to L (for example, for \(a>\sim 6 L\) in the case of beam under end load).



Conjugate Beam Effective Thickness


Enhanced Effective Thickness


Polyvinyl butyral (a safety glass interlayer)

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

xyz :

Axial, through-the-thickness and lateral beam directions

E :

Young’s modulus of glass

G :

Secant shear modulus of the interlayer

\(\nu \) :

Poisson’s ratio

\(A_i\) :

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

\(p(x) = p \) :

Transverse load per unit length acting on the laminated glass beam

F :

Concentrated transverse load acting on the laminated glass beam


Out-of-plane deflection of the laminated glass beam

\(\tau (x)\) :

Shear stress transmitted by the interlayer


Distributed torque per unit length due to the shear coupling


Axial force acting in the glass plies


Bending moment due to the external load


Transverse (shear) force acting in the glass plies

\(\mu \) :

Geometrical nondimensional parameter

\(\alpha \) :

Parameter dependent on glass and interlayer mechanical properties

\({\widetilde{P}}\) :

Effective axial load

\({\widetilde{p}}(x)\) :

Effective distributed transverse load

\({\widetilde{F}}\) :

Effective concentrated transverse load

\({\widetilde{v}}(x)\) :

Out-of-plane deflection of the conjugate beam


Fictitious transverse load per unit length acting on the laminated glass beam

\(q_{i}(x)\) :

Fictitious transverse load per unit length acting on the ith glass ply

\(n_{i}(x)\) :

Fictitious axial load per unit length acting on the ith glass ply

\(M_i(x)\) :

Bending moment associated to the fictitious load on the ith glass ply

\(I_{eff;w}\) :

Effective moment inertia

\(v_{eff}(x)\) :

Out-of-plane deflection of the effective beam

\(y_i\) :

Distance from the mid-plane of the ith glass ply

\(\sigma _i(x,y_i)\) :

Surface stress along the ith ply

\(\sigma _{eff}\) :

Axial stress along the effective beam

\(\lambda \) :

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


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Galuppi, L., Nizich, A.J. Cantilevered laminated glass balustrades: the Conjugate Beam Effective Thickness method—part I: the analytical model. Glass Struct Eng 6, 377–395 (2021).

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  • Laminated glass
  • Balustrades
  • Glass design
  • Effective thickness
  • Analytical methods