Consistency In Glass Design

Introduction: Sustainability, Reliability and Glass Design

Structural engineers often achieve sustainable designs by minimizing material usage and using material with low-embodied energy and/or carbon. Outside of concrete and steel, facades (glass and aluminum) comprise the largest environmental, economic, and social impact in the building industry (UNEP 2022). While new facade materials seek to mitigate climate impacts by reducing operational energy and carbon throughout the product lifecycle, minimizing the quantity of glass used in engineering applications remains one direct way to improve the sustainability of structural and architectural glass systems through direct reduction of material.

This research compares two common glass design methods used in the United States through a set of original worked examples. Comparison of results shows the importance of accurately modeling actual material behavior to 1) maximize sustainability through materially efficient designs and 2) produce designs with consistently reliable levels of risk. Results show that maximal sustainability of glass (via minimal material usage) correlates to consistent probability of failure metrics.

Current State of Glass Design: Probabilistic and Allowable Stress Methods

Glass structural elements have become increasingly common; however, there currently is not a universally recognized and codified glass design standard in the United States. As a result, while there are several efforts at developing design guides for structural glass (ASTM International 2016, NGA 2022, Feldman et al. 2014, Feldman et al. 2023, Wuest et al. 2023, March and Lancaster 2018) each has somewhat different design methodologies and concomitant assumptions. With the variety of design references, there is a potential for accidental confusion of valid application of these methods. Moreover, most structural glass in the United States is currently designed by either direct or indirect reference to ASTM E1300 (ASTM International 2016), which uses a probabilistic glass failure prediction methodology (GFPM) first developed by Beason and Morgan (1984) for annealed glass and later extended to heat treated glass (Morse and Norville 2012; Soules et al. 2020). GFPM design methodology applies probabilistic-based performance metrics to specifications of glass layups that meet consistent failure thresholds (i.e., 0.008 probability of breakage, accepted as “safe” design throughout the industry). Due to the perceived complexity of the GFPM, a significant portion of glass design still uses “strength design”, or allowable stress design (ASD). In the United States, the ASD method, as recently summarized in NCSEA Engineering Structural Glass Design Guide (March and Lancaster 2018), can also provide designs that appear to meet reliability indices, but result in overdesign and less sustainable structures. Moreover, sole reliance on the single largest maximum principle tensile stress (SLMPTS) is not always representative of actual performance as surface flaws often precipitate failure at lower stresses and different locations from the SLMPTS. A common mistake often takes values derived from probabilistic-based design (PBD), and with existing guidance from standards like the appendices of ASTM E1300, incorrectly applies those to an ASD method. Some examples of design elements commonly adopted from E1300 include procedures for developing load factors, equivalent thickness design, base stresses, and use of design factors in lieu of safety factors.

Allowable Stress Design

Examples 8.1 and 8.4 of the NCSEA Structural Engineering Glass Design Guide were taken as a representative ASD approach (March and Lancaster 2018) and used as a baseline for comparison with the PBD results. This ASD method begins with a base allowable stress, σ_base, which varies with each of the three main glass types and is used to calculate an allowable design stress, σ_allowable, for each possible load duration as:

σ_allowable = ψ_Pb∙ ψ_LDF∙ ψ_surface∙ σ_base

where Ψ_Pb = probability of breakage factor, Ψ_LDF = load duration factor, and Ψ_surface = surface treatment factor. A maximum moment, M, for the specimen is extracted and used in conjunction with σ_allowable to determine the minimum required section modulus, S, where S = M/σ_allowable. The largest required section modulus for each specimen given the set of load durations is then selected to determine the required buildup of the glass plate. Once the section parameters have been determined, σ_service is calculated from the greatest value of maximum moment at each load combination and the selected section modulus. The utilization of the design is determined as a percent of σ_allowable:

utilization_service = σ_service/σ_allowable

The plate geometry can be modified through iterative design to result in greater utilization or to adhere closer to standard glass plate sizes.

Probabilistic-Based Design

The probabilistic-based GFPM method utilizes finite-element analysis to generate the maximum and minimum principal stresses at the nodes of each of the mesh points, which were spaced at 25 mm (Soules et al. 2020). These field stresses, σ_i, and associated areas, A_i­­­­, were exported for calculation in excel, where each node was assigned its tributary area and consequently the probability of breakage (POB) according to the GFPM. The POB for an entire glass specimen is calculated by:

P_b = 1 – e^-B

where B, the risk function is found by the following summation,

B = k ∙ Σ[(c_i∙(t_d/60)^(1/n) ∙(σ_max - RCSS))^m∙A_i]

and where k = surface flaw parameter, equal to 2.86 × 10^-53 N^-7m¹², N = number of nodes where stresses were extracted, t_d = duration of loading = 3s (for analysis of 3s duration), n = static fatigue constant, 16, σ_max,i = maximum principal stress at i^th node (Pa), RCSS = 69 × 10⁶ Pa for FT glass, A_i = tributary area at i^th node. The biaxial stress correction factor, c­_i, is calculated by:

c_i = -0.005 ∙r_i⁶ + 0.022 ∙ r_i⁵ + 0.055 ∙ r_i⁴ + 0.039 ∙ r_i³ + 0.031 ∙r_i² + 0.06 ∙r_i + 0.08

with

r_i = (σ_min,i – RCSS)/( σ_max,i – RCSS)

where σ_min,i = minimum principal stress at i^th node (Pa).

For each plate, repeating the process of modelling, extracting, and calculating the POB until the POB was at, or just over 0.008 provides the necessary thickness given loading and geometric parameters. This procedure also established an envelope of required thicknesses for the POB to lie between 0 and 1.

Method of Comparison

The representative ASD method was taken from NCSEA (March and Lancaster 2018), while the representative PBD method was taken from ASTM E1300 (ASTM International 2016). A parametric analysis for 2-sided simply supported (i.e., 1-way span) rectangular glass plates used geometry with constant width of 1 m while varying the length from 0.5 to 3 m. The plates were modelled as single-ply thermally toughened float glass, with parameters Young’s Modulus, E = 70 GPa, Poisson’s Ratio, ν = 0.23, RCSS = 69 × 10⁶ Pa for FT glass and a density, ρ = 2500 kg/m³. A uniform live load of 1.91 kN/m applied over the face of the plate was used in all aspect ratios, and the only dead load was self-weight. The loads were adjusted to be compatible with the GFPM by using a load duration of 3 s, at a POB of 0.008. The stress factors used in ASD method then became Ψ_Pb = 1.000, Ψ_LDF.FT.3s = 1.000, and Ψ_surface = 1.000. In accordance with the ASD method for example 8.4 in March and Lancaster (2018), iterations began with a plate thickness 10% of the width and continued until thicknesses converged. The resulting plate thicknesses were rounded to the next 0.1 mm of thickness for comparison with the PBD method.

Results

Figures 1 and 2 show the minimum thickness for structural design of the glass plates given the uniform live load, for an equivalent duration of 3 s according to each method.

Figure 1 also includes a POB envelope that shows the range of required thicknesses for a POB between 0 and 1 as determined by the PBD methods. All values for glass plate thicknesses that lie above the POB envelope would have a POB of 0, and all those below the envelope would have a POB of 1, which is certain failure. For example, at an aspect ratio of 2.0, the glass has a POB of 100% at values below 6.5 mm, and a POB of 0 for values greater than 9.7 mm. Figure 2 shows the next closest commercially available nominal thickness for the plate at each aspect ratio for each design method. The solid lines near the top of the plot indicate the nominal values, while the dashed/dotted lines below mirror those found in Figure 1 for the calculated thickness values.

Discussion

Figures 1 and 2 show, in all aspect ratios of the glass plates, the methods incorporating ASD methods required glass of the same or greater nominal thickness than those based in strictly PBD. Figure 1 shows a linear relationship for both methods between required thickness and aspect ratio. The plate thicknesses calculated from ASD methods lie at the upper limit of the POB envelope as calculated per GFPM, but well above the required thicknesses for a probability of 0.008. This means that the POB for the ASD thickness is above 0, but unnecessarily conservative compared to the PBD method. The average ASD plate thickness is 16.4% greater than corresponding thicknesses calculated using GFPM.

In Figure 2, the required nominal plate thickness reveals a notable difference in required standard glass plate sizes. Examination of available nominal thicknesses indicates that on average, ASD glass plates required 20% greater material than PBD plates. The largest difference in necessary nominal thickness was at an aspect ratio of 2.5, in which the ASD method resulted in a plate 4 mm, or 33% thicker than the PBD method. Accounting for the same loading, more material would be required to meet the demands of the ASD, whereas PBD resulted in a thinner, more efficient design even after assigning the designs to available nominal sizes.

Conclusions and Future Work

Confusion may arise when a designer sees ASD methods employing aspects of PBD as a hybrid approach (as found in the NCSEA method). The NCSEA examples use some of the data and coefficients from E1300 but applies them to a standard FEA/mechanics model based in ASD. The result incorporates mechanistic methods with probabilistic methods, and this combination of incompatible processes lead to glass designs with inconsistent reliability. For example, the allowable stresses used in the ASD method (March and Lancaster 2018) are taken from E1300 and are associated with edge stresses for a maximum POB less than or equal to 0.008 for a 3s load duration. However, edges have more severe flaws (through cutting and handling) than the general plate surface, resulting in more stringent limiting stresses. Thus, the ASD design may result in a greater thickness than needed when considering surface stresses.

PBD is distinguished from ASD methods because it incorporates not only maximum stresses, but also field stresses and surface flaws. By calculating the POB across the entirety of each glass surface, the GFPM accounts for surface flaws that are known to cause a breakage at values below the SLMPTS. This summation of POBs at discrete points differentiates probabilistic design from elastic design, and cannot be scaled due to its nonlinear base equations.

The PBD method requires greater analytical complexity but should be used because it 1) more accurately models actual physical behavior 2) results in more efficient material usage (~15% less material on average than ASD) and 3) provides designs that have consistent reliability of failure. Acceptance of this probabilistic-based philosophy acknowledges the value in establishing a design that reduces the risk of failure to a consistent and acceptably low level. Moreover, this approach complements existing material standards (e.g., ACI 318, AISC 360, AWC NDS, TMS 402/602) to provide designs that consistently clear failure thresholds while minimizing material usage (and consequently, improving sustainability).

References

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ASTM International. ASTM E1300-16 Standard practice for determining load resistance of glass in buildings. West Conshohocken, Pennsylvania, USA. ASTM; 2016. www.astm.org p. 1-62.

Beason, W. Lynn, and Morgan, J. "Glass failure prediction model." Journal of Structural Engineering 110, no. 2 (1984).

Feldmann, M., et al. 2014. "Guidance for European structural design of glass components." Publications Office of the European Union. p 1-196.

Feldmann, M., et al. 2023. "The new CEN/TS 19100: Design of glass structures." Glass Structures & Engineering. p 1-21.

March, M., and Lancaster, F. 2018. NCSEA Engineering Structural Glass Design Guide. Chicago, IL: NCSEA.

Morse, S., and Norville, H. 2012. “Design methodology for determining the load resistance of heat-treated window glass.” Journal of architectural engineering, 18(1), 42-51.

NGA (National Glass Association with GANA). 2022. NGA’s GANA Glazing Manual. Vienna, VA.

Soules, J., Morse, S., and Norville, H. 2020. “Application of the Glass Failure Prediction Model to Flat Glass Using Finite-Element Modeling.” Journal of Architectural Engineering. 26 (2). 10.1061/(ASCE)AE.1943-5568.0000395.

Soules, J., Morse, S., and Norville, H. 2020. “Application of the Glass Failure Prediction Model to Flat Odd-Shaped Glass Using Finite-Element Modeling.” Journal of Architectural Engineering. 26. 10.1061/(ASCE)AE.1943-5568.0000437.

UNEP (United Nations Environment Program). 2022. “Global Status Report for Buildings and Construction: Towards a Zero‑emission, Efficient and Resilient Buildings and Construction Sector”. Nairobi.

Wüest, T., Fildhuth, T., and Luible, A. 2023. "New Swiss technical specification SIA 2057 for glass structures and its post failure limit state concept." Glass Structures & Engineering. p 1-13.