Engineering Analysis and Design Method for Blast-Resistant Laminated Glass Composite Systems

Abstract

Laminated glass (LG) composite systems are increasingly being utilized in architectural and security applications due to their enhanced strength and safety features. Understanding the structural response of LG systems is crucial for optimizing their performance under blast loads. This paper presents a comprehensive study of an analytical model for predicting the static and dynamic resistance functions of various LG systems used in blast-resistant designs to advance engineering analysis and design methods. The proposed analytical model integrates the strain-rate-dependent interlayer behavior with the glass dynamic increase factors to generate a physically consistent post-fracture membrane resistance, offering a unified framework for deriving the static and dynamic resistance functions directly applicable to single-degree-of-freedom (SDOF) analyses across different LG layups. The developed models were validated statistically using full-scale water chamber results and dynamically against experimental blast field data and the results from shock tube testing. We validated the model’s accuracy for various LG layup configurations, including variations in the glass and interlayer sizes, types, and thicknesses. The established dynamic resistance model was developed by incorporating a strain-rate-dependent interlayer material model. The energy absorption of LG panels, influenced by factors like interlayer thickness and type, is critical for blast design, as it determines the panels’ ability to withstand and dissipate energy, thereby reducing the transmitted forces and deformations to a building’s structure. The dynamic model closely matched the dynamic deflection time histories, with a maximum difference of 6% for all the blast experiments. The static resistance validations across the various LG configurations consistently demonstrated reliable prediction results. The energy absorption comparisons between the analytical and quasi-static LG panel responses ranged from 1% to 17%. These advancements provide higher-fidelity SDOF predictions and clear guidance for selecting the interlayer type and thickness to optimize energy absorption. This will result in enhanced blast resistance and contribute to more effective blast mitigation in glazing system design.

1. Introduction

Blast events can cause significant threats to structures and their occupants, making the study of blast-resistant materials crucial for enhancing safety and mitigating damage. Laminated glass (LG) has emerged as a key component in blast-resistant designs due to its ability to withstand high-energy impacts. LG involves two or more layers of glass stuck together with interlayers, typically made of polymeric materials like polyvinyl butyral (PVB) or ethylene–vinyl acetate (EVA). Such interlayers enhance the structural integrity of the glass, making it more resistant to penetration and shattering during blast events. Understanding the performance of LG under blast events is essential for optimizing its performance in blast-resistant applications. Various design parameters for LG were investigated to determine their influence on both the static and blast resistance [1,2]. Parameters, such as glass type, interlayer type, and interlayer thickness, were analyzed to understand their effect on the response of LG systems [1,2].

The mechanical behavior of LG is predominantly influenced by the interlayer material, emphasizing its critical role when designing and estimating the structural elements comprising LG. The interlayers display viscoelastic properties, meaning that their stiffness and mechanical responses change with the load duration and temperature [3,4,5,6,7]. Consequently, the mechanical properties of LG are influenced by these factors [8,9,10,11]. In tensile testing, SentryGlas (SG) exhibited an elastoplastic response under varying strain rates, with higher strain rates resulting in higher yield stresses and elongation, whereas the strain rate had no significant effect on the failure strength [12,13,14].

Glass may shatter during a blast or hurricane event, resulting in many sharp pieces that may be launched into the interior of buildings at high speeds, potentially causing harm to their occupants. In recent years, there has been a substantial increase in interest in designing civilian structures that are capable of withstanding blast loads. Evaluating the behavior of glazing is crucial, as the majority of blast-related injuries result from flying shattered glass [15]. In a full-scale quasi-static water chamber, laminated panels were tested using three unique interlayers: PVB, EVA, and SG. The EVA interlayers exhibited higher levels of deformation compared to the SG, which showed lower deformation. Additionally, size increases in the panels with SG and PVB interlayers led to the most significant reduction in static strength [1,16].

The large deformation theory was employed to derive approximate solutions based on the numerical results [17]. Two distinct finite element (FE) models, a physical and smeared model, were compared and analyzed for simulating impact tests on LG windows [17]. Several studies have studied the resistance of LG systems employing PVB interlayers under varying strain rates. In blast design, simplified dynamic models are commonly utilized to predict the behavior of structures or components, like walls, beams, and windows. The resistance function of LG systems can be utilized in a single-degree-of-freedom (SDOF) model to anticipate the dynamic response [18]. Another study evaluated the experimental and theoretical responses of monolithic glass and LG across temperatures ranging from 0 to 77 °C [19]. Researchers have also investigated the influences of interlayer thickness and the effects of 1 hr sustained loading at various temperatures [20].

During the pre-glass-cracking stage, the polymeric interlayers exhibit limited flexural stiffness; however, they still play a crucial role in the behavior of LG by providing the shear stresses [21]. These shear stresses help to restrain or limit the comparative slipping of the glass panes, which assists in maintaining the integrity of an LG panel [21]. A study conducted experiments [22] to compare the maximum failure stress of LG and monolithic glass with identical geometries and total thicknesses. The research showed that at room temperature, the failure strengths of annealed LG matched those of annealed monolithic glass of the same thickness, but decreased to 75% of monolithic glass’s resistance at 77 °C [22]. Under blast loading, another study found that there was no obvious differences in the deflection or maximum principal stresses between monolithic glass and LG plates of the same total thickness [23].

One well-known approach is the First-Order Shear Deformation theory, which assumes that the planes normal to the middle plane stay straight, although not certainly perpendicular to it, beyond deformation. This theory has been widely used by researchers and has been shown to provide good results, especially when the shearing rigidity is appropriately chosen [24,25]. ASTM E1300 [26] highlights a procedure for calculating the load resistance of glass in buildings. It provides methods for determining the maximum allowable load that a glass lite can withstand before breakage occurs, considering various factors, such as glass type, thickness, size, support conditions, and probability of breakage. Another approach, referred to as the Enhanced Effective Thickness method, includes LG beams with three layers or multiple equally thick layers of glass, demonstrating its applicability to a wider range of configurations. The method’s accuracy, confirmed through comparisons with experiments, suggests its potential for accurately predicting the behavior of laminated glass composite structures in various practical applications [27].

The existing design software programs adjust the static resistance of LG using dynamic increase factors. There are various programs capable of determining window responses to blast loads. There are three primary codes in the U.S.; most of these analysis codes use the SDOF method, some of them depend on analytical methods only, and others depend on both analysis and experimental results. A comparison of the different software tools is provided in

Table 1. Comparison among LG Windows Design Software.

Software	Analysis Method	Analysis Modules	Analysis Output
HazL (v 1.2)	SDOF. Fragment flight model.	Monolithic glass, plastic windows, laminated windows. Insulated glass units, windows retrofitted with anti-shatter film.	The hazard levels. Glazing response parameters. Reaction loads and frame bite. Fragment trajectory. Hazard rating. Pressure–impulse diagrams.
WINGARD (v 6.2)	Analytical methods.	Laminated windows.	Glazing response parameters.
SBEDS-W (v 5.1)	SDOF. Fragment flight model.	Glazing, frame members, window mullions. Laminated, filmed, annealed, heat-strengthened, tempered glazing in single or double-pane windows.	The window hazard level. Fragment throw, deflection. In-plane and out-of-plane reaction loads. Analyze frame and mullion components. P-I diagrams.
WinDAS (v 1.0)	Experimental by agencies in the U.S. and the U.K. Analysis guide.	Laminated windows.	The window hazard level. Glazing response parameters.

.

The software tools include Window Fragment Hazard Level Analysis (HazL) [28], WINdow Glazing Analysis Response and Design (WINGARD) [29], the single-degree-of-freedom Blast Effects Design Spreadsheet for Windows (SBEDS-W) [30], and Window Design and Analysis Software (WinDAS).

Although these tools and prior studies have contributed to LG modeling, several critical limitations remain. First, the existing analytical models assume the polymeric interlayer is a rate-independent material, often using quasi-static properties or empirical dynamic factors. For example, Behr et al. [19,20] and Wei et al. [23] predicted the LG response without accounting for the strain-rate-dependent interlayer behavior. In addition, software programs, such as HazL [28] and SBEDS-W [30], primarily incorporate dynamic increase factors for the glass while applying simplified adjustments for the interlayer. Such assumptions neglect the substantial experimental evidence showing that the interlayer stiffness and energy absorption capacity increase significantly under higher strain rates, directly affecting the post-fracture membrane resistance [6,12,14].

A second limitation lies in the simplified modeling of glass–interlayer interactions after fracture. Many analytical methods, such as the effective thickness approaches [27,31], approximate the cracked system response but overlook the critical role of interlayer elongation and bond transfer between the broken glass fragments. The studies by Timmel et al. [17] and Pelfrene et al. [32] demonstrate that the post-breakage resistance is strongly controlled by the interlayer’s ability to transfer membrane stresses; however, this mechanism is often reduced to oversimplified assumptions in the existing models.

For the SDOF analysis, LG beams were tested experimentally to assess their behavior [33]. Using the same material as the field specimens, pin and roller supports were utilized to test the beams under four-point bending. A pressure–displacement resistance function was formed by converting the force–displacement response of LG beams into an effective thickness function. This function was analyzed using programs like SBEDS, WINGARD, and CWBlast, with SBEDS showing the most accurate predictions, especially when using the laboratory-measured resistance function. However, none of the programs accurately modeled the overall displacement–time history response, and improvements are needed for better predictive capabilities [33].

A comprehensive numerical, analytical, and experimental study is needed to improve the existing engineering analysis of and design methods for LG windows under blast loading. This paper introduces an analytical model for the analysis and prediction of the static and dynamic resistance function of different LG systems used in blast-resistant designs. The models are intended to aid engineers and manufacturers in improving the resistance of critical infrastructure against different blast scenarios. Despite significant progress, the current analytical models do not fully capture the strain-rate dependency of polymeric interlayers and the mechanical interaction mechanisms governing post-breakage load transfer. The present work addresses these limitations by introducing an analytical model that integrates the strain-rate-dependent interlayer behavior with the glass dynamic increase factors to generate a consistent post-fracture membrane resistance. The results generated by the analytical models are validated statically and dynamically against experimental field data and previously published results obtained from the shock tube testing of blast-resistant LG systems. The research presented in this paper is supported by these validations, which evaluate the model’s reliability and accuracy across various configurations and layup LG systems, including different parameters such as glass sizes, types, and thicknesses, and different interlayer material types and thicknesses.

2. Blast Design Methodology

Many researchers have investigated the resistance function of an LG system with a PVB interlayer under different strain rates. Walls, beams, and windows are common structural components whose behaviors are often predicted with simplified dynamic models. An LG system’s dynamic resistance can be predicted using the resistance function of the LG system through an SDOF simulation [18].

There are three different stages in the load–deformation response of a laminated glass composite system, which are idealized in Figure 1. An effective design requires the derivation of the resistance function that can describe this behavior. In this scenario, the behavior of the LG layers under a load is characterized by distinct stages. Initially, the first non-cracked LG layer experiences elastic deformation until it fails, shown by point 1 in Figure 1. This failure leads to a decrease in the pressure without any notable change in the deflection until point 2 in Figure 1. During the second stage, as the load is transferred to the second non-cracked LG layer, there is a subsequent increase in the pressure until point 3 in Figure 1, followed by a decrease without a significant deflection change until point 4 in Figure 1. Finally, the cracked LG layers, supported by the polymeric interlayer, exhibit elastic resistance before ultimately failing at point 5 in Figure 1. This sequential response highlights the complex behavior of LG under uniform loads.

2.1. Response of Glass Under Static and Dynamic Loading

Designing structural elements to withstand incidental explosions, particularly during blast events, requires a deep understanding of how structural materials behave dynamically and how the components and systems respond to such loads. When subjected to dynamic loads from explosions, structural systems and materials exhibit behaviors that differ from those under static loads. One key difference is that materials can display an enhancement in strength under dynamic loading, potentially improving the structural response [34]. As a result, structures exposed to blast events are often designed to experience plastic deformation, allowing them to absorb more energy from the explosion. Static loads are gradually applied to the system and remain constant for a long duration, associated with the structure’s response time. In contrast, blast loads involve a rapid application of force, leading to a rapid increase in the structural element stresses [18].

Materials respond differently under dynamic loads compared to static loads, particularly polymeric interlayer materials [6]. In Figure 2a, rapid loading (high strain rate) increases the stress levels and resistance before the rupture of the interlayer material, demonstrating that the strain-rate dependency is crucial. This phenomenon is significant [34], as the increase in strength enables both the interlayer and the glass to achieve a higher level of resistance, typically ranging from 10 to 30%. In blast design, this effect is accommodated through the use of a dynamic increase factor (DIF) [35]. Therefore, when predicting the static resistance function for LG panels, the static response of the polymeric interlayer, without considering any dynamic increase factors, should be used. On the other hand, to compute the dynamic resistance of LG systems, the predicted resistance function used in the SDOF dynamic analysis should utilize both the dynamic response of the interlayer and the dynamic increase factor (DIF) for the glass. Based on Equation (1), the DIF for the glass can be determined based on the particular value of the strain rate ($\dot{\varepsilon }$) [36]. This equation was derived from a dataset of 30 tensile tests on annealed float glass conducted at strain rates of up to 1000 s⁻¹ [36]. While the dataset was obtained for annealed glass, the same formula has been widely applied in the literature to both annealed and tempered glass, as both exhibit similar strain-rate-strengthening trends, with tempered glass showing a higher baseline tensile strength [36].

$$\begin{array}{cc}DIF=1.137+0.015\mathit{log}\left(\dot{\varepsilon }\right)& \mathrm{When} {1.0}^{-5}\le \dot{\varepsilon } \le 350\\ DIF=-2.911+1.608\mathit{log}\left(\dot{\varepsilon }\right)& \mathrm{When} 350\le \dot{\varepsilon }\end{array}$$

(1)

The normalized static and dynamic resistance functions for a typical LG panel made up of two layers of tempered glass with a PVB interlayer are shown in Figure 2b. The analytical model for the static resistance in Figure 2b was developed using both the quasi-static material properties of tempered glass and PVB shown in Figure 2a. In contrast, the analytical model for the dynamic resistance shown in Figure 2b used a DIF of 16% from Equation (1) to modify the tempered glass response. In addition, the dynamic response of PVB at a strain rate of 45 s⁻¹, shown in Figure 2a, was used to predict the tension membrane resistance of the interlayer. The combined dynamic resistance of the LG system (glass and interlayer) is shown in Figure 2b. The details of the model for developing the analytical resistance function of LG systems are described in Section 2.4. The dynamic resistance of the system in Figure 2b has an energy absorption capability of more than 2.2 times that of the static resistance of the LG system. Understanding these differences is crucial for engineers designing LG systems to withstand blast loads. By considering the dynamic properties of materials and allowing for plastic deformation, engineers can design structures that can absorb and dissipate the energy from explosions, enhancing the safety and resilience of LG systems.

2.2. Single-Degree-of-Freedom (SDOF)

Glazing windows can be simplified and represented using an SDOF model, which combines springs, masses, and a damper. In blast-loading scenarios, damping is typically disregarded because it has minimal impact on the initial peak deflection [30]. For instance, an LG window under a uniform dynamic load can be simulated as a basic spring–mass system, as illustrated in Figure 3. To specify an equivalent SDOF system, it is mandatory to calculate different system parameters, such as the mass, M_e, the spring constant, k_e, and the applied dynamic pressure or force, F_e. The spring constant, k_e, represents the resistance of the system and can be determined from the structural properties of the system. In the case of LG windows, it can be defined as the force–displacement relation that is called the resistance function, R.

Biggs (1964) [18] suggested a list of corresponding transformation factors to equate the equivalent SDOF system to the real structural system. The details of this procedure and the transformation factors can be found in Biggs (1964) [18]. The equation of motion can be formulated for the equivalent SDOF system by applying the concept of dynamic equilibrium.

2.3. Resistance Function

Figure 1 illustrates the load–deflection response of typical LG panels. These panels commonly demonstrate a two-stage response: an initial elastic phase until the glass layers break, known as the pre-breakage stage, followed by the tension membrane response of the broken window, identified as the post-breakage stage. To establish the first point on the static resistance function, the tensile strength of the glass is initially calculated, which can be determined through material testing or standards (e.g., ASTM C158) [37], depending on the glass pane type. After calculating the stress based on the tensile strength of the glass, the failure pressure (P) of the first glass layer can be obtained through the plate theory approach [31]. It is important to note that the fracture of the glass occurs first in the layer directly exposed to the applied pressure, which includes the entire LG cross-section, including its thickness, stiffness, and moment of inertia. Then, to evaluate the corresponding deflection, several researchers have addressed the clamped and simply supported rectangular thin-plate problem under uniform loading, offering approximate solutions [31,38,39,40,41,42,43]. Therefore, the midspan deflection of the first point can be evaluated using the governing Equation (2), with the previously calculated failure pressure, P.

$$\delta ={\displaystyle \frac{\mathrm{\alpha }\ast P\ast a^4}{\mathrm{D}}}$$

(2)

The value of α, a numerical factor dependent on the ratio b/a of the plate edges, can be found in Timoshenko’s book (1959) [31]. Table 8 (page 120) provides the α values for simply supported rectangular plates, while Table 35 (page 202) contains the values for clamped rectangular plates [31]. Following the initial failure of the first glass layer, the response drops at the same deflection but with a new pressure value. This pressure can be computed by applying Equation (2) with the known deflection, considering adjustments to the section properties after the first layer’s failure. This adaptation neglects the failed first layer and focuses on the remaining intact layers, primarily the interlayer and the second glass layer.

After the response drop, the load is transferred to the polymeric interlayer and the second glass layer as the primary components bearing the applied pressure. This transition increases the pressure until the eventual failure of the second glass layer. This failure pattern highlights the nature of LG failure under increasing pressure, emphasizing the need to consider the behavior of individual layers in the overall response of LG to the loading conditions. It is significant to state that, at this stage, the influence of the broken first glass layer on the strength and stiffness becomes negligible; however, its contribution to the mass remains significant and must be considered. When calculating the failure pressure (p) using the same steps as for the first point, the analysis focuses uniquely on the effect of the polymeric interlayer and the second glass layer, as the first glass ply has already failed and no longer contributes significantly to the load resistance. The evaluation of the deflection at this point follows the same procedure as in the previous point, using Equation (2). Subsequently, another drop in the response occurs due to the breakage of the second glass pane at the same deflection. The pressure at this point can be evaluated using Equation (2) as well.

The behavior of LG after cracking is mostly affected by the interlayer material. As soon as both glass layers have cracked in the LG configuration, the interlayer carries the entire load on the window, experiencing tension and generating membrane stresses [32]. Remarkably, the interlayer can continue to bear the load even after glass failure, as it can elongate significantly. Figure 4 explains the development of LG failure criteria, showing the post-break phase where the interlayer initially behaves elastically, followed by yielding, showing an elastoplastic response [32]. While the impact of broken glass fragments on the strength and stiffness can be disregarded at this stage, their contribution to the mass must be considered. The total mass of the window in the post-break phase includes the protective interlayer, as well as any remaining glass fragments still attached to these elements after the glass breakage. As the displacements increase, membrane forces develop in the window due to the axial stretching of the polymeric interlayer. For the short rectangular membrane, the deflection at the center of such a membrane for any pressure value (P) can be obtained through Equation (3), referring to the Stress Analysis Manual by the Air Force Flight Dynamics Laboratory [44].

$$\delta =n_1\ast a\ast \sqrt[3]{{\displaystyle \frac{P\ast a}{E\ast t}}}$$

(3)

The value of n₁ is determined by calculating the ratio a/b, where (a) represents the longer side of the rectangle and (b) represents the shorter side. It can be obtained from Figure 5 or the generated polynomial equation below, Equation (5).

After conducting many trials based on the method described above for calculating the membrane deflection, we identified a significant limitation of and errors with this approach. The method assumes a constant Young’s modulus for the interlayer material, overlooking its actual variation throughout the material’s response, especially under dynamic loads with high strain rates. This limitation becomes particularly apparent when considering the material’s strain-rate dependency. The strain rate has a substantial influence on the stiffness and the energy of the interlayer materials; however, the current method does not account for this dependency. The method, as described, involves using Equation (3) to evaluate the maximum deflection at the center of a short rectangular membrane for any given pressure value (P). However, this equation does not account for the changes in the Young’s modulus, which can lead to inaccurate predictions.

$$\begin{array}{c}{f}_x=n_2\ast \sqrt[3]{P^2\ast E\ast {\left.\left({\displaystyle \frac{a}{t}}\right.\right)}^2}\\ f_y=n_3\ast \sqrt[3]{P^2\ast E\ast {\left.\left({\displaystyle \frac{a}{t}}\right.\right)}^2}\end{array}$$

(4)

n₁ = 0.7057 − 0.5333x + 0.1833x² − 0.03002x³ + 0.001894x⁴

(5)

n₂ = 0.0721 + 0.3847x − 0.2609x² + 0.0741x³ − 0.009967x⁴ + 0.0005233x⁵

(6)

n₃ = 0.884 − 1.049x + 0.5611x² − 0.1538x³ + 0.02102x⁴ − 0.001134x⁵

(7)

To address these limitations, it is crucial to consider that the Young’s modulus of the interlayer material can vary with the strain rate. Therefore, to incorporate the variation in Young’s modulus, the stress at the center of the membrane in both the x and y directions can be calculated first using Equation (4), as outlined in the Stress Analysis Manual by the Air Force Flight Dynamics Laboratory [44]. Coefficients n₂ and n₃ can be found in Figure 5 or from the polynomial equations (Equations (6) and (7)) derived from Figure 5. After calculating the stress in both the x and y directions, the maximum stress value is selected based on the geometry of the membrane. By selecting the point on the dynamic interlayer response curve that corresponds to the specific stress value, the secant modulus (E_sc) is calculated. With this secant modulus, the modified equation below (Equation (8)) can be applied to accurately calculate the membrane deflection (δ_m):

$${\delta }_m=n_1\ast a\ast \sqrt[3]{{\displaystyle \frac{p\ast a}{E_{sc}\ast t}}}$$

(8)

This approach ensures that the variation in the Young’s modulus is taken into account, providing a more precise prediction of the response of the membrane under dynamic loading scenarios. Incorporating the strain-rate-dependent material properties is essential for more accurate and reliable analyses of LG systems subjected to dynamic loading conditions. Overall, this process leads to the development of a static resistance function for LG, which can be utilized in the SDOF model to analyze and predict the performance of LG under different loading conditions.

2.4. Analytical Model for Resistance Function

Developing an accurate analytical model for LG panels under uniform loading requires several key steps, as shown in Figure 6. First, an initial assumption is made regarding the deformed shape of the panel, typically based on the expected behavior of the material (glass and interlayer) under a load. This assumption forms the basis for further calculations and analysis. Next, the material properties of the LG, including the characteristics of the interlayer material, are considered. These properties play a crucial role in determining the panel’s response to applied loads. Once the initial assumptions and material properties are established, the next step is to evaluate the response through the governing equations, as described in Section 2.2, for both the glass and the polymeric interlayer that describe the behavior of the LG panel. These equations are derived based on the fundamental principles of mechanics and are used to calculate the stress and strain distribution within the panel. The final step is to obtain the pressure in terms of deflection for the static resistance function of LG systems. After this process, the dynamic response of the LG panels can be predicted through the equivalent SDOF system.

When deriving the resistance function, several assumptions were adopted to ensure analytical computability while maintaining consistency with field conditions. The applied blast load was assumed to be uniformly distributed across the LG panel, which is a standard assumption in blast design, since the reflected pressures typically act over the entire glazing surface. The boundary condition was modeled as simply supported, reflecting the realistic restraint of LG windows in frames, where the bite size is relatively small compared to the overall panel dimensions. This assumption has also been employed in prior studies [31,39,40,41,42,43]. Furthermore, the bond between the glass plies and the polymeric interlayer was assumed to be perfect (fully bonded) to capture the composite action and avoid introducing additional uncertainties from debonding, which is typically mitigated through manufacturing quality control in laminated glazing systems. These assumptions are widely accepted in the literature and provide a reasonable representation of practical glazing performance under blast loads.

The model, as currently formulated, does not account for environmental effects, such as temperature, humidity, or long-term aging. These factors were excluded to maintain the focus on the development of the mechanical resistance functions, but they can be incorporated into future work to extend the applicability of the model to real-world service conditions.

3. Static Resistance Function Calibration

This section provides a comparison between the predicted static laminated glass responses using the developed analytical model and the experimental tests. The experimental tests employed various sizes of LG panels and different types of glass and polymeric interlayers.

3.1. Water Chamber Static Testing Validation

A static resistance function calibration was conducted to validate the accuracy of the developed static analytical model for predicting the static response of LG. The calibration involved comparing the predicted static LG response with the experimental tests performed in a full-scale water chamber with two LG panel configurations: square LG panels with a size of 457 × 457 mm (18 × 18 inches), and rectangular LG panels with a size of 965 × 1676 mm (38 × 66 inches).

3.1.1. Square LG Panels

Figure 7 compares annealed LG panels, with a size of 457 × 457 mm (18 × 18 inches), using two different polymeric interlayers: PVB and UVEKOL-S. The experimental test was conducted in a water chamber (as shown in Figure 8), with the rate of loading controlled at a center deflection rate of 6.35 mm/min (0.25 in/min). The details of the quasi-static testing using the water chamber can be found in [1,45,46,47]. The static LG matrix for these panels is presented in

Table 2. Static LG square (S-LGS) panels test matrix.

#	Specimen ID	Interlayer Type	Interlayer and Glass Thickness		Glass Type **
			Interlayer, mm (in)	Inner and Outer * Layers, mm (in)
S-LGS-1	1/16A-0.09 UVEKOL-1/16A	UVEKOL-S	2.29 (0.09)	1.59 (1/16)	A
S-LGS-2	1/8A-0.12PVB-1/8A	PVB	3.05 (0.12)	3.18 (1/8)

* Inner glass layer is on the side where the load is applied. ** A = annealed glass. # Specimen label.

. The analytical resistance functions for the LG panels were used to predict the static response under quasi-static uniform loading. The SDOF analytical model utilized the static resistance of the interlayer materials, developed in a previous work [6], used in the LG panels.

Figure 7a shows the experimental results for the 1.59 mm (0.0625 in) annealed LG panel with a 2.29 mm (0.09 in) UVEKOL-S interlayer. The analytical model accurately predicts the peak stress of the first glass layer. However, it underestimates the maximum pressure on the second glass layer by approximately 20%. Additionally, there is a 24% difference in the interlayer’s response between the model’s prediction and the experimental behavior. The quasi-static response for the 3.18 mm (0.125 in) thick annealed LG panel utilizing a 3.05 mm (0.12 in) PVB polymeric interlayer is presented in Figure 7b. The predicted static resistance function is generally higher than the experimental response, particularly for the first cracked glass layer by 26%. Also, the model overestimates the interlayer resistance by 32% in the post-breakage phase, which decreases to 13% at interlayer failure.

3.1.2. Rectangular LG Panels

Table 3. Static LG rectangular (S-LGR) panels test matrix.

#	Specimen ID	Interlayer Type	Interlayer and Glass Thickness		Glass Type **
			Interlayer, mm (in)	Inner and Outer * Layers, mm (in)
S-LGR-1	0.375T-0.09 SG-0.375T	SG	2.29 (0.09)	9.53 (0.375)	T
S-LGR-2	0.375T-0.09 EVA-0.375T	EVA
S-LGR-3	0.375T-0.09 EVA-0.5T	EVA		12.7 (0.5) and 9.53 (0.375)
S-LGR-4	0.375HS-0.09 SG-0.375HS	SG		9.53 (0.375)	HS

* Inner glass layer is on the side where the load is applied. ** HS = heat-strengthened glass; T = tempered glass. # Specimen label.

presents the 965 × 1676 mm (38 × 66 inches) rectangular LG panels used in this study. These panels incorporated two types of glass, tempered and heat-strengthened glass, along with two different interlayer materials, SG and EVA. The experimental results were obtained from the same full-scale water chamber [1,45,46,47], see Figure 7, using a consistent displacement rate of 6.35 mm/min (0.25 in/min). The static response of the interlayer materials, SG and EVA, was utilized in the SDOF analytical model, which was conducted by a previous work [6]. Figure 9 shows the static analytical model used in the validation with the water chamber experimental test.

As illustrated in Figure 9a,b, the analytical and experimental results for the pressure–deflection response of the symmetric 9.53 mm (0.375 in) tempered glass panels with 2.29 mm (0.09 in) SG and EVA agreed satisfactorily, within 20% and 12% of each other, respectively. For the non-symmetric panel with 2.29 mm (0.09 in) EVA, shown in Figure 9c, the predicted static resistance function overestimated the experimental results by 5%. Remarkably, the analytical model exhibited a typical failure pattern of LG, characterized by failure in both the first and second glass layers. However, the experimental results showed that the two glass layers cracked simultaneously, indicating a different failure mode than what was predicted. This difference was attributed to several factors, including the simplifications and assumptions made in the analytical model. The analytical model’s assumption of sequential failure of the glass layers suggests a certain behavior under a load, where one layer fails before the other. However, the experimental result of simultaneous cracking in both layers for specimen S-LGR-3 indicates a more complex behavior not fully captured by the model. Furthermore, Figure 9d compares the experimental and analytical model results for the 9.53 mm (0.375 in) symmetric heat-strengthened panel with a 2.29 mm (0.09 in) SG interlayer. The predicted static resistance function exceeds the experimental response by 12%, and the model overestimates the interlayer resistance by 15% during the post-breakage phase.

Table 4. Energy absorption for the static LG panels.

#	Experimental (kPa-mm)	Analytical (kPa-mm)	Differences %
S-LGS-1	3208	2834	−11.5%
S-LGS-2	568	627	+10%
S-LGR-1	10,264	10,477	+2%
S-LGR-2	10,090	10,742	+6.5%
S-LGR-3	5905	4880	−17%
S-LGR-4	9840	9986	+1%

# Specimen label.

demonstrates the energy absorption capability of each LG panel, comparing the results from the experimental tests and the predicted static analytical model. The energy absorption capability of LG panels is crucial for blast design because it directly affects the panel’s ability to withstand and dissipate energy during an explosion. The energy absorbed by LG panels can be influenced by several factors, including the interlayer thickness and type, and the layup configuration of the panel. Panels with higher energy absorption capacities can help reduce the transmitted forces and deformations to the building structure, enhancing its overall blast resistance.

Overall, the analytical models demonstrate reliable prediction results for the quasi-static response of the LG panels, following the validations with various LG configurations (symmetric and non-symmetric) and parameters, such as different glass sizes, types, and thicknesses, as well as different interlayer material types and thicknesses. The observed energy absorption differences of 1–17% are mainly attributed to the post-breakage membrane elongation and localized glass fragmentation. The analytical model idealizes these effects, while the experiments naturally capture their variability, leading to some deviations that remain within an acceptable range and confirm the model’s reliability for practical blast-resistant design.

From the design perspective, these effects can be mitigated by selecting interlayer types and thicknesses that maximize the membrane capacity; combining different interlayer materials (e.g., PVB with SG) to balance stiffness, ductility, and energy absorption; ensuring consistent lamination quality; and applying appropriate frame restraint details to support energy absorption.

4. Dynamic Response Verification

This section aims to compare the predictions from the developed dynamic analytical model to the field blast response results and shock tube blast tests [48,49]. The comparison focuses on the displacement–time histories measured in the tests and those calculated by the analytical model, aiming to assess the model’s reliability and accuracy. Additionally, the analytical resistance functions for the LG panels were employed to predict the dynamic response under hypothetical blast-loading scenarios. The SDOF analytical model utilized the high-strain response of the interlayer materials, developed in previous work [6], with the DIF for the glass that can be determined based on the particular value of the strain rate. Figure 10 shows the typical dynamic behavior of an LG system under blast loading, demonstrating the different stages of the blast response.

4.1. Field IGU Panels

The validations are presented in Figure 11, including the annealed insulated glass unit (IGU) panels with different sizes, featuring two distinct polymeric interlayers: EVA and PVB [48]. The field experimental tests covered a wide range of pressure and impulse combinations, which were utilized for validating the dynamic response of the developed analytical model. Different layup LG systems were also used to demonstrate the model’s ability to predict the dynamic responses of various configurations. The dynamic IGU matrix is detailed in

Table 5. Dynamic annealed insulated glass unit (D-IGU) panels test matrix.

#	Specimen ID	Panel Size, mm × mm (in × in)	Interlayer Type	Interlayer and Glass Thickness			Blast Loading
				Interlayer, mm (in)	Inner and Outer * Layers, mm (in)	Middle Layer, mm (in)	Pressure (kPa)	Impulse (kPa-msec)
D-IGU-1	0.25A-0.5AIR-0.25A-0.12EVA-0.25A	711 × 1321 (28 × 52)	EVA	3.05 (0.12)	6.35 (0.25)	-	192	635
D-IGU-2	0.25A-0.5AIR-0.5A-0.12EVA-0.5A	1321 × 1321 (52 × 52)	EVA	3.05 (0.12)	12.7 (0.5)	-	195	929
D-IGU-3	0.25A-0.5AIR-0.5A-0.18EVA-0.5A	2540× 1321 (100 × 52)	EVA	4.57 (0.18)	12.7 (0.5) and 9.53 (0.375)	-	171	909
D-IGU-4	0.25A-0.5AIR-0.5A-0.06PVB-0.5A-0.06PVB-0.5A	1727 × 1118 (68 × 44)	PVB	1.53 (0.06)	12.7 (0.5)	12.7 (0.5)	427	821
D-IGU-5	0.25A-0.5AIR-0.5A-0.06PVB-0.5A-0.06PVB-0.5A-0.06PVB-0.5A						1059	1772

* Inner glass layer is on the side where the load is applied. # Specimen label.

.

Figure 11a illustrates the field blast results for the 6.35 mm (0.25 in) rectangular annealed IGU with a 12.7 mm (0.5 in) air gap and a 3.05 mm (0.12 in) EVA interlayer. The analytical model did not fully predict the response, especially the failure, in this specimen. This is because the developed model did not stop at the last resistance point but assumed that the resistance continued to increase, based on the interlayer’s membrane resistance. To address this, the model’s response was limited by an ultimate deflection based on the interlayer material’s tensile strength. On the other hand, Figure 11b shows the second IGU panel, which was a square panel using the same EVA interlayer but with a different glass thickness of 12.7 mm (0.5 in). The analytical model closely predicted the full response up to the peak deflection of this panel, with only a 1.2% difference.

The dynamic response of the third IGU panel, presented in Figure 11c, involved another rectangular panel utilizing a 4.57 mm (0.18 in) EVA interlayer. The predicted analytical response overestimates the field blast results by 2.1%, however, due to the excessive damage that occurred in this specimen, as shown in Figure 11f. Since the model did not consider rebound, it was unable to accurately predict the post-peak behavior. Figure 11d demonstrates the response of a multilayer IGU panel with three 12.7 mm (0.5 in) glass thicknesses and a 1.53 mm (0.06 in) PVB interlayer. The analytical model closely predicted the full response but underestimated the peak deflection of this panel by 5.7%. Another multilayer rectangular IGU panel with four layers of 12.7 mm (0.5 in) glass thickness and the same PVB interlayer was also analyzed to further validate the analytical model, shown in Figure 11e. The model was able to predict the full dynamic response for the deflection history with a difference of 6%.

4.2. Shock Tube LG Panels

This section validates the dynamic analytical model using four field test results from a series conducted for the National Institute of Building Science (NIBS) and Protective Design Center (PDC) [49]. The LG systems tested were single units measuring 1219 mm (48 in) in width and 1524 mm (60 in) in height, with two types of polymeric interlayer materials, PVB and SG, as detailed in

Table 6. Dynamic annealed LG (D-LG) panels test matrix.

#	Specimen ID	Interlayer Type	Interlayer and Glass Thickness		Blast Loading
			Interlayer, mm (in)	Inner and Outer * Layers, mm (in)	Pressure (kPa)	Impulse (kPa-msec)
D-LG-1	1/8A-0.06PVB-1/8A	PVB	1.53 (0.06)	3.18 (1/8)	33	248
D-LG-2	1/8A-0.06PVB-1/8A				40	310
D-LG-3	1/8A-0.06PVB-1/8A				56	414
D-LG-4	1/8A-0.1SG-1/8A	SG	2.54 (0.1)	3.18 (1/8)	48	379

* Inner glass layer is on the side where the load is applied. # Specimen label.

. Figure 12 illustrates the validation of the dynamic analytical model, showing that the predicted deflections closely matched the field results for all the panels.

Figure 12a–c present the same 3.18 mm (0.125 in) annealed LG panel with a 1.53 mm (0.06 in) thick PVB interlayer, subjected to three different blast-loading scenarios, shown in Table 6. The first panel experienced a pressure of 33 kPa and an impulse of 248 kPa-msec, with the analytical model underestimating the dynamic response by only 1%, as shown in Figure 12a. The second blast-loading scenario exhibited a difference of +2.5%, as illustrated in Figure 12b. Furthermore, Figure 12c shows the third LG panel, with the analytical model’s dynamic response exceeding the experimental results by 2%. For the last LG panel, utilizing a 2.54 mm (0.1 in) SG interlayer, the predicted dynamic response by the developed model showed a 10 mm difference compared to the shock tube test response; see Figure 12d.

At a high strain rate, 45 s⁻¹, both the experimental results and analytical model indicates that using SG instead of PVB provides an approximate 35–40% increase in the post-fracture resistance. This improvement reflects the higher stiffness and reduced viscoelastic compliance of SG, which enhances the overall blast resistance of laminated glass systems.

In summary, the dynamic analytical model predicted the experimental blast response for all the different LG systems employing various panel sizes and thicknesses, interlayer types and thicknesses, and layup configurations. The analytical model was able to closely predict the dynamic response for a wide range of pressure–impulse combinations. In addition, the developed dynamic model closely matched the blast dynamic deflection history throughout the whole response up to the peak values. It is important to note that, in general, the model was not reliable for predicting the post-peak response because the model did not predict the rebound if severe damage occurred, as shown in Figure 11f for specimen D-IGU-3. When failure occurred, such as in specimen D-IGU-1, the analytical model did not predict such a response. This can be attributed to the resistance function, which did not terminate at the last resistance point when using the developed analytical model. Instead, the analysis assumed that the resistance kept increasing based on the membrane resistance of the interlayer. To account for this, an upper deflection limit was imposed on the analytical model’s response based on the tensile strength of the polymeric interlayer material; see Figure 11a. This limit was calculated using a localized length of the interlayer based on the shorter dimension of the panel [50,51,52].

The predictive accuracy of the proposed model is primarily governed by the interlayer properties, particularly the type and thickness, which control the post-fracture membrane capacity and energy absorption. The panel size and boundary conditions also influence the deformation response, underscoring the importance of carefully selecting these parameters in both modeling and design applications.

5. Conclusions

This paper presented an analytical model to improve predictions of the static and dynamic responses of various LG systems and their components for blast design purposes. The developed model was utilized to predict the static resistance function of different LG systems under uniform pressure up to failure. In addition, the dynamic resistance function of LG systems was used in the blast analysis, highlighting its capabilities. The engineering analysis and design method was used to predict the response of different LG systems. This paper investigated the various parameters known to influence the components and system capacities, such as the glass tensile strength, polymeric interlayer response (both static and dynamic), panel aspect ratio, the DIF for glass, the variation in the interlayer’s Young’s modulus, and the LG boundary conditions. The predicted resistance function of LG systems was statically verified against experimental data using full-scale water chamber results. The dynamic model was validated against results from experimental field tests conducted on IGU panels and shock tube tests performed on LG panels. Based on the findings of this study, the following conclusions are drawn:

• LG panels typically exhibit a two-stage response: an initial elastic phase until the glass breaks, followed by the tension membrane response.
• LG systems subjected to blasts undergo plastic deformation due to the polymeric interlayer, enabling them to develop higher energy absorption capabilities.
• The interlayers exhibit different dynamic and static material responses.
• For the static resistance function, the static material response of the interlayer was used, whereas, for the dynamic resistance, the high-strain-rate material response of the interlayer and the DIF for the glass were employed.
• In the analytical model, the failure pressures of both the first and second glass layers were obtained through the plate-bending approach. After the glass breakage, the resistance of the LG system is dominated by the tension membrane resistance of the interlayer.
• The static energy absorption comparisons between the analytical and quasi-static LG panel responses varied from 1% to 17%.
• Throughout the entire LG responses up to the peak values, the blast dynamic deflection time history was closely aligned with the dynamic model.
• If severe damage occurs during a blast event, the dynamic model cannot accurately predict the post-peak response since it does not account for rebound.
• The dynamic model assumes the tension membrane resistance continues to increase, and thus, an upper deflection limit based on the interlayer’s tensile strength should be incorporated into the blast analysis and design methods presented in this paper.

For engineering practice, the findings suggest that selecting an interlayer thickness to balance the stiffness and ductility is the key to enhancing blast resistance. Thicker or stiffer interlayers, such as SG, enhance the post-breakage membrane capacity, while more ductile interlayers, such as EVA, enhance the energy absorption through elongation. Hybrid layups (e.g., PVB with SG) can further optimize performance by improving both the ductility and strength. Combined with consistent lamination quality and proper frame restraint detailing, this provides a practical strategy to optimize LG systems for blast mitigation. Environmental factors, such as thermal variation, humidity, and aging, were not been incorporated into the present model. Future extensions may address these factors to align the model more closely with field performance conditions.