Dealing with experimental data, inherently prone to a variety of possible approximations and errors, a critical approach in the selection of the target points was preferred to a blind implementation of the procedure. Consequently, target points yielding an overall smoother interpolation of the general trend of the mastercurve were chosen, rather than slavishly connecting the terminal parts of all the connected experimental branches. This approach also took into consideration that the terminal parts of the experimental curves generally have a higher regularity than the initial ones. This phenomenon is also discussed in §4.2.2 and further highlighted in Figure 5.14 by the juxtaposition of branches which do not strictly require a linking function

lar than the following trend). With this approach, target points and linking functions

:

curves prove to be less regu-

$$t\_L = \mathbf{1} \mathbf{0}^{t\prime} \qquad , \qquad J\_L^{-1} = \mathbf{1} \mathbf{0}^{l\prime}$$

Substituting the general equation yields the final function :

$$\begin{aligned} f\_L^{-1}(t) &= 10^\wedge \left( -2.84 \cdot 10^{-3} \ln^3 t + 0.142 \ln^2 t - 2.42 \ln t + 15.6 \right) \\ &\text{for } t \in [t\_A, t\_B]. \end{aligned}$$

The same procedure has been applied in an analogous manner for all the gaps in all moduli for the three materials in different intervals.

Dealing with experimental data, inherently prone to a variety of possible approximations and errors, a critical approach in the selection of the target points was preferred to a blind implementation of the procedure. Consequently, target points yielding an overall smoother interpolation of the general trend of the mastercurve were chosen, rather than slavishly connecting the terminal parts of all the connected experimental branches. This approach also took into consideration that the terminal parts of the experimental curves generally have a higher regularity than the initial ones. This phenomenon is also discussed in §4.2.2 and further highlighted in Figure 5.14 by the juxtaposition of branches which do not strictly require a linking function (e.g. and , where the initial part of the curves prove to be less regular than the following trend). With this approach, target points and linking functions are presented in Figure 5.19.

*Figure 5.19 - target points for linking branches*

175

174

regression was carried out with a linear least squares method, henceforth generating

*Figure 5.18 - scheme of the linking procedure27*

slopes of the curves and were worked out, to be then used as boundary con-

Coefficients , , and were found by solving the linear system granting the observance of the established boundary conditions. Numerical results in this exam-

>

> >

> >

 and

>

of the target points and

> for

in the

From the linear regression, ordinates

ditions for the polynomial interpolation. Given:

ple are given to be later implemented in the general equation:

The plain black curve in Figure 5.18 is the polynomial solution

the subscript is used to indicate a linking function. The solution

the gap between target points and in the transformed variables , , where

original over environment can be found by applying the following

>

inverse transformations:

the dotted lines in the figure.

Portions of the linking branches have later been cut, to avoid superposition with the experimental data, in the - small - portions where the curves were overlapping (experimental data were always preferred to points of the linking functions).

The final operation on the data at this stage, to enhance both reliability and computational load of Prony series interpolations, was to re-sample the data sets among a fixed number of equally distributed points in logarithmic time scales. Provided with a large distribution of points (where stands for either or ), dense in some areas and sparser in others, a new time array is defined to uniformly distribute the points along the curve:

Accordingly, re-sampled values of the moduli were calculated as the average of the nearest ten points to the re-sampled abscissa in the original data set. An example of re-sampled data is presented in the following figure for 30 points only, whereas two thousand points were used in the actual data analysis.

*Figure 5.20 - example of a thirty-points re-sampling*

#### **<IP_ADDRESS> WLF regression (A proposed fitting procedure)**

*Mastercurves, produced and perfected in previous paragraphs (cfr. §<IP_ADDRESS>, Table 5.6 and Figure 5.15), are inherently tied to the set of shift factors. Within the iterative procedure described in*

Figure 5.13, to allow for the use of experimental results in terms of relaxation and creep moduli for arbitrary temperatures within the investigated range, a generalization of the punctual temperature-shift correlations needs to be provided.

176

177

A now popular formulation for time-temperature superposition was proposed in 1980 by Williams, Landel and Ferry [114], in the form of an equation with two coef-

 A regression procedure is herein proposed for shift-factor distributions and cor-

The procedure is based on the WLF equation and aims to find the best fitting

of

The outcome allows to compute the normalized sum of square differences Γ

First, a three-dimensional array of values was created, where

The space can be presented as a two-dimensional matrix28 in , in which each element holds a pair of values for and for and . For , values for the two constants in each element of the matrix are set to be linearly ranging among predefined arbitrary limits, so that and are spaces centred around initial best-

For a given set of shift factors and temperatures , the regression base

For each element in (i.e. for each pair of constants , ), the value of the right part of the WLF function is calculated for all temperatures , yielding

The sum of square differences for all elements was then arranged in

28 The shape of the matrix arbitrary: any different space can have been adopted, without jeopardizing the discussed procedure but possibly affecting its accuracy. In the following, will be assumed, a choice moti-

For each iteration , the pair yielding minimum value of is found and

vated by the fact that there is no a-priori insight on the best fitting values of any of the two constants.

associated to a specific couple of constants and in .

ficients and (cfr. §<IP_ADDRESS>).

responding test temperatures.

the WLF regression (cfr. §<IP_ADDRESS>).

designates the progression in the iteration.

coefficients and .

guess values and :

is described as:

.

 × .

A now popular formulation for time-temperature superposition was proposed in 1980 by Williams, Landel and Ferry [114], in the form of an equation with two coefficients and (cfr. §<IP_ADDRESS>).

$$-\log\_N a\_T = \alpha = \frac{-\mathcal{C}\_1 \{T - T\_{ref}\}}{\mathcal{C}\_2 + \{T - T\_{ref}\}}$$

A regression procedure is herein proposed for shift-factor distributions and corresponding test temperatures.

The procedure is based on the WLF equation and aims to find the best fitting coefficients and .

The outcome allows to compute the normalized sum of square differences Γ of the WLF regression (cfr. §<IP_ADDRESS>).

First, a three-dimensional array of values was created, where designates the progression in the iteration.

The space can be presented as a two-dimensional matrix28 in , in which each element holds a pair of values for and for and . For , values for the two constants in each element of the matrix are set to be linearly ranging among predefined arbitrary limits, so that and are spaces centred around initial bestguess values and :

$$\mathcal{C}\_{\mathbf{1},\min} = \mathcal{C}\_{\mathbf{1},0} - a \qquad \qquad \mathcal{C}\_{\mathbf{1},\max} = \mathcal{C}\_{\mathbf{1},0} + a$$

$$\mathcal{C}\_{\mathbf{2},\min} = \mathcal{C}\_{\mathbf{1},0} - b \qquad \qquad \mathcal{C}\_{\mathbf{2},\max} = \mathcal{C}\_{\mathbf{1},0} + b$$

For a given set of shift factors and temperatures , the regression base is described as:

$$\mathbf{p} = (T\_u; \delta\_u)$$

For each element in (i.e. for each pair of constants , ), the value of the right part of the WLF function is calculated for all temperatures , yielding .

The sum of square differences for all elements was then arranged in × .

$$\mathbf{D} = D\_{ij} = \sum\_{u=1}^{U} \left( \alpha\_{ij,u} - \delta\_u \right)^2 \qquad \text{with} \qquad \alpha\_{ij,u} = \frac{-\mathcal{C}\_{1,i} \left( T\_u - T\_{ref} \right)}{\mathcal{C}\_{2,j} + T\_u - T\_{ref}}.$$

For each iteration , the pair yielding minimum value of is found and associated to a specific couple of constants and in .

177

176

alization of the punctual temperature-shift correlations needs to be provided.

Figure 5.13, to allow for the use of experimental results in terms of relaxation and creep moduli for arbitrary temperatures within the investigated range, a gener-

*Figure 5.20 - example of a thirty-points re-sampling*

*Mastercurves, produced and perfected in previous paragraphs (cfr. §<IP_ADDRESS>, Table 5.6 and Figure 5.15), are inherently tied to the set of shift factors. Within the iterative procedure described in*

**<IP_ADDRESS> WLF regression (A proposed fitting procedure)**

Portions of the linking branches have later been cut, to avoid superposition with the experimental data, in the - small - portions where the curves were overlapping

The final operation on the data at this stage, to enhance both reliability and computational load of Prony series interpolations, was to re-sample the data sets among a fixed number of equally distributed points in logarithmic time scales. Provided with a large distribution of points (where stands for either or

), dense in some areas and sparser in others, a new time array is defined to

Accordingly, re-sampled values of the moduli were calculated as the average of the nearest ten points to the re-sampled abscissa in the original data set. An example of re-sampled data is presented in the following figure for 30 points

(experimental data were always preferred to points of the linking functions).

only, whereas two thousand points were used in the actual data analysis.

uniformly distribute the points along the curve:

28 The shape of the matrix arbitrary: any different space can have been adopted, without jeopardizing the discussed procedure but possibly affecting its accuracy. In the following, will be assumed, a choice motivated by the fact that there is no a-priori insight on the best fitting values of any of the two constants.

For the iteration of , limit of and are re-centred around the new best estimation and of the WLF constants. New limits are set as:

$$\begin{aligned} \mathcal{C}^{k+1}\_{1,\text{min}} &= \mathcal{C}^{k}\_{1,i^\*} - \frac{a}{2^{k-1}} \qquad & \mathcal{C}^{k+1}\_{1,\text{max}} = \mathcal{C}^{k}\_{1,i^\*} + \frac{a}{2^{k-1}}\\ \mathcal{C}^{k+1}\_{2,\text{min}} &= \mathcal{C}^{k}\_{2,i^\*} - \frac{b}{2^{k-1}} \qquad & \mathcal{C}^{k+1}\_{2,\text{max}} = \mathcal{C}^{k}\_{2,i^\*} + \frac{b}{2^{k-1}} \end{aligned}$$

Such procedure is repeated until convergence is deemed satisfactory. The scheme of the algorithm advancement is:

178

179

*Figure 5.21 - WLF functions for a) relaxation and b) creep mastercurves*

Figure 5.13, for all the shift-factor distributions tested within our analyses. The procedure has proven to be always swiftly convergent, provided that the best-fitting

Choosing , , , and a maximum number of 20 iterations, the procedure returns the final values for the WLF coefficients

Graphical results, displaying that the defined WLF functions are a good regression of the experimental shift values, are presented in Figure 5.21, whereas Figure 5.22 provides insight on the convergence of the procedure for the six functions in

*Table 5.7 - WLF coefficients*

equations must be graphically fitting the trend of each regression base.

 12.91 29.46 -53.36 -21.96 -40.12 -103.82 -83.46 -118.95 228.43 38.70 198.31 315.54 Curves in Figure 5.22 for the final shift-factor distributions show that the functions are not all convergent to the same value. This ought to be expected, for the inherent differences among the six regression bases : distributions which are more prone to be fitted by a curve belonging to the WLF family will yield lower final values of . Prior to the acquisition of final values for and two checks were performed: all error curves must be asymptotic to a final value and the WLF

**SG DG41 PVB relax. creep relax. creep relax. creep**

values are simply enclosed within the initial limits.

terms of the minimum value in per iteration .

displayed in Table 5.7.

*Although the absolute convergence of the method has not been demonstrated with mathematical proof, the convergence has been verified within the general algorithm described in*

Figure 5.13, for all the shift-factor distributions tested within our analyses. The procedure has proven to be always swiftly convergent, provided that the best-fitting values are simply enclosed within the initial limits.

Choosing , , , and a maximum number of 20 iterations, the procedure returns the final values for the WLF coefficients displayed in Table 5.7.

Graphical results, displaying that the defined WLF functions are a good regression of the experimental shift values, are presented in Figure 5.21, whereas Figure 5.22 provides insight on the convergence of the procedure for the six functions in terms of the minimum value in per iteration .

*Table 5.7 - WLF coefficients*

Curves in Figure 5.22 for the final shift-factor distributions show that the functions are not all convergent to the same value. This ought to be expected, for the inherent differences among the six regression bases : distributions which are more prone to be fitted by a curve belonging to the WLF family will yield lower final values of . Prior to the acquisition of final values for and two checks were performed: all error curves must be asymptotic to a final value and the WLF equations must be graphically fitting the trend of each regression base.

*Figure 5.21 - WLF functions for a) relaxation and b) creep mastercurves*

178

*Although the absolute convergence of the method has not been demonstrated with mathematical proof, the convergence has been verified within the general algorithm described in*

,

1. *(preliminary)*, take the regression base and set ,

using the right part of the WLF equation,

,

6. find the minimum value of associated to indexes ,

,

4. *(start the iteration)*, set and compute the values of **,** 5. for and calculate the sum of square differences

the new best estimation and of the WLF constants. New limits are set as:

and

and

and

Such procedure is repeated until convergence is deemed satisfactory. The

2. arbitrarily define , , , , and , along with a maximum number

,

,

minimum

are re-centred around

and repeat from

For the iteration of , limit of

of iterations , 3. define the initial limits

7. take and in , 8. define the new limits

point 4 until .

scheme of the algorithm advancement is:

Due to the nature of the algorithm, notably due to the progressively increasing density of the inquired mesh of points in . If and are the best possible values of and respectively, the accuracy of the results at the end of the iteration ( ) is:

$$\mathcal{C}\_1^\* = \mathcal{C}\_1^k \pm \frac{2a}{m2^k} = \mathcal{C}\_1^k \pm 0.002 \qquad \text{and} \qquad \mathcal{C}\_2^\* = \mathcal{C}\_2^k \pm \frac{2b}{n2^k} = \mathcal{C}\_2^k \pm 0.0025$$

Therefore, coefficients in Table 5.7 are accurate up to the second decimal.

*Figure 5.22 - convergence of the WLF regression algorithm*

#### **<IP_ADDRESS> Prony series corfficients for mastercurves**

Experimental mastercurves, produced and discussed in § <IP_ADDRESS>, have to be conveniently fitted with appropriate functions, to be implemented in analytical or numerical laminated glass design problems.

Several models, techniques and functions can be used to adequately fit exponentially decaying data sets, the kind of those obtained with the experimental campaign; some have been previously mentioned and discussed in §3.1.3. All things considered, the choice of a Wiechert model, associated with a Prony series regression, have been deemed to be the optimum choice for the presented fitting problem. The series are therefore searched in the form:

$$M(t) = M\_0 + \sum\_{i=1}^{N} M\_i \exp\left(\frac{t}{\tau\_i}\right)$$

where is either the relaxation modulus or the inverse compliance , and , , are the constants to be found.

An effective and versatile technique, capable of performing the Prony series regression on experimental data, has been articulated in 1997 in a research paper by Gant and Bower as the "*domain of influence method (DOI)"* [243]. The paper extensively provides instructions to implement the method in a regression algorithm; as neither the series themselves nor the regression method are innovative contributions

180

181

*Figure 5.23 – detail of data polish procedure outcome*

An illustration of the polish procedure is shown in Figure 5.23, highlighting how this method occasionally causes the data sets to encompass a few gaps among sub-

to this work, the method will not be further discussed here. Nonetheless, original remarks are listed, regarding properties and limits of the procedure emerged throughout the analyses (details on the algorithm are provided in Appendix I):

dure, adding some level of arbitrariness,

ates upon a meaningful amount of points,

mental time point,

the general algorithm described in Figure 5.13.

sequent points.

efficients, regardless the regularity of the curve,

point, this may affect significantly the accuracy.

1. the regression algorithm requires to be fed a monotonic descending array of data, therefore experimental results may be required a refinement proce-

2. in the arbitrary selection of the value of (cfr. Figure 5.24), a verification criterion is needed, to ensure that each exponential term always oper-

3. the method being based on a natural exponentials sum (a choice which has arguably several mathematical advantages), the number of terms required to perform an accurate fit depends on the shape of the function to be regressed, 4. experimental data on long time spans generally require large number of co-

5. due to the presence of a final, constant, "steady state" term within the series, resulting curves may be soon unfit to extrapolate data after the last experi-

6. the first point of the regression is always equal to the first experimental

For the experimental nature of data and data analysis processes described in previous paragraphs, average experimental mastercurves, produced in §<IP_ADDRESS> and displayed in Figure 5.20, did not always present a strictly monotonic descending trend. Therefore, a preliminary polishing procedure was needed to allow for the implementation of the DOI method. Starting from the first data point and proceeding in ascending time order, the procedure simply removed any data point which, for higher time values, had a higher modulus than any of the previous points. Similarly to the procedure for linking branches, the polish procedure is also an integral component of

to this work, the method will not be further discussed here. Nonetheless, original remarks are listed, regarding properties and limits of the procedure emerged throughout the analyses (details on the algorithm are provided in Appendix I):

For the experimental nature of data and data analysis processes described in previous paragraphs, average experimental mastercurves, produced in §<IP_ADDRESS> and displayed in Figure 5.20, did not always present a strictly monotonic descending trend. Therefore, a preliminary polishing procedure was needed to allow for the implementation of the DOI method. Starting from the first data point and proceeding in ascending time order, the procedure simply removed any data point which, for higher time values, had a higher modulus than any of the previous points. Similarly to the procedure for linking branches, the polish procedure is also an integral component of the general algorithm described in Figure 5.13.

*Figure 5.23 – detail of data polish procedure outcome*

An illustration of the polish procedure is shown in Figure 5.23, highlighting how this method occasionally causes the data sets to encompass a few gaps among subsequent points.

180

An effective and versatile technique, capable of performing the Prony series regression on experimental data, has been articulated in 1997 in a research paper by Gant and Bower as the "*domain of influence method (DOI)"* [243]. The paper extensively provides instructions to implement the method in a regression algorithm; as neither the series themselves nor the regression method are innovative contributions

Due to the nature of the algorithm, notably due to the progressively increasing

ues of and respectively, the accuracy of the results at the end of the iteration

Therefore, coefficients in Table 5.7 are accurate up to the second decimal.

*Figure 5.22 - convergence of the WLF regression algorithm*

Experimental mastercurves, produced and discussed in § <IP_ADDRESS>, have to be conveniently fitted with appropriate functions, to be implemented in analytical or

Several models, techniques and functions can be used to adequately fit exponentially decaying data sets, the kind of those obtained with the experimental campaign; some have been previously mentioned and discussed in §3.1.3. All things considered, the choice of a Wiechert model, associated with a Prony series regression, have been deemed to be the optimum choice for the presented fitting problem. The series

 where is either the relaxation modulus or the inverse compliance

and

are the best possible val-

density of the inquired mesh of points in . If

**<IP_ADDRESS> Prony series corfficients for mastercurves**

, and , , are the constants to be found.

numerical laminated glass design problems.

are therefore searched in the form:

( ) is:

Serviceability and post-failure behaviour of laminated glass structural elements

*Figure 5.24 –final mastercurve and elements of the series with a) 4-terms or b) 8-terms*

*Figure 5.25 - Prony fit comparison with different term series*

183

*Figure 5.26 - error for a) 5- , b) 10- , c) 15 -, d) 20- and e) 40-term series*

Gaps are meaningful to the DOI method, to the extent that the higher the number of terms in the series, the smaller are the domains among which the constants and are computed. If the domains are small enough and encompass one or several wide gaps, the number of points in that specific domain may drop dramatically, therefore severely affecting the accuracy of the regression in that portion of the

curve. For each interpolation, a limit to the minimum width of each domain

has been imposed as follow:

Gaps are meaningful to the DOI method, to the extent that the higher the number of terms in the series, the smaller are the domains among which the constants and are computed. If the domains are small enough and encompass one or several wide gaps, the number of points in that specific domain may drop dramatically, therefore severely affecting the accuracy of the regression in that portion of the curve. For each interpolation, a limit to the minimum width of each domain has been imposed as follow:

*Figure 5.26 - error for a) 5- , b) 10- , c) 15 -, d) 20- and e) 40-term series*

183

182

*Figure 5.25 - Prony fit comparison with different term series*

*Figure 5.24 –final mastercurve and elements of the series with a) 4-terms or b) 8-terms*

$$
\delta \log(t) \ge 3 \cdot \max(\log(t\_i) - \log(t\_{i+1}), \forall i \in [1, \text{size}(t)])
$$

Figure 5.24 shows an example of the regression of DG41 relaxation mastercurve with 4-terms and 8-terms series, highlighting the effects on the number of terms on the accuracy. The number of terms in increased by reducing the value of , connected to the width of each DOI.

For internal consistency among final results produced in this work, it was decided to provide Prony series with equal number of terms for all moduli of all materials. The number was then chosen to successfully fit all the curves with appropriate accuracy and also avoid inappropriate "jiggling" (a technical term used in [243] and visible in Figure 5.24) around the mean value.

For increasing number of terms, the accuracy was first estimated as the percental error between experimental data29 and the generated Prony series, along the entire time span of each mastercurve. For increasing number of terms, curves are shown in Figure 5.25 and percental errors in Figure 5.26. Error curves show that the accuracy initially increases if the number of terms also increases: the maximum error between 5- and 15-terms series diminishes from about to mere (cfr. Figure 5.26.a, b & c); nonetheless, data also suggest that the error does not monotonically decrease with increasing number of terms (cfr. Figure 5.26.d e). To evaluate the accuracy of the Prony series fit to experimental data, a slightly refined criterion may be used to knowingly choose the final number of terms for the series. The normalized average of square root of square differences Γ , can be defined as:

$$\bar{\Gamma}\_{\rm m} = \frac{\mu}{\max(\mu)} \qquad \text{where} \qquad \Gamma\_{\rm m} = \frac{\sqrt{\sum\_{i=1}^{n} (M\_{\rm P}(t\_i) - M\_i)^2}}{n}$$

 being the number of elements in each data set and the value of the modulus at time , calculated using the appropriate Prony series. Figure 5.27 shows that a very high number of terms is not suited, if the purpose is to maximise the accuracy of the fit. All things considered, the choice of the conclusive number of elements for the series has been chosen mediating among the three different criteria:

A reasoned choice among several options finally led to the choice:

$$\aleph = 20$$

#### *This criterion was implemented in the general algorithm shown in*

Figure 5.13. With the final mastercurves coefficients, the normalized sum of square differences between the experimental data of shifted branches and interpolat-

184

185

*Figure 5.28 – final 20-terms Prony series mastercurves*

was found, allowing to find and compare the overall

The shape of the 20-terms Prony series functions is displayed in Figure 5.28,

*Figure 5.27 - error evaluation for Prony series with increasing number of terms*

ing Prony mastercurves Γ

measure Γ of shift-factors accuracy among various distributions.

compared to experimental data and coefficients are shown in Table 5.8.

29 The data as they are collected prior to the polish procedure, which has been used to the sole purpose of running the fitting DOI algorithm and generate the series.

ing Prony mastercurves Γ was found, allowing to find and compare the overall measure Γ of shift-factors accuracy among various distributions.

The shape of the 20-terms Prony series functions is displayed in Figure 5.28, compared to experimental data and coefficients are shown in Table 5.8.

*Figure 5.27 - error evaluation for Prony series with increasing number of terms*

*Figure 5.28 – final 20-terms Prony series mastercurves*

185

184

29 The data as they are collected prior to the polish procedure, which has been used to the sole purpose of running

Figure 5.13. With the final mastercurves coefficients, the normalized sum of square differences between the experimental data of shifted branches and interpolat-

Figure 5.24 shows an example of the regression of DG41 relaxation mastercurve with 4-terms and 8-terms series, highlighting the effects on the number of terms on the accuracy. The number of terms in increased by reducing the value of ,

For internal consistency among final results produced in this work, it was decided to provide Prony series with equal number of terms for all moduli of all materials. The number was then chosen to successfully fit all the curves with appropriate accuracy and also avoid inappropriate "jiggling" (a technical term used in [243] and

For increasing number of terms, the accuracy was first estimated as the percental error between experimental data29 and the generated Prony series, along the entire time span of each mastercurve. For increasing number of terms, curves are shown in Figure 5.25 and percental errors in Figure 5.26. Error curves show that the accuracy initially increases if the number of terms also increases: the maximum error between 5- and 15-terms series diminishes from about to mere (cfr. Figure 5.26.a, b & c); nonetheless, data also suggest that the error does not monotonically decrease with increasing number of terms (cfr. Figure 5.26.d e). To evaluate the accuracy of the Prony series fit to experimental data, a slightly refined criterion may be used to knowingly choose the final number of terms for the series. The

 being the number of elements in each data set and the value of the modulus at time , calculated using the appropriate Prony series. Figure 5.27 shows that a very high number of terms is not suited, if the purpose is to maximise the accuracy of the fit. All things considered, the choice of the conclusive number of elements for the series has been chosen mediating among the three different crite-

significant "jiggling" around experimental values must be avoided,

 *This criterion was implemented in the general algorithm shown in*

A reasoned choice among several options finally led to the choice:

, can be defined as:

Γ

the fitting DOI algorithm and generate the series.

ria:

connected to the width of each DOI.

visible in Figure 5.24) around the mean value.

normalized average of square root of square differences Γ

Γ

series must be accurate over the entire time span,

the number of elements should be as low as possible.

Serviceability and post-failure behaviour of laminated glass structural elements *Table 5.8 - mastercurves coefficients, , time in seconds, moduli in MPa*

187

30 Tabulated times have been chosen according to the design instructions for load durations described in

The analytical solution is arguably not always an optimal tool from a LG design perspective. In preliminary design phases, to envisage general design choices, an order of magnitude for interlayer modulus is usually plenty just to grasp the properties

Values for short-term actions at low temperature and for long-term ones at high

**<IP_ADDRESS> Analytical and Tabulated results**

tion of time at an arbitrary temperature for any of tested materials.

With the experimental outcomes presented in the previous paragraphs, results for both relaxation and creep compliance can be given as a func-

A case study is shown on the procedure for extracting from mastercurves and WLF equations the solution for a specific material, time, temperature and action (i.e. imposed stress or strain). For a DG41 interlayer, whose relaxation modulus is needed after 10 minutes of imposed strain at a temperature , the WLF coefficients and can be retrieved from Table 5.7 p.179, and the shift factor can be computed by rearranging the WLF equation as follows:

yielding . The coefficients , and for the DG41 relaxation

As mastercurve coefficients are given with in and in ,

mastercurve are in Table 5.8 p.186, while the time-shifted Prony series is:

*Figure 5.29 - DG41 relaxation modulus over time for*

converting to the appropriate dimensions ultimately yields:

of laminated elements. Simplified tabulated results30

prEN16612-2013 and used in CNR-DT 210/2013.

are in Appendix G.

#### **<IP_ADDRESS> Analytical and Tabulated results**

With the experimental outcomes presented in the previous paragraphs, results for both relaxation and creep compliance can be given as a function of time at an arbitrary temperature for any of tested materials.

A case study is shown on the procedure for extracting from mastercurves and WLF equations the solution for a specific material, time, temperature and action (i.e. imposed stress or strain). For a DG41 interlayer, whose relaxation modulus is needed after 10 minutes of imposed strain at a temperature , the WLF coefficients and can be retrieved from Table 5.7 p.179, and the shift factor can be computed by rearranging the WLF equation as follows:

$$a\_T \{ T, T\_{ref} \} = 10^A \qquad \qquad \text{with} \qquad A = \frac{C\_1 \{ T - T\_{ref} \}}{C\_2 + \{ T - T\_{ref} \}}$$

yielding . The coefficients , and for the DG41 relaxation mastercurve are in Table 5.8 p.186, while the time-shifted Prony series is:

$$G(t, T) = G\_0 + \sum\_{i=1}^{20} \mathcal{G}\_i \exp\left(-\frac{t}{a\_T \{T, T\_{ref}\} \cdot \tau\_i}\right).$$

As mastercurve coefficients are given with in and in , converting to the appropriate dimensions ultimately yields:

#

*Figure 5.29 - DG41 relaxation modulus over time for*

The analytical solution is arguably not always an optimal tool from a LG design perspective. In preliminary design phases, to envisage general design choices, an order of magnitude for interlayer modulus is usually plenty just to grasp the properties of laminated elements. Simplified tabulated results30 are in Appendix G.

Values for short-term actions at low temperature and for long-term ones at high

187

186

*Table 5.8 - mastercurves coefficients, , time in seconds, moduli in MPa*

9

0,881

1,58E+09

2,791

1,47E+10

3,701

8,88E+04

5,224

5,04E+03

0,118

6,20E+03

0,154

7,56E+03

8

2,488

2,89E+08

4,398

2,78E+09

5,780

2,56E+04

5,220

6,50E+02

0,279

1,43E+03

0,452

1,15E+03

7

6,650

5,24E+07

8,698

3,53E+08

8,562

8,66E+03

3,533

7,88E+01

0,807

4,29E+02

0,565

1,55E+02

6

9,531

1,27E+07

8,004

5,90E+07

7,320

2,50E+03

3,282

8,30E+00

1,101

1,22E+02

0,720

1,87E+01

5

11,13

2,94E+06

8,660

4,73E+06

5,686

7,57E+02

3,148

6,91E-01

1,385

3,10E+01

1,379

1,85E+00

4

10,98

5,13E+05

9,119

1,18E+06

5,282

2,11E+02

2,703

7,01E-02

2,119

8,21E+00

2,301

2,78E-01

3

13,56

1,15E+05

11,85

1,76E+05

5,437

6,60E+01

2,112

8,29E-03

2,439

2,31E+00

2,741

3,94E-02

2

11,52

2,22E+04

13,11

2,83E+04

2,719

2,12E+01

2,287

8,32E-04

2,929

6,01E-01

2,505

5,44E-03

1

21,90

3,50E+03

24,46

4,08E+03

2,570

5,06E+00

4,027

9,35E-05

6,609

1,47E-01

3,553

4,81E-04

0,098

0,083

0,325

0,245

1,415

0

0,845

i

PVB

DG41

SG

20

0,091

5,92E+16

0,189

2,14E+19

0,029

5,94E+10

0,053

2,63E+14

0,003

1,45E+10

0,011

5,02E+13

19

0,157

1,30E+16

0,224

4,00E+18

0,038

1,74E+10

0,041

3,77E+13

0,021

4,28E+09

0,025

1,08E+13

18

0,203

2,75E+15

0,338

5,56E+17

0,043

5,39E+09

0,093

4,70E+12

0,033

1,22E+09

0,023

1,44E+12

17

0,219

5,55E+14

0,338

8,52E+16

0,044

1,69E+09

0,074

3,40E+11

0,024

2,88E+08

0,030

1,57E+11

16

0,247

1,12E+14

0,397

1,10E+16

0,045

4,54E+08

0,095

3,68E+10

0,035

6,90E+07

0,038

2,07E+10

15

0,315

2,20E+13

0,454

1,67E+15

0,083

1,31E+08

0,117

3,02E+09

0,055

1,79E+07

0,048

2,63E+09

14

0,356

4,66E+12

0,454

2,32E+14

0,056

3,91E+07

0,754

2,46E+08

0,053

4,96E+06

0,054

3,43E+08

13

0,234

8,99E+11

0,657

3,72E+13

0,088

1,17E+07

2,040

4,00E+07

0,057

1,29E+06

0,054

4,15E+07

12

0,495

1,66E+11

0,570

7,10E+12

0,154

3,38E+06

3,521

4,04E+06

0,082

3,40E+05

0,083

5,07E+06

11

0,523

4,40E+10

0,811

7,44E+11

0,445

9,25E+05

5,544

5,15E+05

0,093

9,11E+04

0,089

6,71E+05

10

0,687

7,53E+09

1,378

1,10E+11

2,104

2,79E+05

5,506

5,77E+04

0,104

2,44E+04

0,097

8,09E+04

30 Tabulated times have been chosen according to the design instructions for load durations described in prEN16612-2013 and used in CNR-DT 210/2013.

temperatures have not been shown in the tables for two different reasons, related either to the range of the experimental analyses or the nature of the analytical model used to build the mastercurves.

Limits of both analytical and tabulated results will be further discussed in §<IP_ADDRESS>, along with suggested safety factors that should be considered for a safe LG design.

#### **<IP_ADDRESS> Limits and reliability of results**

Results gathered using Prony series mastercurves should always be looked at with a critical eye to the whole range of materials properties. As the original theory for time-temperature superposition has been developed for dynamic tests (cfr. §<IP_ADDRESS>) [114,291], application to direct shear tests as those performed in this work must account for the actual testing conditions.

In chapter Chapter 4, it was discussed how experimental curves for relaxation and creep (cfr. Figure 4.35 to 4.37) start to be reliable after a finite time interval. While short-term effects for high and intermediate temperatures can be drawn through the mastercurve gathering from experimental results at lower temperatures, the same cannot be said for the lowest temperatures (i.e. for SG, for DG41 and for PVB). Lowest temperatures must account for their own reliability, so short-term properties cannot not be extrapolated from the series prior the reliability limit (this justifies blank values in the top-left corner of Tables in Appendix G).

*Figure 5.30 - reliability limits for mastercurves Prony series*

188

189

31 Coefficients and in Table 5.8 stand for this infinite modulus value (cfr. §<IP_ADDRESS>), for relaxation and creep

Instantaneous elastic properties are always superimposed to the viscoelastic effects [114,282,291,292], therefore outcomes of viscoelastic analyses should always be looked at with a critical eye. Results given in terms of Prony series mastercurves and WLF coefficients are not meant to describe short-term properties of the materi-

Besides, for long term effects on the right side of the curves, a limit to the reliability of Prony series results must also be provided. The Wiechert model (cfr. §<IP_ADDRESS>) on which the Prony series are based, implies the presence of a finite value of each modulus upon which the material settles for an infinite amount of time31 [243]. Nonetheless, the shape of experimental mastercurves suggest that, if tests had been performed for higher temperatures or if the same tests had been carried out for longer times, the shape of the curves may have continued a descending trend, rather than reaching a steady value (Figure 5.30). To avoid extracting inaccurate value from the series, reliability limits for which values are reliable are given as a function

of time and temperature :

the WLF coefficients.

functions respectively.

where and are the maximum and minimum reliability times for the mastercurve at the reference temperature (Figure 5.30 and Table 5.9) and , are

*Table 5.9 – reliability times in seconds*

**SG DG41 PVB relax. creep relax. creep relax. creep**

Note how the shape of the reliability limits is closely correlated to the shape of the WLF equation; in fact, the same technique used to define the reliability limits can be applied to plot the results on simple graphs, correlating time, temperature and

An important aspect to evaluate concerns short-term properties of tested materials, namely their known load/strain rate dependent properties (cfr. §3.2 and §4.2). Analyses performed with LG specimens are dedicated to study viscous effects, which are increasingly important with passing of time. Hence, the goal of proposed viscoelastic models is to reliably replicate the effects of long-term constant actions, while the rate at which these actions are applied has a limited influence on the test outcome. Conversely, analyses described in §5.1.1 for the tensile response of thermoplastic interlayers showed that the stiffness of such materials for short-term effects is highly non-linear and influenced by the rate at which actions are applied.

materials moduli. These graphical results are also shown in Appendix G.

Besides, for long term effects on the right side of the curves, a limit to the reliability of Prony series results must also be provided. The Wiechert model (cfr. §<IP_ADDRESS>) on which the Prony series are based, implies the presence of a finite value of each modulus upon which the material settles for an infinite amount of time31 [243]. Nonetheless, the shape of experimental mastercurves suggest that, if tests had been performed for higher temperatures or if the same tests had been carried out for longer times, the shape of the curves may have continued a descending trend, rather than reaching a steady value (Figure 5.30). To avoid extracting inaccurate value from the series, reliability limits for which values are reliable are given as a function of time and temperature :

$$\left| 10^{\wedge} \right| ^{\left[ \log\_{10} t\_1 - \frac{-C\_1 \left( T - T\_{ref} \right)}{C\_2 + \left( T - T\_{ref} \right)} \right]} \le t \le 10^{\wedge} ^{\left[ \log\_{10} t\_2 - \frac{-C\_1 \left( T - T\_{ref} \right)}{C\_2 + \left( T - T\_{ref} \right)} \right]}$$

where and are the maximum and minimum reliability times for the mastercurve at the reference temperature (Figure 5.30 and Table 5.9) and , are the WLF coefficients.

*Table 5.9 – reliability times in seconds*

Note how the shape of the reliability limits is closely correlated to the shape of the WLF equation; in fact, the same technique used to define the reliability limits can be applied to plot the results on simple graphs, correlating time, temperature and materials moduli. These graphical results are also shown in Appendix G.

An important aspect to evaluate concerns short-term properties of tested materials, namely their known load/strain rate dependent properties (cfr. §3.2 and §4.2). Analyses performed with LG specimens are dedicated to study viscous effects, which are increasingly important with passing of time. Hence, the goal of proposed viscoelastic models is to reliably replicate the effects of long-term constant actions, while the rate at which these actions are applied has a limited influence on the test outcome. Conversely, analyses described in §5.1.1 for the tensile response of thermoplastic interlayers showed that the stiffness of such materials for short-term effects is highly non-linear and influenced by the rate at which actions are applied.

Instantaneous elastic properties are always superimposed to the viscoelastic effects [114,282,291,292], therefore outcomes of viscoelastic analyses should always be looked at with a critical eye. Results given in terms of Prony series mastercurves and WLF coefficients are not meant to describe short-term properties of the materi-

189

188

*Figure 5.30 - reliability limits for mastercurves Prony series*

temperatures have not been shown in the tables for two different reasons, related either to the range of the experimental analyses or the nature of the analytical model

Limits of both analytical and tabulated results will be further discussed in §<IP_ADDRESS>, along with suggested safety factors that should be considered for a safe LG

Results gathered using Prony series mastercurves should always be looked at with a critical eye to the whole range of materials properties. As the original theory for time-temperature superposition has been developed for dynamic tests (cfr. §<IP_ADDRESS>) [114,291], application to direct shear tests as those performed in this work

In chapter Chapter 4, it was discussed how experimental curves for relaxation and creep (cfr. Figure 4.35 to 4.37) start to be reliable after a finite time interval. While short-term effects for high and intermediate temperatures can be drawn through the mastercurve gathering from experimental results at lower temperatures, the same cannot be said for the lowest temperatures (i.e. for SG, for DG41 and for PVB). Lowest temperatures must account for their own reliability, so short-term properties cannot not be extrapolated from the series prior the reliability limit (this justifies blank values in the top-left corner of Tables in Appendix

used to build the mastercurves.

**<IP_ADDRESS> Limits and reliability of results**

must account for the actual testing conditions.

design.

G).

31 Coefficients and in Table 5.8 stand for this infinite modulus value (cfr. §<IP_ADDRESS>), for relaxation and creep functions respectively.

als (indicatively, not much shorter than 5-10 minutes). Therefore, short-term relaxation and creep moduli found with a linear viscoelastic approach should be considered as conservative lower bound values of both creep and relaxation, as if actions were applied at an infinitesimal stress or strain rate. In reality, the stiffness of the materials for short-term effects is influenced by the strain-rate hardening response described with the GR model in <IP_ADDRESS>. In fact, the actual short-term response of LG elements can possibly be better anticipated using the results of short-term tensile tests on interlayer specimens, neglecting all viscous effects.

# **5.1.3 Temperature-depentent adhesive properties**

Adhesive properties of interlayers in glass lamination are usually investigated with TCT tests 32: the ends of two-layered LG specimens, whose glass has been artificially cracked on both sides, are pulled with a constant displacement rate. The peak load measured during these tests can be correlated to the adhesion among the materials. Nonetheless, this interpretation is intrinsically correlated to the geometry of specimens and is not prone to be easily generalized [125,293]. While adhesion is represented in many ways, what is measured is really the "practical adhesion" which is defined as the force or the work required to detach a film or coating from its substrate [294]. K. L. Mittal wrote [295]:

*"A frequently asked question is: What is the best method for measuring adhesion? The answer is: the best, rather most appropriate, method is one that simulates usage stress conditions."*

To my knowledge, full loss of adhesion of glass-interlayers interface has never been reported as an issue in the LG market. In most cases, a partial debonding (i.e. a loss of adhesion which only affects a potion of the interface) is enough to deem that the element has essentially failed its purpose: aesthetics and transparency, trait features of LG, are henceforth compromised. Also, the perception of security of the structural element is compromised from the end-user point of view, even though the mechanical properties may not have been deeply affected. Therefore, a study is shown on the conditions under which the adhesion has been observed to be undermined in LG safety glass.

Interlayer materials are known to have crucial temperature-dependent properties (cfr. §1.1.2), which are yet to be effectively correlated with the strength of the adhesion bond with glass. Even though tests on LG specimens described in §4.2 were not purposely aimed to this analysis, results discussed in §<IP_ADDRESS> and described in Appendix C allow for some considerations. Far from attempting to a detailed dissertation on this very articulate and complex topic, contents of this paragraph can be looked at as a first step for the design a specific investigation, should the circumstances demand for a better understanding of LG delamination. Recent researches

190

191

An analysis in that direction has been performed for all specimens with numerical integration of recorded force and displacement data. Figure 5.31 presents a

*Figure 5.31 - a) displacement and b) load over time for specimen S12 (55°C), c) instantaneous power and d) supplied energy by numerical integration*

highlighted that the strain rate (i.e. the instant power distribution over which the strain energy builds up) is a relevant factor in the onset and progression of delamina-

*"Using the TCT test, we have shown that steady state delamination can be obtained only in a limited range of temperature and applied velocity."*

ing phase, similarly to more traditional tensile tests [125].

Alas, for specimens where delamination was measured at the end of tests, there is no reliable data indicating the exact time frame when delamination process has started, and how it has evolved over time. Nonetheless, in some tests the climatic chamber was open for very brief periods after the end of the loading phase (i.e. when presses are loaded by fastening of the loading bolt Figure 4.11 p.100) to perform small operations and to check the setup. Observations during those operations showed that some delamination had already occurred shortly after the end of the loading phase (about 10 minutes to one hour). While the extent of the delamination could not be measured accurately in those phases, neither compared to the final delamination described in Appendix C, the simple observation that some had occurred allows to ascribe to the loading phase itself a role in setting the conditions for the surpass of the fracture toughness threshold. With this consideration, an estimate of adhesion thoughtless can be given in terms of supplied energy throughout the load-

tion [293]:

32 Through Crack Tensile test

highlighted that the strain rate (i.e. the instant power distribution over which the strain energy builds up) is a relevant factor in the onset and progression of delamination [293]:

## *"Using the TCT test, we have shown that steady state delamination can be obtained only in a limited range of temperature and applied velocity."*

Alas, for specimens where delamination was measured at the end of tests, there is no reliable data indicating the exact time frame when delamination process has started, and how it has evolved over time. Nonetheless, in some tests the climatic chamber was open for very brief periods after the end of the loading phase (i.e. when presses are loaded by fastening of the loading bolt Figure 4.11 p.100) to perform small operations and to check the setup. Observations during those operations showed that some delamination had already occurred shortly after the end of the loading phase (about 10 minutes to one hour). While the extent of the delamination could not be measured accurately in those phases, neither compared to the final delamination described in Appendix C, the simple observation that some had occurred allows to ascribe to the loading phase itself a role in setting the conditions for the surpass of the fracture toughness threshold. With this consideration, an estimate of adhesion thoughtless can be given in terms of supplied energy throughout the loading phase, similarly to more traditional tensile tests [125].

*Figure 5.31 - a) displacement and b) load over time for specimen S12 (55°C), c) instantaneous power and d) supplied energy by numerical integration*

An analysis in that direction has been performed for all specimens with numerical integration of recorded force and displacement data. Figure 5.31 presents a

191

190

als (indicatively, not much shorter than 5-10 minutes). Therefore, short-term relaxation and creep moduli found with a linear viscoelastic approach should be considered as conservative lower bound values of both creep and relaxation, as if actions were applied at an infinitesimal stress or strain rate. In reality, the stiffness of the materials for short-term effects is influenced by the strain-rate hardening response described with the GR model in <IP_ADDRESS>. In fact, the actual short-term response of LG elements can possibly be better anticipated using the results of short-term tensile

Adhesive properties of interlayers in glass lamination are usually investigated

ficially cracked on both sides, are pulled with a constant displacement rate. The peak load measured during these tests can be correlated to the adhesion among the materials. Nonetheless, this interpretation is intrinsically correlated to the geometry of specimens and is not prone to be easily generalized [125,293]. While adhesion is represented in many ways, what is measured is really the "practical adhesion" which is defined as the force or the work required to detach a film or coating from its sub-

*"A frequently asked question is: What is the best method for measuring adhesion? The answer is: the best, rather most appropriate, method is one that*

To my knowledge, full loss of adhesion of glass-interlayers interface has never been reported as an issue in the LG market. In most cases, a partial debonding (i.e. a loss of adhesion which only affects a potion of the interface) is enough to deem that the element has essentially failed its purpose: aesthetics and transparency, trait features of LG, are henceforth compromised. Also, the perception of security of the structural element is compromised from the end-user point of view, even though the mechanical properties may not have been deeply affected. Therefore, a study is shown on the conditions under which the adhesion has been observed to be under-

Interlayer materials are known to have crucial temperature-dependent properties (cfr. §1.1.2), which are yet to be effectively correlated with the strength of the adhesion bond with glass. Even though tests on LG specimens described in §4.2 were not purposely aimed to this analysis, results discussed in §<IP_ADDRESS> and described in Appendix C allow for some considerations. Far from attempting to a detailed dissertation on this very articulate and complex topic, contents of this paragraph can be looked at as a first step for the design a specific investigation, should the circumstances demand for a better understanding of LG delamination. Recent researches

32: the ends of two-layered LG specimens, whose glass has been arti-

tests on interlayer specimens, neglecting all viscous effects.

**5.1.3 Temperature-depentent adhesive properties**

strate [294]. K. L. Mittal wrote [295]:

mined in LG safety glass.

32 Through Crack Tensile test

*simulates usage stress conditions."*

with TCT tests

Serviceability and post-failure behaviour of laminated glass structural elements graphical interpretation of this analysis for one of the specimens33.

$$E\_{\rm ext} = \int\_{\delta\_0}^{\delta\_1} F(\delta)d\delta \cong \frac{1}{2} \sum\_{i=1}^{N-1} \left[ (\delta\_{i+1} - \delta\_i)(F\_{i+1} + F\_i) \right] \qquad ; \qquad P = \frac{d}{dt} E\_{\rm ext}$$

Figure 5.31.a shows the progression of the loading phase with the four imposed displacement steps performed by design, to progressively load the specimens; Figure 5.31.c highlights that the energy is supplied in the same number of "chunks" corresponding to the manual bolts tightening.

*Figure 5.32 - average delamination correlated with average loading energy density*

Results show that most of the energy is supplied in the loading phase. The energy density of the glass-interlayer interfaces can be computed dividing the supplied energy for the area of the interlayers34. Details are in Appendix H.

192

193

A lower bound prediction of PDLG bending stiffness can be obtained by fully neglecting the mechanical contribution of broken plies. Nonetheless, experimental evidence suggests that after tempered glass shatters, it is still able to give a contribution to the overall bending stiffness, through its adhesion to the interlayers. Compar-

Figure 5.32 condenses average data from the delamination analysis (Appendix C) and outcomes of load energy density calculations (Appendix H); the resulting correlation helps to draw some general conclusion on the fracture toughness of the glass-interlayer interface. The maximum imposed shear stress in the loading phase

*Table 5.10 – glass-interlayer fracture toughness estimate for varying temperatures*

 **PVB DG41 SG** >165 *n/d n/d* >264 >100 *n/d n/d* >100 *n/d n/d* >97 *n/d* >639 >106 >55 *n/d* >118 *n/d* 170 196 >788 >126 *n/d n/d* >148 <274 <518 >185 *n/d n/d* <115 <199 <333 160 190 *n/d n/d* <215 *n/d n/d* <161

Results agree with outcomes of other researches on PVB interlayers: about at for (Elzière, 2017 [293]), stressing how temperature plays a key role in defining the fracture toughness of the interface. Information on the energy density alone seems to be insufficient to assess whether delamination will arise: the interface in able to take in significant energy at lower temperatures without compromising the integrity of the adhesion, on the other hand, while tem-

Experimental evidence discussed in §4.3 shows that both the increased stiffness provided by damaged glass plies in PDLG and residual stiffness of FDLG beams is significant. For given climatic conditions, because of the higher short-term stiffness of new interlayer materials (cfr. §4.2 and §4.1), post-failure performance of LG elements can be substantially enhanced compared to more traditional structures, lami-

perature rises, lower energy levels are capable to onset partial delamination.

**5.2 Post-failure properties of LG**

**5.2.1 Analysis of the tension-stiffening effect**

nated with PVB interlayers.

**Temp. fracture toughness**

to the average extent of delamination for each material.

33 To help correlating these results to other parts of this manuscript, data is shown for specimen S12 (one of the SG-laminates), one among those that had been previously chosen as a case study for the data analysis in §4.2.2 and respective subparagraphs.

34 approximately 2000 mm2 , varying from one specimen to another for manufacturing tolerances. This calculation requires once again the hypothesis that, glass being order of magnitudes stiffer than interlayers, it can be considered relatively undeformable (cfr. §4.2.1). This hypothesis implies that interlayer shear stress and strain can be approximated with uniform distributions.

Figure 5.32 condenses average data from the delamination analysis (Appendix C) and outcomes of load energy density calculations (Appendix H); the resulting correlation helps to draw some general conclusion on the fracture toughness of the glass-interlayer interface. The maximum imposed shear stress in the loading phase to the average extent of delamination for each material.

*Table 5.10 – glass-interlayer fracture toughness estimate for varying temperatures*

Results agree with outcomes of other researches on PVB interlayers: about at for (Elzière, 2017 [293]), stressing how temperature plays a key role in defining the fracture toughness of the interface. Information on the energy density alone seems to be insufficient to assess whether delamination will arise: the interface in able to take in significant energy at lower temperatures without compromising the integrity of the adhesion, on the other hand, while temperature rises, lower energy levels are capable to onset partial delamination.

#### **5.2 Post-failure properties of LG**

Experimental evidence discussed in §4.3 shows that both the increased stiffness provided by damaged glass plies in PDLG and residual stiffness of FDLG beams is significant. For given climatic conditions, because of the higher short-term stiffness of new interlayer materials (cfr. §4.2 and §4.1), post-failure performance of LG elements can be substantially enhanced compared to more traditional structures, laminated with PVB interlayers.

#### **5.2.1 Analysis of the tension-stiffening effect**

192

, varying from one specimen to another for manufacturing tolerances. This calculation

*Figure 5.32 - average delamination correlated with average loading energy density*

33 To help correlating these results to other parts of this manuscript, data is shown for specimen S12 (one of the SG-laminates), one among those that had been previously chosen as a case study for the data analysis in §4.2.2 and

requires once again the hypothesis that, glass being order of magnitudes stiffer than interlayers, it can be considered relatively undeformable (cfr. §4.2.1). This hypothesis implies that interlayer shear stress and strain can be

plied energy for the area of the interlayers34. Details are in Appendix H.

Results show that most of the energy is supplied in the loading phase. The energy density of the glass-interlayer interfaces can be computed dividing the sup-

graphical interpretation of this analysis for one of the specimens33.

Figure 5.31.a shows the progression of the loading phase with the four imposed displacement steps performed by design, to progressively load the specimens; Figure 5.31.c highlights that the energy is supplied in the same number of "chunks" corre-

sponding to the manual bolts tightening.

respective subparagraphs. 34 approximately 2000 mm2

approximated with uniform distributions.

A lower bound prediction of PDLG bending stiffness can be obtained by fully neglecting the mechanical contribution of broken plies. Nonetheless, experimental evidence suggests that after tempered glass shatters, it is still able to give a contribution to the overall bending stiffness, through its adhesion to the interlayers. Compar-

ing experimental results to the minimum stiffness expected can provide useful insight for understanding this *tension stiffening* (TS) effect, quantifying the mechanical contribution of damaged glass plies.

In PDLG specimens, a fictitious equivalent elastic modulus for fractured glass plies can be estimated under two hypotheses: first, that fractured plies confined by adherent interlayers can still be looked at as an isotropic material, able to bear tensile and compressive stresses; the second, that the moment of inertia along the strong axis of damaged and undamaged plies is the same. The validity the first hypothesis is confirmed by experimental outcomes on the position of the neutral axis in PDLG, at least when quasi-static imposed loads are modest (cfr. §<IP_ADDRESS>, Table 4.9 p.143). For a simply supported LG beam having span , assembled with three glass plies of thickness and inertia about the strongest axis, with a concentrated load at mid-span, the equivalent modulus for one ( ) or two ( ) shattered plies is:

$$E\_{fg,eq,1} = \frac{P\_m L^3}{48f\_{1,g}\eta\_m} - 2E\_g$$

$$gP \cdot L^3 = E$$

$$E\_{fg,eq,2} = \frac{a p\_m L^\nu}{96 f\_{1,g} \eta\_m} - \frac{E\_g}{2}$$

*Table 5.11 - fictitious equivalent Young modulus [GPa] of damaged glass plies*

† cfr. Table 4.8

where is a correction factor accounting for torsional effects. With two broken plies, PDLG beams are bent laterally because of the asymmetric glass failure. As a result, applying a vertical force at midspan generates non-negligible torsional effects along with flexural ones. Throughout tests III-5 (Table 4.7), torsional restraints35 (cfr. Figure 4.42 p129) were used to keep specimens in vertical position. Accounting for the exact position of the restraints and the extent of the lateral sag, the correction factor has been calibrated with a 3D FE linear elastic model.

194

195

With the same simplified approach used for the analysis of three-point bending tests, the lower bound of expected frequencies for specimens in PDLG conditions can be anticipated, by neglecting the mechanical contribution of broken glass plies

is known for the frequency of the first modal form of a uniform beam of span and modulus , simply supported at both ends with uniform weight per unit length

The expected natural frequency of specimens can be calculated in Table 5.12, using experimental values of the glass elastic modulus from quasi-static three-point

in §4.3.2. Such values have been compared to experimental data from bending tests. An analysis of the influence of boundary conditions is performed in Appendix F.

*Table 5.12 - expected frequencies based on ULG three-point bending tests stiffness*

**A2** 67,8 75,1

**B2** 65,8 86,5 **B3** 68,5 88,3

**C2** 67,1 85,3 **C3** 69,1 86,6

*\* an equivalent area of homogeneous DG41 interlayer is used, to account for the different density of PC embed-*

36 The exact weight of glass and interlayers has been determined experimentally with density tests performed after

 and DG41 . The density of PC could not be assessed experimentally, the literature value of was used [350] (although, for this specific case study the volume of PC is so small that it could have been assimilated to DG41 interlayer with very limited consequences). 37 Validation of this analytical formulation, with calibration of the correction factor for this case study is pro-

of all the analyses described in §4.3. Density results are: glass , SG

† **L freq**

5,35 8,40 0,85 2,44 1,08 0,212 2,7

5,35 8,40 1,83\* 2,44 1,08 0,225 2,7

1,71 6,48 0,55 2,44 9,68 0,164 2,35 74,3

36. An analytical formulation

and geometries described

86,9

85,1

and interlayer while accounting for both of their masses

(or mass per unit length ) with 37 [107,296]:

bending tests (Table 4.8), along with material properties36

**5.2.2 Dynamic tests**

**A1** 66,3

**B1** 66,4

**C1** 66,7

*† cfr. Table 4.8.*

vided in Appendix F.

*ded within the interlayer matrix.*

35 which have been installed as close as possible to midspan for those tests.

#### **5.2.2 Dynamic tests**

With the same simplified approach used for the analysis of three-point bending tests, the lower bound of expected frequencies for specimens in PDLG conditions can be anticipated, by neglecting the mechanical contribution of broken glass plies and interlayer while accounting for both of their masses 36. An analytical formulation is known for the frequency of the first modal form of a uniform beam of span and modulus , simply supported at both ends with uniform weight per unit length (or mass per unit length ) with 37 [107,296]:

$$f\_1 = \frac{9.87a}{2\pi} \sqrt{\frac{E\_g J g}{\overline{w} L^4}}$$

The expected natural frequency of specimens can be calculated in Table 5.12, using experimental values of the glass elastic modulus from quasi-static three-point bending tests (Table 4.8), along with material properties36 and geometries described in §4.3.2. Such values have been compared to experimental data from bending tests. An analysis of the influence of boundary conditions is performed in Appendix F.

*Table 5.12 - expected frequencies based on ULG three-point bending tests stiffness*

*† cfr. Table 4.8.*

194

ing experimental results to the minimum stiffness expected can provide useful insight for understanding this *tension stiffening* (TS) effect, quantifying the mechani-

In PDLG specimens, a fictitious equivalent elastic modulus for fractured glass plies can be estimated under two hypotheses: first, that fractured plies confined by adherent interlayers can still be looked at as an isotropic material, able to bear tensile and compressive stresses; the second, that the moment of inertia along the strong axis of damaged and undamaged plies is the same. The validity the first hypothesis is confirmed by experimental outcomes on the position of the neutral axis in PDLG, at least when quasi-static imposed loads are modest (cfr. §<IP_ADDRESS>, Table 4.9 p.143). For a simply supported LG beam having span , assembled with three glass plies of thickness and inertia about the strongest axis, with a concentrated load at mid-span, the equivalent modulus for one ( ) or two

cal contribution of damaged glass plies.

( ) shattered plies is:

**stage I**

**stage II**

**stage III**

† cfr. Table 4.8

factor has been calibrated with a 3D FE linear elastic model.

35 which have been installed as close as possible to midspan for those tests.

*Table 5.11 - fictitious equivalent Young modulus [GPa] of damaged glass plies*

† 66.3 67.8 66.4 65.8 68.5 66.7 67.1 69.1

 29.9 28.8 18.2 16.3 22.3 16.3 22.9 14.3 stiffness increase *%* 22.5 21.2 13.7 12.4 16.3 12.2 17.1 10.3

 16.3 16.9 16.7 15.3 13.6 9.09 10.7 11.0 stiffness increase *%* 49.2 50.0 50.3 46.5 39.7 27.3 31.9 31.8

where is a correction factor accounting for torsional effects. With two broken plies, PDLG beams are bent laterally because of the asymmetric glass failure. As a result, applying a vertical force at midspan generates non-negligible torsional effects along with flexural ones. Throughout tests III-5 (Table 4.7), torsional restraints35 (cfr. Figure 4.42 p129) were used to keep specimens in vertical position. Accounting for the exact position of the restraints and the extent of the lateral sag, the correction

**unit A1 A2 B1 B2 B3 C1 C2 C3**

*\* an equivalent area of homogeneous DG41 interlayer is used, to account for the different density of PC embedded within the interlayer matrix.*

36 The exact weight of glass and interlayers has been determined experimentally with density tests performed after of all the analyses described in §4.3. Density results are: glass , SG and DG41 . The density of PC could not be assessed experimentally, the literature value of was used [350] (although, for this specific case study the volume of PC is so small that it could have been assimilated to DG41 interlayer with very limited consequences). 37 Validation of this analytical formulation, with calibration of the correction factor for this case study is pro-

vided in Appendix F.

The approach used to compute the expected natural frequency in the ULG state can be tweaked, using additional hypotheses, and applied for both PDLG states (Stages II&III).

The minimum frequency expected, shown in Figure 5.33, can be computed by neglecting the mechanical contribution of damaged plies and interlayers (i.e. neglecting the TS effect), while accounting for all masses.

A better estimate of the experimental PDLG natural frequencies can be produced, based on data from three-point bending tests elaborated in §5.2.1, which quantify the TS effect. With an equivalent Young modulus for damaged plies (Table 5.11), an equivalent modulus can be computed for the entire cross section by homogenising material properties according to proportions shown below. Using instead of in the previous formulation for yields results of the Equivalent Stiffness Model (ESM) for stages II & III shown in Figure 5.33.

$$\begin{aligned} E\_{eq} &= E\_g \left( \frac{2}{3} + \frac{E\_{fg,eq,1}}{E\_g} \right) \\ E\_{eq} &= E\_g \left( \frac{1}{3} + \frac{2E\_{fg,eq,2}}{3E\_g} \right) \end{aligned} \quad \text{for PDLG stage II (1 broken lyly)}$$

Although based on a small number of specimens and lacking statistical accuracy, results of the comparison shown in Figure 5.33 exhibit several consistent patterns, prompting to wide-ranging considerations.

*Stage I – Undamaged Laminated Glass (ULG)*

*Stage II & III– Partially Damaged Laminated Glass (PDLG)*

196

197

*Figure 5.33 - experimental and model frequencies comparison*

† *cfr. Table 4.10*

*Figure 5.33 - experimental and model frequencies comparison*

197

196

The approach used to compute the expected natural frequency in the ULG state can be tweaked, using additional hypotheses, and applied for both PDLG states

The minimum frequency expected, shown in Figure 5.33, can be computed by neglecting the mechanical contribution of damaged plies and interlayers (i.e. ne-

A better estimate of the experimental PDLG natural frequencies can be produced, based on data from three-point bending tests elaborated in §5.2.1, which quantify the TS effect. With an equivalent Young modulus for damaged plies (Table 5.11), an equivalent modulus can be computed for the entire cross section by homogenising material properties according to proportions shown below. Using instead of in the previous formulation for yields results of the Equivalent

Although based on a small number of specimens and lacking statistical accuracy, results of the comparison shown in Figure 5.33 exhibit several consistent pat-

 The experimental natural frequency of undamaged LG specimens (Stage I) is consistently lower than expected, based on the ESM with glass properties descending from experimental three-point bending tests. Causes of this phenomenon were investigated thoroughly, but the fundamental reason for the discrepancy among the two measurements could not be tracked down. Recorded data and instruments were double-checked multiple times showing no evidence of

 Comparison of dynamic and ESM for undamaged SG-laminated (Type A) seem to be closer one another, with a difference of about 3%. On the other hand, results for DG41-laminates (Type B & C), display a greater average difference, around 7%, regardless the presence of the PC foil and interlayer thick-

 With the only exception of specimen C1, whose ESM frequency in Stage II is slightly lower compared to experimentally measured one ( ), dynamic tests experimental results can be consistently found within the minimum ex-

ESM model is more accurate for Stage II than for any other damage state, with

for PDLG stage II (1 broken ply)

for PDLG stage III (2 broken plies)

glecting the TS effect), while accounting for all masses.

Stiffness Model (ESM) for stages II & III shown in Figure 5.33.

terns, prompting to wide-ranging considerations.

*Stage I – Undamaged Laminated Glass (ULG)*

acquisition errors or unintended setup response.

Dynamic test frequencies are always lower than the ESM ones.

*Stage II & III– Partially Damaged Laminated Glass (PDLG)*

pected frequency and equivalent stiffness model predictions.

predictions consistently within 4% relative error.

(Stages II&III).

ness.

 Results for specimens B & C in Stage III are consistently close to the average of the ESM and minimum expected frequencies (e.g. for specimen B2 )

# **5.2.3 Equivalent thermal expansion model**

Tempered glass has strong compressive stresses on outer surfaces and tensile stresses inside. For undamaged tempered glass, compressive and tensile stresses are in equilibrium (cfr. §1.1.1); the outer surface of glass is plane and each portion exchanges equal and opposite actions with all neighboring ones. Due to the elastic behaviour of glass, when the glass fails, fragments release part of the stored mechanical energy by expanding at the edges, releasing compressive stresses, and contracting in the central part, releasing tensile stresses.

If a system, like an interlayer with good adhesion, can keep fragments from scattering away, the observed macroscopic behaviour of the fractured glass ply is its tendency to expand in its own plane.

*Figure 5.34 - fragmentation of broken tempered glass plies*

An expansion originated by glass failure, such as the one illustrated in Figure 5.35, is arguably not easy to model starting from a microscopic point of view. Repulsion mechanisms originating the expansion are dependent on a variety of factors such as the strength and depth of the tempering process, thickness of glass plies, crack density and distribution, shape of the fragments, size of contact points between fragments, etc.

198

199

*Figure 5.36 - flexibility method model for an expanding central fractured glass ply*

38 The FE model was built using an equivalent non-isotropic temperature gradient: a numerical gradient only ap-

plied in the two main dimensions of the specimen, but not across the thickness. 39 Also known as method of consistent deformations.

An equivalent temperature variation can be defined as a function of the previously defined fictitious equivalent modulus (cfr. §5.2.1) of the fractured

To model the mutual effects of broken tempered glass plies on undamaged ones, a simple model was put to the test for PDLG specimens, reproducing the effects measured with the onset of cracks using an Equivalent Thermal Expansion Model (ETEM)38. One can postulate that for the transition between Stage I and Stage II, if the interlayer shear stiffness was zero, the fractured glass size would increase freely, not involving the adjacent plies (Figure 5.36.a). Besides, if the hypothesis that plane sections remain plane still holds [164], a solution can be found with the flexibility

*Figure 5.35 - expansion of a confined tempered glass ply upon failure*

method39, as presented in Figure 5.36.

To model the mutual effects of broken tempered glass plies on undamaged ones, a simple model was put to the test for PDLG specimens, reproducing the effects measured with the onset of cracks using an Equivalent Thermal Expansion Model (ETEM)38. One can postulate that for the transition between Stage I and Stage II, if the interlayer shear stiffness was zero, the fractured glass size would increase freely, not involving the adjacent plies (Figure 5.36.a). Besides, if the hypothesis that plane sections remain plane still holds [164], a solution can be found with the flexibility method39, as presented in Figure 5.36.

*Figure 5.35 - expansion of a confined tempered glass ply upon failure*

*Figure 5.36 - flexibility method model for an expanding central fractured glass ply*

An equivalent temperature variation can be defined as a function of the previously defined fictitious equivalent modulus (cfr. §5.2.1) of the fractured

198

*Figure 5.34 - fragmentation of broken tempered glass plies*

An expansion originated by glass failure, such as the one illustrated in Figure 5.35, is arguably not easy to model starting from a microscopic point of view. Repulsion mechanisms originating the expansion are dependent on a variety of factors such as the strength and depth of the tempering process, thickness of glass plies, crack density and distribution, shape of the fragments, size of contact points between

 Results for specimens B & C in Stage III are consistently close to the average of the ESM and minimum expected frequencies (e.g. for specimen B2

Tempered glass has strong compressive stresses on outer surfaces and tensile stresses inside. For undamaged tempered glass, compressive and tensile stresses are in equilibrium (cfr. §1.1.1); the outer surface of glass is plane and each portion exchanges equal and opposite actions with all neighboring ones. Due to the elastic behaviour of glass, when the glass fails, fragments release part of the stored mechanical energy by expanding at the edges, releasing compressive stresses, and contracting in the central part, releasing tensile stresses. A graphical interpretation of this

If a system, like an interlayer with good adhesion, can keep fragments from scattering away, the observed macroscopic behaviour of the fractured glass ply is its

)

**5.2.3 Equivalent thermal expansion model**

phenomenon on a small scale is shown in Figure 5.34.

tendency to expand in its own plane.

fragments, etc.

38 The FE model was built using an equivalent non-isotropic temperature gradient: a numerical gradient only applied in the two main dimensions of the specimen, but not across the thickness. 39 Also known as method of consistent deformations.

glass ply. If the stiffness of interlayers is negligible compared with glass, fractured or not, and provided with the experimental measure of the axial strain on one lateral – undamaged - glass ply40, it follows that:

$$
\Delta T\_{eq}(E\_{fg,eq}) = \frac{1}{\alpha\_g} \left( \frac{A\_g}{A\_{fg}} \frac{E\_g}{E\_{fg,eq}} + 1 \right) \varepsilon\_{11,cf}
$$

where and are cross-sectional areas of undamaged and fractured plies respectively, is glass thermal expansion coefficient [5,19,38] and is the longitudinal measured strain at mid-span on the outer undamaged plies upon failure of the central ply41.

 values have been calculated using results for of PDLG specimens with one or two broken plies (Table 5.11) 42.

*Table 5.13 - equivalent modulus and gradient for PDLG models*

† *cfr. Table 5.11* ‡ *cfr. Figure 4.53*

201

*Figure 5.38 - numerical model compared to experimental strain measures* Numerical results have been validated in Figure 5.38 with experimental measurements gathered with tests II-1 (cfr. Table 4.7 p.132) for Type B specimens. Results of this comparison are consistent with a swiftly relaxing material: strain gauges and , at mm distance from the lateral edges of the beams, show considerably high strains at failure, rapidly dropping in the a few minutes. This entails that on the outer edges the interlayer material rapidly loses the ability to transfer a signif-

*Figure 5.37 - Stage II linear elastic finite element results using ETEM*

With ETEM, a prediction of effects transferred trough the interlayers from shattered glass plies on undamaged ones is possible. With the calibrated thermal expansion described above for Stage II, Figure 5.37 shows results of different strain patterns on the outer surface of the external glass ply (Figure 5.37.a) and within the interlayer (Figure 5.37.b), varying the stiffness of the interlayer itself.

Numerical results allow to study shear stresses transfer mechanism through the interlayers. Regardless the stiffness of the material, the shear transfer zone from the expanding central ply to the outer undamaged plies shows a peak between 50- 100mm from the edge ((Figure 5.37.b) This peak shrink over time, while the transfer zone progressively expands towards the inner regions, for diminishing interlayer shear modulus. The steady value of strain at midspan proves that the total shear transfer is consistent over time (i.e. for a relaxing interlayer).

40 cfr. strain gauges for specimens Type B & C shown in Figure 4.45 p.104. 41 Conventionally, the strain measured by instrument after thirty seconds from the failure of the central ply was used. Figure 4.53 p.115 shows the progression of . 42 With the aforementioned hypotheses, such procedure is only rigorous for Stage II, a symmetrical damage state.

The same procedure is an approximation for the asymmetric Stage III, the curvature-induced effects as with two broken plies are neglected.

*Figure 5.37 - Stage II linear elastic finite element results using ETEM*

*Figure 5.38 - numerical model compared to experimental strain measures*

Numerical results have been validated in Figure 5.38 with experimental measurements gathered with tests II-1 (cfr. Table 4.7 p.132) for Type B specimens. Results of this comparison are consistent with a swiftly relaxing material: strain gauges and , at mm distance from the lateral edges of the beams, show considerably high strains at failure, rapidly dropping in the a few minutes. This entails that on the outer edges the interlayer material rapidly loses the ability to transfer a signif-

201

200

40 cfr. strain gauges for specimens Type B & C shown in Figure 4.45 p.104. 41 Conventionally, the strain measured by instrument after thirty seconds from the failure of the central ply

was used. Figure 4.53 p.115 shows the progression of . 42 With the aforementioned hypotheses, such procedure is only rigorous for Stage II, a symmetrical damage state. The same procedure is an approximation for the asymmetric Stage III, the curvature-induced effects as with two

glass ply. If the stiffness of interlayers is negligible compared with glass, fractured or not, and provided with the experimental measure of the axial strain on one lateral

where and are cross-sectional areas of undamaged and fractured plies respectively, is glass thermal expansion coefficient [5,19,38] and is the longitudinal measured strain at mid-span on the outer undamaged plies

values have been calculated using results for of PDLG specimens with

*Table 5.13 - equivalent modulus and gradient for PDLG models*

† 66.4 65.8 68.5 66.7 67.1 69.1 ‡ 63.1 63.5 64.9 67.4 68.6 66.5

 stage II 58 64 52 69 52 79 stage III 63 68 80 117 103 100

terlayer (Figure 5.37.b), varying the stiffness of the interlayer itself.

transfer is consistent over time (i.e. for a relaxing interlayer).

With ETEM, a prediction of effects transferred trough the interlayers from shattered glass plies on undamaged ones is possible. With the calibrated thermal expansion described above for Stage II, Figure 5.37 shows results of different strain patterns on the outer surface of the external glass ply (Figure 5.37.a) and within the in-

Numerical results allow to study shear stresses transfer mechanism through the interlayers. Regardless the stiffness of the material, the shear transfer zone from the expanding central ply to the outer undamaged plies shows a peak between 50- 100mm from the edge ((Figure 5.37.b) This peak shrink over time, while the transfer zone progressively expands towards the inner regions, for diminishing interlayer shear modulus. The steady value of strain at midspan proves that the total shear

† 18.2 16.3 22.3 16.3 22.9 14.3

† 16.7 15.3 13.6 9.09 10.7 11.0

**unit B1 B2 B3 C1 C2 C3**

† *cfr. Table 5.11* ‡ *cfr. Figure 4.53*

42.

– undamaged - glass ply40, it follows that:

upon failure of the central ply41.

broken plies are neglected.

one or two broken plies (Table 5.11)

icant amount of stress from the central fractured ply to the lateral ones. On the other hand, instruments , and are noticeably steady, an observation in full agreement with results presented in Figure 5.37.a.

Numerical model results for the positions corresponding to instruments , and are in perfect agreement with experimental results around after the failure of the class ply. This should come to no surprise, as the ETEM with whom the numerical analysis was carried was calibrated with experimental readings at after failure of the glass ply.

The slightly rising trend of those three instruments in the first minutes has to be ascribed to the delay in the formation of some of the cracks: once tempered ply fails, most cracks are generated almost instantly; however, experimental observations showed that the number and density of cracks grew slightly over a few hours. In point of fact, this agrees with the idea that interlayers exhibit a stronger action in immediately after failure, meaning that they are more effective in holding back the fragments. As the time passes, viscous deformations originate a progressive drop in their ability to clasp the fragments and contain partially-formed cracks from expanding further. Consequently, a few new cracks form over time, contributing to the overall expansion of the fractured ply. From a macroscopic point of view, this behaviour results in a delayed expansion, a phenomenon which was not taken into account in our model. Finally, the gradual reduction of recorded strain over time, which was recorded after the first hour up to 24 hours in Figure 4.51 p.138, can be instead ascribed to the relaxation of the shear modulus of interlayer materials, due to viscous effects.

Consistently with results for creep and relaxation discussed in §5.1.2, this comparison proves once again that DG41 interlayer modulus at room temperature is very high for short and impulsive actions (about 1 to 2 GPa), but rapidly diminishes for its inclination to viscous relaxation.

Furthermore, the ETEM can be used to analyze non-symmetrical damage states, like Stage III specimens with central and one of lateral plies broken (cfr. Table 4.7 p.132). With data in Table 5.13, insight on the lateral deflection can be obtained and compared with experimental measurements (Test III-2) Figure 5.39.

*Figure 5.39 – Stage III lateral deflection comparison (view from above)*

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203

43 Either for conspicuous viscous relaxation or a choice of a different, softer, interlayer material. 44 The strain rate of has been calculated knowing the speed rate ( ) and the

Results from the numerical model highlight that a decrease of the interlayer's stiffness43, does not translate in a significant variation of the lateral deflection, which is mainly affected by geometrical properties (i.e. the numerical model showed that the thickness of the plies and the length of the specimens are the main factors affecting the lateral deflection). Nonetheless, while the numerical model shows a progressively diminishing sag for diminishing interlayer stiffness, experimental measurements show instead a slight sag increase. This inconsistency may be explained by pointing out that the ETEM model doesn't account for the delayed increase in the number of cracks, which in fact contributes to expanding the lateral and central bro-

Uniaxial tests performed on SG-laminated samples (cfr. §4.3.4 p.147), cut out of Type A specimens after all tests had been performed, allow for a better understand-

*Figure 5.40 - Experimental results of tensile tests on FDLG specimens*

Experimental results described in Figure 4.65 can be compared with simple models known in literature [38,55,63] and experimental outcomes on interlayer specimens analysed in §4.1, whose results can be seen in Figure 4.4 p.93. The latter comparison is shown in Figure 5.40. While interlayer results are shown for a strain rate which is almost double the strain rate for FDLG44 (i.e. results of dotted curves are overestimating the response one would record for the same specimens at ), a very significant TS effect is measured, due to the presence of adherent glass fragments. While tests on FDLG could not be carried out up to very high strain to preserve the instruments (cfr. §4.3.4), plausible forecasts can be hypothesized, if glass fragments adhesion is eventually lost and FDLG ultimately dete-

ing of both TS effect and compressive response of glass fragments.

ken plies, thus inducing a further sag increase.

**5.2.4 Fully damaged LG models**

riorates to the interlayer only.

span of dog-bone specimens between the clamps ( , cfr. §4.1.1).

Results from the numerical model highlight that a decrease of the interlayer's stiffness43, does not translate in a significant variation of the lateral deflection, which is mainly affected by geometrical properties (i.e. the numerical model showed that the thickness of the plies and the length of the specimens are the main factors affecting the lateral deflection). Nonetheless, while the numerical model shows a progressively diminishing sag for diminishing interlayer stiffness, experimental measurements show instead a slight sag increase. This inconsistency may be explained by pointing out that the ETEM model doesn't account for the delayed increase in the number of cracks, which in fact contributes to expanding the lateral and central broken plies, thus inducing a further sag increase.

## **5.2.4 Fully damaged LG models**

Uniaxial tests performed on SG-laminated samples (cfr. §4.3.4 p.147), cut out of Type A specimens after all tests had been performed, allow for a better understanding of both TS effect and compressive response of glass fragments.

*Figure 5.40 - Experimental results of tensile tests on FDLG specimens*

Experimental results described in Figure 4.65 can be compared with simple models known in literature [38,55,63] and experimental outcomes on interlayer specimens analysed in §4.1, whose results can be seen in Figure 4.4 p.93. The latter comparison is shown in Figure 5.40. While interlayer results are shown for a strain rate which is almost double the strain rate for FDLG44 (i.e. results of dotted curves are overestimating the response one would record for the same specimens at ), a very significant TS effect is measured, due to the presence of adherent glass fragments. While tests on FDLG could not be carried out up to very high strain to preserve the instruments (cfr. §4.3.4), plausible forecasts can be hypothesized, if glass fragments adhesion is eventually lost and FDLG ultimately deteriorates to the interlayer only.

202

*Figure 5.39 – Stage III lateral deflection comparison (view from above)*

icant amount of stress from the central fractured ply to the lateral ones. On the other hand, instruments , and are noticeably steady, an observation in full

Numerical model results for the positions corresponding to instruments , and are in perfect agreement with experimental results around after the failure of the class ply. This should come to no surprise, as the ETEM with whom the numerical analysis was carried was calibrated with experimental readings

The slightly rising trend of those three instruments in the first minutes has to be ascribed to the delay in the formation of some of the cracks: once tempered ply fails, most cracks are generated almost instantly; however, experimental observations showed that the number and density of cracks grew slightly over a few hours. In point of fact, this agrees with the idea that interlayers exhibit a stronger action in immediately after failure, meaning that they are more effective in holding back the fragments. As the time passes, viscous deformations originate a progressive drop in their ability to clasp the fragments and contain partially-formed cracks from expanding further. Consequently, a few new cracks form over time, contributing to the overall expansion of the fractured ply. From a macroscopic point of view, this behaviour results in a delayed expansion, a phenomenon which was not taken into account in our model. Finally, the gradual reduction of recorded strain over time, which was recorded after the first hour up to 24 hours in Figure 4.51 p.138, can be instead ascribed to the relaxation of the shear modulus of interlayer materials, due to

Consistently with results for creep and relaxation discussed in §5.1.2, this comparison proves once again that DG41 interlayer modulus at room temperature is very high for short and impulsive actions (about 1 to 2 GPa), but rapidly diminishes for

Furthermore, the ETEM can be used to analyze non-symmetrical damage states, like Stage III specimens with central and one of lateral plies broken (cfr. Table 4.7 p.132). With data in Table 5.13, insight on the lateral deflection can be obtained and

compared with experimental measurements (Test III-2) Figure 5.39.

agreement with results presented in Figure 5.37.a.

at after failure of the glass ply.

its inclination to viscous relaxation.

viscous effects.

43 Either for conspicuous viscous relaxation or a choice of a different, softer, interlayer material. 44 The strain rate of has been calculated knowing the speed rate ( ) and the span of dog-bone specimens between the clamps ( , cfr. §4.1.1).

A different comparison can be performed using the effective modulus formula [63,104], allowing to estimate the proportionality coefficient for the maximum load bearing capacity of SG-laminated fractured glass at room temperature:

$$E\_{eff} = k \frac{l^\*}{\lambda^\*} E\_p$$

*Table 5.14 – effective modulus in FDLG with SG interlayer [MPa]*

*Figure 5.41 – FDLG compression tests and proposed constitutive law*

The size of a typical -thick tempered glass ply fragment was measured between on average. Up to the maximum stress (around strain), the effective modulus appears to be higher than SG interlayer alone; comparing numerical data on tensile tests for FDLG with the photographic follow-up carried in §4.3.4, one can estimate the depth of the delaminated surface around the size of the glass fragments, yielding . Table 5.14 lists values of the modulus for SG-laminated FDLG.

Compression tests performed on FDLG specimens are also described in §4.3.4, showing a remarkable hardening behaviour. As experimental curves presented an initial nonlinear branch, followed by a linear one (cfr. Figure 4.67 p.150 ), it was observed that such stress-strain correlations could be well approximated through a hardening polynomial up to , followed by a linear function for higher strains, up to the ultimate failure.

204

205

*Figure 5.42 - FDLG bending tests comparison to analytical model* For Type A specimens, both initial stiffness and ultimate strength decreased with storage time: mechanical properties of FDLG beams appeared to have substantially changed after five-month storage, compared to properties of observed a few hours after the failure of the last glass ply. The decrease in stiffness and strength can be explained by considering that glass cracks allowed for a widespread exposition of

The hardening behaviour, represented by the progressive change of slope of the curves, is arguably due to closing of cracks and subsequent transmission of compressive stresses through direct contact between glass fragments. Table 5.15 lists main mechanical parameters obtained from tensile and compression tests on FDLG

*Table 5.15 – mechanical parameters for SG-FDLG made with tempered plies*

Stage IV tests (cfr. Table 4.7 p.132) highlighted that the load-bearing capacity of FDLG beams is not entirely lost. Significant detrimental effects of time have been observed, as they triggered a drastic reduction of both the stiffness and ultimate strength in SG-laminated specimens (Type A) and total loss of shape and loadcarrying capacity for DG41-laminated ones (Type B). Reinforced DG41-laminates

Yielding stress †

specimens for tensile and compressive actions.

Yielding strain Long term stress asymptote

Ultimate stress Ultimate strain † *The higher value refers to higher strain rate.*

(Type C) showed an intermediate behaviour.

**Tensile tests**

Secant modulus at

**Compression tests** Secant modulus at

The hardening behaviour, represented by the progressive change of slope of the curves, is arguably due to closing of cracks and subsequent transmission of compressive stresses through direct contact between glass fragments. Table 5.15 lists main mechanical parameters obtained from tensile and compression tests on FDLG specimens for tensile and compressive actions.

*Table 5.15 – mechanical parameters for SG-FDLG made with tempered plies*

† *The higher value refers to higher strain rate.*

Stage IV tests (cfr. Table 4.7 p.132) highlighted that the load-bearing capacity of FDLG beams is not entirely lost. Significant detrimental effects of time have been observed, as they triggered a drastic reduction of both the stiffness and ultimate strength in SG-laminated specimens (Type A) and total loss of shape and loadcarrying capacity for DG41-laminated ones (Type B). Reinforced DG41-laminates (Type C) showed an intermediate behaviour.

*Figure 5.42 - FDLG bending tests comparison to analytical model*

For Type A specimens, both initial stiffness and ultimate strength decreased with storage time: mechanical properties of FDLG beams appeared to have substantially changed after five-month storage, compared to properties of observed a few hours after the failure of the last glass ply. The decrease in stiffness and strength can be explained by considering that glass cracks allowed for a widespread exposition of

205

204

A different comparison can be performed using the effective modulus formula [63,104], allowing to estimate the proportionality coefficient for the maximum

>

*strain rate 5 s 10 s 30 s 60 s*

*strain rate 0.0025 0.005 0.015 0.03*

*Table 5.14 – effective modulus in FDLG with SG interlayer [MPa]*

*slow test* 9300 8110 5864 4283 *fast test* 6320 4464 2153 1133

*slow test* 4662 3292 1686 899 *fast test* 6320 4464 2153 1133

*Figure 5.41 – FDLG compression tests and proposed constitutive law*

The size of a typical -thick tempered glass ply fragment was measured

 on average. Up to the maximum stress (around strain), the effective modulus appears to be higher than SG interlayer alone; comparing numerical data on tensile tests for FDLG with the photographic follow-up carried in §4.3.4, one can estimate the depth of the delaminated surface around the size of the glass fragments, yielding . Table 5.14 lists

Compression tests performed on FDLG specimens are also described in §4.3.4, showing a remarkable hardening behaviour. As experimental curves presented an initial nonlinear branch, followed by a linear one (cfr. Figure 4.67 p.150 ), it was observed that such stress-strain correlations could be well approximated through a hardening polynomial up to , followed by a linear function for higher

load bearing capacity of SG-laminated fractured glass at room temperature:

**as a function of total elapsed time**

between

**as a function of total accumulated strain**

values of the modulus for SG-laminated FDLG.

strains, up to the ultimate failure.

the interlayer to moisture during the storage interval, triggering a decay in the chemical bonds between glass and interlayers around each glass fragment [45].

A simple FDLG analytical beam model was defined using stress-strain curves drawn from tensile and compression tests. For tensile stresses, "slow tests" were used (cfr. Figure 4.65.a p.148), as the strain rate of three-point tests on FDLG beams was the same imposed in those tests. The FDLG model curve in Figure 5.42.a was obtained by computing the moment-curvature diagram with experimental constitutive equations for tensile and compressive actions (Figure 5.42.b), then integrating the curvature twice over the beam axis. Boundary conditions of a simply supported three-point bending test were considered.

Figure 5.42 shows how the analytical model can accurately predict the response of five-month old FLDG beams (which ought to be expected, as the material specimens used to perform tensile and compressive tests and draw the experimental stress-strain correlations were extracted from those LG beams and tested a few days after). On the other hand, the analytical model seems to overestimate the ultimate load-bearing capacity and stiffness after the initial part of the curves.

Differences between the model and 5-month experimental results can be explained by comparing the experimentally recorded strain at the bottom and at the top of beams (through "omega" transducers in Figure 4.44 p.130) with those of "material specimens" in tension and compression tests (Figure 5.43).

*Figure 5.43 – experimental compressive and tensile strain on top and bottom of the beam at midspan, compared to FDLG analytical model results*

In the tensile region (to the right in Figure 5.43), the model quite accurately predicts the strain pattern in 5-month old FDLG beams. On the contrary, the model is only accurate in the initial part of the compressed region (to the left in Figure 5.43), for loads approximately on third the ultimate capacity, while it vastly underestimates the strain at top fibres for higher loads. As an example, at a load of (85% of the ultimate capacity) the error is around in the tensile region, but is about one order of magnitude higher in the compressed region.

Very high differences in the compressed region can be explained considering the

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207

*Figure 5.45 - increment in SG stiffness vs. delamination ratio from FE elastic analyses*

*Figure 5.44 - Schematic crack patterns: a) symmetric layout, b) maximum offset*

greater confinement of glass fragments in uniaxial compression tests, rather than in the compressed region on top of the FDLG beams in three-point tests. In the latter case, a progressive expulsion of shards and fragments was observed from top of the beams, which can explain both the lower stiffness and the reduced value of the ulti-

As the depth of the delamination grows, the tensile stiffness of the composite material (glass fragments + interlayer) tends to the stiffness of the interlayer alone. On the other hand, the interlayer exhibits an increased stiffness because of the migration of stresses between glass fragments and the interlayer. This pattern creates stress concentrations at the edge of the delamination surface, inducing more delamination over time. To simulate the effect of the loss of bond adhesion on the TS effect, an elastic FE model was developed. The width of glass fragments in tested specimens has been observed to be comparable with the thickness of glass plies, so fragments have been imagined as perfect cubes of side length 10 mm for modelling purposes. Numerical simulations have been performed considering two simplified cracking patterns. Figure 5.44 shows symmetric layout and maximum offset be-

mate load capacity with respect to the model.

**5.2.5 Delamination and TS effect**

tween cracks of the two glass plies.

greater confinement of glass fragments in uniaxial compression tests, rather than in the compressed region on top of the FDLG beams in three-point tests. In the latter case, a progressive expulsion of shards and fragments was observed from top of the beams, which can explain both the lower stiffness and the reduced value of the ultimate load capacity with respect to the model.

## **5.2.5 Delamination and TS effect**

206

Very high differences in the compressed region can be explained considering the

*Figure 5.43 – experimental compressive and tensile strain on top and bottom of the beam at midspan, compared to FDLG analytical model results* In the tensile region (to the right in Figure 5.43), the model quite accurately predicts the strain pattern in 5-month old FDLG beams. On the contrary, the model is only accurate in the initial part of the compressed region (to the left in Figure 5.43), for loads approximately on third the ultimate capacity, while it vastly underestimates the strain at top fibres for higher loads. As an example, at a load of (85% of the ultimate capacity) the error is around in the tensile region, but is about one

the interlayer to moisture during the storage interval, triggering a decay in the chemical bonds between glass and interlayers around each glass fragment [45].

supported three-point bending test were considered.

load-bearing capacity and stiffness after the initial part of the curves.

rial specimens" in tension and compression tests (Figure 5.43).

order of magnitude higher in the compressed region.

A simple FDLG analytical beam model was defined using stress-strain curves drawn from tensile and compression tests. For tensile stresses, "slow tests" were used (cfr. Figure 4.65.a p.148), as the strain rate of three-point tests on FDLG beams was the same imposed in those tests. The FDLG model curve in Figure 5.42.a was obtained by computing the moment-curvature diagram with experimental constitutive equations for tensile and compressive actions (Figure 5.42.b), then integrating the curvature twice over the beam axis. Boundary conditions of a simply

Figure 5.42 shows how the analytical model can accurately predict the response of five-month old FLDG beams (which ought to be expected, as the material specimens used to perform tensile and compressive tests and draw the experimental stress-strain correlations were extracted from those LG beams and tested a few days after). On the other hand, the analytical model seems to overestimate the ultimate

Differences between the model and 5-month experimental results can be explained by comparing the experimentally recorded strain at the bottom and at the top of beams (through "omega" transducers in Figure 4.44 p.130) with those of "mate-

As the depth of the delamination grows, the tensile stiffness of the composite material (glass fragments + interlayer) tends to the stiffness of the interlayer alone. On the other hand, the interlayer exhibits an increased stiffness because of the migration of stresses between glass fragments and the interlayer. This pattern creates stress concentrations at the edge of the delamination surface, inducing more delamination over time. To simulate the effect of the loss of bond adhesion on the TS effect, an elastic FE model was developed. The width of glass fragments in tested specimens has been observed to be comparable with the thickness of glass plies, so fragments have been imagined as perfect cubes of side length 10 mm for modelling purposes. Numerical simulations have been performed considering two simplified cracking patterns. Figure 5.44 shows symmetric layout and maximum offset between cracks of the two glass plies.

*Figure 5.44 - Schematic crack patterns: a) symmetric layout, b) maximum offset*

*Figure 5.45 - increment in SG stiffness vs. delamination ratio from FE elastic analyses*

*Figure 5.45* shows results of FE analyses, where it can be observed that mean TS effect may be approximated using an exponential interpolation.

For each delamination ratio, the figure shows the upper and lower values of the stiffness increase of the interlayer due to TS, corresponding to the two cracking patterns of Figure 5.44. For instance, the stiffness of the interlayer is about twice as big for debonded surface ratios of ~40% and four times as big for ratios of ~15%. It can be observed that at increasing debonded surface ratio the two limit values of the stiffness increment approach one another and for high delamination ratios (more than 80%) they are almost zero.

# **5.2.6 Synthesis of post failure LG properties**

Table 5.16 presents general information useful from a designer point of view to understand and compare effects of progressive damage on LG elements.

*Table 5.16 - noteworthy effects of progressive damage on LG beams*

† *generally negligible, provided there are no stiff constraints to elongation at both ends of the beam,*

‡ *because of the significant lateral deflection, the longitudinal relative displacement of beam ends can result in a net contraction rather than an elongation.*

\* *grades are assigned by comparison among the three specimen types.*

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209

*Figure 6.1 - external rendering and sketch of the semi-circular ramp*

*To highlight and clarify some of the results discussed within this manuscript, some examples are provided, showing how experimental findings can be used to improve LG design. The following studies will try to focus on the main aspects of novelty, whereas standard and wellestablished procedures may be referenced and summarized but not*

In the following paragraphs, two case-studies are shown: the first, dealing with a cold-bending application of large glazing panes, the second with the post-failure

A recent concept design for a museum in Parma, Italy is taken as a case study. A ramp which contains part of an historical automotive exhibition grants access from the ground floor to the 1st floor of the building. This part of the structure is designed around two semi-circles: one for the external profile and one for an internal court

**Chapter 6 Applications**

*dealt with extensively.*

**6.1 Cold-bent glazing**

safety assessment of a compressed LG strut.

which serves as a light well (Figure 6.1).

# **Chapter 6 Applications**

*To highlight and clarify some of the results discussed within this manuscript, some examples are provided, showing how experimental findings can be used to improve LG design. The following studies will try to focus on the main aspects of novelty, whereas standard and wellestablished procedures may be referenced and summarized but not dealt with extensively.*

In the following paragraphs, two case-studies are shown: the first, dealing with a cold-bending application of large glazing panes, the second with the post-failure safety assessment of a compressed LG strut.

# **6.1 Cold-bent glazing**

208

*Figure 5.45* shows results of FE analyses, where it can be observed that mean TS

For each delamination ratio, the figure shows the upper and lower values of the stiffness increase of the interlayer due to TS, corresponding to the two cracking patterns of Figure 5.44. For instance, the stiffness of the interlayer is about twice as big for debonded surface ratios of ~40% and four times as big for ratios of ~15%. It can be observed that at increasing debonded surface ratio the two limit values of the stiffness increment approach one another and for high delamination ratios (more

Table 5.16 presents general information useful from a designer point of view to

**Type A** (2x1.52 SG)

**Type B** (2x1.52 DG41)

**Type C** (2x3.04 DG41 +2x0.2PC)

*Table 5.16 - noteworthy effects of progressive damage on LG beams*

Interlayer role for in-plane bending negligible negligible negligible

Longitudinal expansion limited† limited† limited† Lateral deflection none none none Effect on undamaged plies unk. non-negligible non-negligible In-plane bending interlayer influence negligible negligible negligible Tensile stresses relaxation rate unk. very slow slow

Longitudinal expansion limited/null ‡ limited/null ‡ Limited/null ‡ Lateral deflection non-negligible non-negligible non-negligible Effect on undamaged plies unk. negligible negligible Rotation at the end supports non-negligible non-negligible non-negligible In-plane bending interlayer influence small negligible small Tensile stresses relaxation rate unk. very slow slow

In-plane bending interlayer influence\* high limited considerable Long-term shape-preserving ability\* good very limited good Long-term fragments adhesion\* good very limited good

‡ *because of the significant lateral deflection, the longitudinal relative displacement of beam ends can result in a net con-*

† *generally negligible, provided there are no stiff constraints to elongation at both ends of the beam,*

\* *grades are assigned by comparison among the three specimen types.*

understand and compare effects of progressive damage on LG elements.

effect may be approximated using an exponential interpolation.

than 80%) they are almost zero.

**Symmetrical PDLG (Stage II)**

**Asymmetrical PDLG (Stage III)**

**FDLG (Stage IV)**

*traction rather than an elongation.*

**ULG**

**5.2.6 Synthesis of post failure LG properties**

A recent concept design for a museum in Parma, Italy is taken as a case study. A ramp which contains part of an historical automotive exhibition grants access from the ground floor to the 1st floor of the building. This part of the structure is designed around two semi-circles: one for the external profile and one for an internal court which serves as a light well (Figure 6.1).

*Figure 6.1 - external rendering and sketch of the semi-circular ramp*

209 Lorenzo Ruggero Piscitelli, *Serviceability and post-failure behaviour of laminated glass structural elements*, © 2020 Author(s), content CC BY 4.0 International, metadata CC0 1.0 Universal, published by Firenze University Press (www.fupress.com), ISSN 2612-8020 (online), ISBN 978-88-6453-985-0 (online PDF)

The original design proposed the use of hot-bent LG panes (cfr. §1.1.1) to cover both the internal and external glazing surfaces. The hot-bending-process being very expensive and detrimental to the optical properties of glass, the aim of the following analysis is to propose solutions for alternative cold-bent LG solution.

The geometry of both glazing elements is presented in Figure 6.2, Figure 6.3 and Table 6.1.

*Figure 6.2 - layout of the case study with details on LG plates sizes and curvatures*

*Figure 6.3 - geometry of LG panes for the internal and external glazing*

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211

4 For external glazing elements that are exposed to direct sunlight, this hypothesis is highly conservative, consider-

ing thatsurface temperatures are likely to rise above ambient temperature.

**6.1.1 Design hypotheses and material properties**

ported pane1 (Figure 6.4.b),

for these analyses, both assumptions going towards safety.

are shown:

wards the supports on the vertical edges (cfr. Figure 6.4.a),

and performed at ambient temperature higher than 2

hours above elapse between installation and use3,4

Some hypotheses are made to perform an analytical verification of both elements and reliably apply the experimental results discussed in the previous chapters: LG panes are simply supported in the vertical direction, with supports placed at the top and bottom edges, bending is induced by pulling the LG elements to-

 to evaluate the effects of horizontal actions, vertical portions of the cold-bent element LG panes are extracted and studied as an infinitely long simply sup-

LG elements are made with tempered glass which is polished, has triangular fin-

the duration of the cold-bending phase is supposed to be higher than minutes

 installation must be performed before the end of June and the opening to the public must not be allowed earlier than September or it must be checked that over

 the building is a category C3 according to Eurocode 1, BSI 6399-1 and CNR-DT210/2013, so the horizontal design loads are: a uniformly distributed and a linear load at a height ,

1 The additional boundary effects on vertical edges and the bi-dimensional structure of the LG plates are neglected

2 This prescription as a design choice, to work with known values of the relaxation modulus. 3 This prescription in terms of months is intended to guarantee a minimum amount of time above 30°C between installation and opening to public. To this end, a comparison of weather station LIMP (Parma Airport, N44°49.33' / E10°17.72', elevation 49.1 m MSL) data over the years 2015, 2016 and 2017 was performed. Daily logs for 2017

,

,

ishing on the edges and is thicker than 5 mm (UNI-EN-12150 [297]),

# **6.1.1 Design hypotheses and material properties**

Some hypotheses are made to perform an analytical verification of both elements and reliably apply the experimental results discussed in the previous chapters:

1 The additional boundary effects on vertical edges and the bi-dimensional structure of the LG plates are neglected for these analyses, both assumptions going towards safety.

2 This prescription as a design choice, to work with known values of the relaxation modulus. 3 This prescription in terms of months is intended to guarantee a minimum amount of time above 30°C between installation and opening to public. To this end, a comparison of weather station LIMP (Parma Airport, N44°49.33' / E10°17.72', elevation 49.1 m MSL) data over the years 2015, 2016 and 2017 was performed. Daily logs for 2017 are shown:

4 For external glazing elements that are exposed to direct sunlight, this hypothesis is highly conservative, considering thatsurface temperatures are likely to rise above ambient temperature.

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210

*Figure 6.3 - geometry of LG panes for the internal and external glazing*

*Table 6.1 - median axis geometry for the cold-bent laminated elements*

height mm mm arc length mm mm vertical supports distance mm mm maximum sag mm mm radius mm mm arc rad rad

**external glazing internal glazing**

The original design proposed the use of hot-bent LG panes (cfr. §1.1.1) to cover both the internal and external glazing surfaces. The hot-bending-process being very expensive and detrimental to the optical properties of glass, the aim of the following

The geometry of both glazing elements is presented in Figure 6.2, Figure 6.3 and

*Figure 6.2 - layout of the case study with details on LG plates sizes and curvatures*

analysis is to propose solutions for alternative cold-bent LG solution.

Table 6.1.

The design maximum tensile stress in the glass plies is [38,300]:

$$f\_{g,d} = \frac{k\_{mod}\ k\_{ed}\ k\_{sf}\ \lambda\_{gA}\ \lambda\_{gl}\ f\_{g,k}}{R\_M\ \chi\_M} + \frac{k\_{ed}^{'}\ k\_{\nu}\ \left(f\_{b,k} - f\_{g,k}\right)}{R\_{M;\nu}\ \chi\_{M;\nu}}$$

with fixed , , , , , , and ; other coefficients may vary for different load durations and are shown in Table 6.2 along with the final result for .

*Table 6.2 – coefficients and values for the design tensile stress of tempered glass plies*

\* *a nominal duration of 30" is used forthe crowd actions on balustrades in* [38]

† *results from Linear Elastic Fracture Model (LEFM) are preferred to prEN16612*

‡ *values are different for the internal and external glazing; with a precautionary approach, the lowest value is considered*

As an original contribution to the design, the shear modulus of the three interlayer materials can now be calculated using the experimental results described in §0, verifying the compliance to the limits in §<IP_ADDRESS>. Table 6.3 shows analytical results compared to tabulated results provided in Appendix G of this manuscript.

*Table 6.3 – interlayer relaxation modulus and tabulated results (in brackets) in [MPa]*

The design values of the interlayer shear modulus have been calculated as:

$$\mathcal{G}\_{\text{int,}d} = \frac{\mathcal{G}\_{\text{int}}}{\mathcal{\nu}\_G}$$

where the safety coefficient was defined as:

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For load cases 2, 3 and 4, cold bending effects are accounted for as pre-stress: tensile effects induced by cold-bending add up with those produced by the live

*Figure 6.4 - a) scheme of cold-bent LG and b) an extracted vertical stripe "beam", used to compute the effects of horizontal loads accounting for c) cold-bending pre-existing stresses*

For the Poisson effect, cold-bending generates compressive and tensile stresses in perpendicular directions ( and in Figure 6.4c). Such stresses diminish over time as an effect of the decaying shear modulus and coupling abilities of all interlay-

ers.

**6.1.2 Design procedure and verifications**

the LG elements. Four cases are considered:

Verifications are made in terms of maximum tensile stress in the glass plies of

wards at 1.2m from the level of the floor plating, applied for 30" and ac-

*Case 2:* linear distribution of forces simulating the push of a crowd (pushing out-

*Case 3:* distributed pressure simulating the push of a crowd (pushing outwards, applied for 30" and accounting for cold-bending effects after 2 months), *Case 4:* distributed pressure simulating the effects of wind (pushing inwards, applied for 5" and accounting for cold-bending effects after 2 months).

*Case 1:* installation (progressive cold-bending in 30 minutes),

counting for cold-bending effects after 2 months),

#### **6.1.2 Design procedure and verifications**

Verifications are made in terms of maximum tensile stress in the glass plies of the LG elements. Four cases are considered:

*Case 1:* installation (progressive cold-bending in 30 minutes),

*Figure 6.4 - a) scheme of cold-bent LG and b) an extracted vertical stripe "beam", used to compute the effects of horizontal loads accounting for c) cold-bending pre-existing stresses*

For the Poisson effect, cold-bending generates compressive and tensile stresses in perpendicular directions ( and in Figure 6.4c). Such stresses diminish over time as an effect of the decaying shear modulus and coupling abilities of all interlayers.

For load cases 2, 3 and 4, cold bending effects are accounted for as pre-stress: tensile effects induced by cold-bending add up with those produced by the live

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212

The design values of the interlayer shear modulus have been calculated as:

distributed loads of wind, including safety factors and dynamic effects, account

 the self-weight only induces a minor in-plane compression, whose effects will be neglected for a safer design (slightly diminishing the tensile stress in glass).

with fixed , , , , , , and ; other coefficients may vary for different load durations and are shown in Table 6.2 along with the final result for .

*Table 6.2 – coefficients and values for the design tensile stress of tempered glass plies*

0.26† 0.65† 0.78† 0.88†

\* *a nominal duration of 30" is used forthe crowd actions on balustrades in* [38] † *results from Linear Elastic Fracture Model (LEFM) are preferred to prEN16612*

 0.80 0.80 1.00 1.00 0.89‡ 0.89‡ 0.89‡ 0.89‡ **52 MPa 58 MPa 79 MPa 82 MPa**

‡ *values are different for the internal and external glazing; with a precautionary approach, the*

*Table 6.3 – interlayer relaxation modulus and tabulated results (in brackets) in [MPa]*

As an original contribution to the design, the shear modulus of the three interlayer materials can now be calculated using the experimental results described in §0, verifying the compliance to the limits in §<IP_ADDRESS>. Table 6.3 shows analytical results compared to tabulated results provided in Appendix G of this manuscript.

**service (50 yrs) bending (30')**\* **crowd (30")**\* **wind gust (5")**

**ture SG DG41 PVB**

8.65 (14.3)

0.46 (0.47)

(5.47)

(13.5)

0.57 (0.66)

0.17 (0.18)

0.57 (0.57)

0.72 (0.72)

(75.6)

(11.5)

for a total of according to Italian standards [298,299],

The design maximum tensile stress in the glass plies is [38,300]:

*lowest value is considered*

**phase/verification time & tempera-**

installation 30 min. at 25°C 65.38

serviceability 200 hrs. at 30°C 10.79

where the safety coefficient was defined as:

crowd push 30 sec. at 35°C 61.5 (61.5) 5.47

wind gust 5 sec. at 35°C 77.3 (77.3) 13.5

loads. On the other hand, to further increase safety, compressive stresses induced by cold-bending will not be accounted for.

*Figure 6.5 - schemes for studying the effects of live-loads in Cases 2, 3 & 4*

Cases 2, 3 and 4 contain both actions of cold-bending and live loads on glass, two effects which refer to different times. In accordance with the CNR standard [38], a procedure analogous to the Palmer-Miner linear damage hypothesis for fatigue is adopted [301,302]. Therefore, verifications are satisfied when:

$$\sum\_{i=1}^{N} \frac{\sigma^i}{f\_{g,d}^i} < 1$$

where and are the -th imposed and resistant stresses respectively.

The Enhanced Effective Thickness (EET) method is used to calculate the equivalent thickness of the LG sandwich structure [113] (cfr. §2.2.2).

An example of the procedure for cases 1 and 2 of the external LG plate is described, all other cases and the internal glazing can be dealt with in an analogous manner.

#### **Case 1**

The LG plate is made with two glass plies of the same thickness , bonded with a thick SG interlayer. For cold-bending, a soft interlayer goes towards safety, hence and . Considering a stripe of LG of unitary amplitude and the glass elastic modulus , the coupling coefficient can be calculated as:

$$\eta = \frac{1}{1 + \frac{E}{G\_{int}} \frac{h\_{int}}{b} \frac{f\_{abs}}{f\_{full}} A^\* \Psi} = 0.9998$$

where:

$$f\_{abs} = \frac{h\_1^3}{12} + \frac{h\_1^3}{12} \quad ; \quad f\_{full} = \frac{(h\_1 + h\_2 + h\_{int})^3}{12} \quad ; \quad A^\* = \frac{h\_1 h\_2}{h\_1 + h\_2} \quad ; \quad \Psi = \frac{168}{17B^2}$$

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Effects of cold-bending after the design relaxation period are calculated with the same procedure described for Case 1, the only difference is the diminished shear re-

Such value of very close to 1, hints that the behaviour of the plate is very close to the monolithic limit. The equivalent thickness which can be used to model the

   The horizontal stress and vertical stress for the -th ply in Case 1 are:

The curvature imposed with the cold-bending is ,

The verification in this case is possible by simple comparison of stresses

and with the design maximum tensile stress of .

The equivalent thickness for calculating the stresses in the -th glass ply is:

bending stiffness of the plate is therefore:

hence the –constant- bending moment induced is:

where:

**Case 2**

laxation modulus which is now:

Such value of very close to 1, hints that the behaviour of the plate is very close to the monolithic limit. The equivalent thickness which can be used to model the bending stiffness of the plate is therefore:

$$\hat{h}\_{\text{w}} = \sqrt[3]{\frac{1}{h\_1^3 + h\_2^3 + 12(h\_1d\_1^2 + h\_2d\_2^2)} + \frac{1 - \eta}{2s^3}} = 17.51 \text{ mm}$$

where:

$$d\_1 = U - \frac{h\_1}{2} \qquad ; \quad U - \frac{h\_2}{2} \qquad ; \quad U = \frac{h\_1 + h\_2 + h\_{int}}{2}$$

The equivalent thickness for calculating the stresses in the -th glass ply is:

$$\hat{h}\_{i,\sigma} = \left| \frac{1}{\frac{2\eta |d\_i|}{h\_1^3 + h\_2^3 + 12(h\_1d\_1^2 + h\_2d\_2^2)} + \frac{h\_i}{\hat{h}\_{\mathbf{w}}^3}} = 17.51 \, mm^3$$

The curvature imposed with the cold-bending is , hence the –constant- bending moment induced is:

$$M\_{cb} = \kappa E I\_{eq} = \kappa E \left(\frac{\hat{h}\_w^3}{12}\right) = 1.06 \cdot 10^5 \text{ Nmm}$$

The horizontal stress and vertical stress for the -th ply in Case 1 are: