Scale modeling of thermo-structural fire tests of multi-orientation wood laminates

Abstract

The stacking sequence of laminated wood significantly impacts the composite mechanical behavior of the material, especially when scaling down thermo-mechanical tests on plywood. In previous research, we developed a scaling methodology for thermo-structural tests on samples with similar cross sections, however this paper focused on testing plywood samples with different stacking sequences between the scales. Plywood samples at ½-scale and ¼-scale were subjected to combined bending and thermal loading, with the loading scaled to have the same initial static bending stresses. While the ¼-scale 4-layer [0°/90°]s laminate and the ½-scale 8-layer [0°/90°/90°/0°]s laminate had an equal number of 0° and 90° layers, as the char front progresses, the sections behave differently. Thus, modeling becomes essential to extrapolating the data from the smaller ¼-scale test to predict the behavior of the larger ½-scale test. Reduced cross-sectional area models (RCAM) incorporating classical laminated plate theory were used to predict the mechanical response of the composite samples as the char front increased. Three methods were proposed for calibrating the RCAM models: Fourier number scaling, from detailed kinetics-based pyrolysis GPyro models, and fitting to data from fire exposure thermal response tests. The models calibrated with the experimental char measurements produced the most accurate predictions. The experimental char models validated to predict the behavior of the ¼-scale tests within 2.5%, were then able to predict the ½-scale test behavior within 4.5%.

Introduction

The fire resistance of all structural members and assemblies has traditionally been experimentally determined in standard large-scale thermo-structural tests, such as the ASTM E119 test method (ASTM 2022). However, standard fire resistance tests require specialized furnaces and large specimens, making them expensive. For example, ASTM E119 has minimum size requirements for loaded wall specimens of at least 100 ft² (9.3 m²) surface area, columns to be at least 9 ft (2.7 m) in length, beams to be at least 12 ft (3.7 m) in length, and floors/roofs to be at least 180 ft² (16.7 m²) in surface area (ASTM 2022). Smaller-scale fire resistance tests are more desirable because they are cost-effective, but currently, no small-scale test exists to evaluate fire resistance. In a previous study, we developed a theoretically based, experimentally validated scaling methodology for reducing the size of the test article while maintaining a similar thermal response as in the large-scale test (Gangi et al. 2023a). A subsequent investigation scaled the mechanical exposure on wood members subject to a combined thermo-structural loading (Gangi et al. 2023b). This study builds upon that work, investigating the additional complexity of multi-orientation materials, specifically the mechanical behavior of wood-based composite members.

In general, composite stacking sequence can significantly influence structural behavior. Previous studies have investigated the effect of ply stacking on the thermo-mechanical behavior of fiber-reinforced polymer (FRP) laminates (Bausano et al. 2006; Grigoriou and Mouritz 2017; Keller et al. 2006a, b; Loh et al. 2021, 2022; Tabaka et al. 2021). Grigoiou and Mouritz (2017) conducted a series of combined fire exposure and axial loading tests on FRP laminates with similar thickness but four different stacking sequences. Under tension, they found that the samples with 0° layers near the surface failed much sooner than the samples with the 0° layers closer to the middle of the sample. Bausano et al. (2006) used modeling to predict the behavior of FRP laminates in a similar set of one-sided fire exposure tests with sustained compression. The temperature rise in the sample was predicted using the finite difference method and the mechanical response was predicted based on both finite element analysis (FEA) and classical laminated plate theory (CLPT). Both the more complex FEA and more simplified CLPT predictions proved reasonable agreement with test data when assessing the times of failure. Keller et al. (Keller et al. 2006a, b) investigated modeling the thermo-mechanical behavior of full-scale GFRP slabs exposed to ISO 834 fire conditions on their underside. 2D thermo-chemical models predicted the temperature progression in the 120 min exposure, while the 3-D thermo-mechanical model accurately predicted the measured mid-span deflections. Additionally, Tabaka et al. (2021) presented a methodology for fire resistance testing of FRP laminates subject to compression in a bench-scale test setup. In all tests of conducted on these FRP laminates, the samples were usually much thinner than a typical wood-based composites.

The primary structural composite panels used in both residential and commercial timber building are wood-based composites like oriented strand board (OSB) and plywood. With the increased use of wood-based composites in building construction, it is important to characterize their fire performance (White and Winandy 2006). While the fire resistance of solid wood members has been well characterized, the performance of wood composites at elevated temperatures is less studied. Previous studies have conducted mechanical tests on laminated timber at elevated temperatures (Bekhta and Niemz 2003; Hirashima et al. 2020; Li 2020; Lie 1977; Malhotra and Rogowski 1967; Sinha et al. 2012; Sinha et al. 2011a, b; Sinha et al. 2011a, b; Soti et al. 2020; Wiesner et al. 2017, 2022). Sinha et al. (2011a, b) conducted a series of four-point bending tests on both OSB and plywood samples at elevated temperatures and found a reduction of strength and increased deformability as temperature increased. However, the tests were conducted at pre-charring temperatures (maximum 200 °C). While not mechanically loaded, Hasburgh et al. (2016) conducted fire exposure tests using ASTM E119 standard time-temperatures to characterize the effect of ply configurations on the fire performance of cross-laminated timber (CLT). That study found that when ply orientation was alternated at each layer (Long-Cross-Long), the cross-ply action prevented deep cracks from forming in the char layer. On the other hand, when two layers were oriented in the same direction (Long-Long-Cross), the char front was very irregular with deep cracks.

Similar to the FRP laminates, the thermo-structural behavior of wood-based composites has been predicted in previous studies with modeling. Under pure mechanical loading, Makowski (Makowski 2015) was able to predict the bending stresses of beech plywood members subject to three-point bending using classical laminated plate theory. Unique to wood-based members, the determination of fire resistance is based on the assumption that the wood Chars and the char layer that develops have zero stiffness (CEN 2015). Analytical models using the reduced cross-sectional area method (RCAM) introduce an effective residual cross-section with material properties at ambient temperature for the design of timber members exposed to fire (Lange et al. 2015; Schmid et al. 2010, 2012, 2014, 2015). The method considers the strength and stiffness reduction beneath the charred layer by adding an additional depth (known as the ‘zero-strength’ layer) to the charring depth (Schmid et al. 2015). Therefore, the RCAM models require the calculation of the charred depth. GPyro is a generalized pyrolysis model developed by Lautenberger and Fernandez-Pello (2009) has been used to simulate the kinetics of the decomposition of wood to char (Ferguson et al. 2013; Richter et al. 2021; Terrei et al. 2022; Vermesi et al. 2017). Richter et al. (2021) used GPyro to simulate wood pyrolysis of particle board samples at a constant heat flux and estimate the charring rate. Our previous study (Gangi et al. 2023b) combined GPyro simulations with the RCAM method to predict the behavior of combined thermo-structural tests of dimensional lumber samples. The advantage of testing dimensional lumber tests products was that they have all the wood grain in the same lengthwise direction, so a scaled-down version of a dimensional lumber section would have the same grain orientation. In contrast, in many engineered wood products, such as cross-laminated timber or plywood, the grain orientation of a thinner product may be vastly different.

Our study builds on previous research (Gangi et al. 2023a, b) on gypsum board and dimensional lumber by experimentally investigating thermo-structural scaling on plywood members. In this study, combined thermo-structural three-point bending tests were conducted on plywood samples at two scales, with the loading scaled to have the same initial static bending stresses. The plywood samples tested had different stacking sequences between two scales to demonstrate the importance of stacking on thermo-structural behavior. A consequence of the different stacking sequences is that the data from the reduced-scale test cannot be directly scaled to predict the behavior of the larger-scale tests. Thus, modeling becomes essential to extrapolating the data from the reduced-scale test to predict the behavior of the larger-scale test. Reduced cross-sectional area models incorporating classical lamination theory were used to predict the mechanical response of the composite samples as the char front increased. The RCAM models incorporated the data from both fire exposure thermal response tests and detailed kinetics-based pyrolysis modeling using GPyro to predict the char levels in the reduced-scale tests. Once validated to the reduced-scale tests, the analytical models were then extended to predicting the behavior of the larger-scale tests.

Thermo-structural test scaling laws for wood-based materials

Our previous study investigated the thermo-structural scaling laws for wood members through a series of thermo-structural three-point bending tests of dimensional lumber samples with one-sided fire exposure (Gangi et al. 2023b). This study will investigate the effectiveness of those same scaling laws when applied to engineered wood-based composites, namely plywood. This section summarizes the thermo-structural test scaling laws used in this study.

Sample geometry

The geometry of the reduced-scale samples in this study was scaled based on the assumption that the heat transfer from the furnace fire exposure will be primarily 1-D through the thickness of the sample. This assumption leads to test samples being designed with the same surface area, with the scale of the sample being determined by its thickness.

The resulting reduced-scale sample has the following dimensions for width, $b_{scale}$, length, $L_{scale}$, and thickness, $d_{scale}$,

$$b_{scale} = b$$

(1)

$$L_{scale} = L$$

(2)

$$d_{scale} = \chi d$$

(3)

where $b$ is full-scale sample width, $L$ is full-scale length, $d$ is full-scale thickness, and $\chi$ is the scaling factor. The scaling factor, $\chi$, is the ratio of the scaled dimensions (reduced-size) to the original dimensions (full-size). For example, for a half-scale test $\chi$ = 0.5. Scaling only the thickness of samples has implications for structural scaling, as maintaining the same surface area results in samples with the same bending stresses but different shear stresses. For that reason, the test loading was designed so that the $L/d$ ratio is high, ensuring that flexure will be the dominant mechanical behavior.

Mechanical loading

The mechanical loading of the reduced-scale test was scaled to produce similar bending stresses as the full-scale test. Stresses determine mechanical failure, and the assumption is that producing similar static stresses in the reduced-scale test will lead to a similar mechanical failure as the sample is damaged by fire exposure. At each scale, the samples will be subject to static three-point bending, with a static point load applied at the mid-span, $L/2$ from either support. In the full-scale tests, the maximum bending stress produced by the three-point bending of a sample, ${\sigma }_{x}$, is given by Eq. (4), while the maximum bending stress in the reduced-scale tests, ${\sigma }_{x,scale}$, is given by Eq. (5). Setting those values equal, the resulting static point load in the reduced-scale test, ${P}_{scale}$, is given by Eq. (6). To produce the same bending stress in a reduced-thickness sample, that the total applied load $P$ would have to be reduced by a factor of ${\chi }^{2}$. Please note that if the goal were to investigate shear failure, the test setup would need to be scaled by ${P}_{scale}=\chi P$ so that shear stress is the same between all scales, with the $L/d$ ratio lowered so shear would control over bending.

$${\text{Bending}}\; {\text{Stress}} = \sigma_{x} = \frac{3PL}{{2bd^{2} }}$$

(4)

$$\sigma_{{x,{\text{scale}}}} = \frac{{3P_{{{\text{scale}}}} L}}{{2bd_{{{\text{scale}}}}^{2} }} = \frac{{3P_{{{\text{scale}}}} L}}{{2b\left( {\chi d} \right)^{2} }}$$

(5)

$$P_{{{\text{scale}}}} = \chi^{2} P$$

(6)

Thermal exposure

In a previous study, we developed and demonstrated test scaling laws for one-sided fire exposure (Gangi et al. 2023a). Thermal scaling laws for one-sided exposure combined Fourier number time-scaling with boundary condition matching to produce similar thermal behavior. Once the thickness of the specimen is reduced by a factor $\chi$, time scaling is required to have similarity between the 1-D heat transfer through the thickness of the specimen. The Fourier number controls the solution to the 1-D heat equation,

$$Fo = \frac{\alpha t}{{d^{2} }} = \frac{{\alpha t_{scale} }}{{\left( {\chi d} \right)^{2} }} \Rightarrow t_{scale} = \chi^{2} t$$

(7)

where $d$ is the thickness, $\alpha$ is its thermal diffusivity, and $t$ is an appropriate characteristic time (e.g., the total test time). Consequently, Fourier number time scaling will reduce the fire exposure time in the reduced-scale test by a factor of ${\chi }^{2}$.

Boundary condition matching ensures the heat flux boundary condition on the furnace side, where the exposure occurs in this 1-D heat transfer, is maintained in the scaled test. On the exposed surface, the boundary condition is both radiation and convection from the furnace, dictated by the fire curve. The temperature of the scaled test fire curve can be modified to give the same heat flux as the full-scale tests. Standard fire time–temperature curves define full-scale furnace gas temperatures, such as ASTM E119 or ISO 834 (1999). At each time step, the modified furnace temperature must be calculated to give the same heat flux as in the full-scale test. The iterative process which uses calculus to ensure that the reduced-scale sample maintains the same surface temperature as in the full-scale case is discussed in more detail in our previous study (Gangi et al. 2023a). From calculations done in the previous study, the modified ASTM E119 fire curves at ½-scale and ¼-scale are shown in Table 1.

Table 1 Summary of modified scaled furnace time–temperature at different scales (Gangi et al. 2023a)

Char time-scale reduction factors

Previous studies have investigated the kinetics of charring in wood-based materials (Abu-Bakar and Moinuddin 2012; Alves and Figueiredo 1989; Branca and Di Blasi 2003; Chen et al. 2020; Mishra et al. 2015; Richter et al. 2019; Richter and Rein 2019; Wagenaar et al. 1993). Richter and Rein (2019) found that the char yield is diminished at higher heating rates. Detailed pyrolysis models have been used to explore the thermal response and charring rates of wood-based materials. In our previous study (Gangi et al. 2023b), thermal-only fire exposure tests were initially conducted on wood-based samples at three different scales to investigate the thermal response with the complexities of wood decomposition and charring. Those fire exposure thermal response tests, combined with detailed kinetics-based pyrolysis modeling using GPyro, assisted in developing guidance on achieving equivalent char levels in reduced-scale tests. The experimental data for the fire exposure tests and the corresponding models showed a similar trend of more char in the full-scale test than in the ½-scale and ¼-scale tests. The charring rate is lower because, at higher heating rates, wood is converted to char at a higher temperature due to char kinetics.

When applied to wood-based materials, the thermo-structural test scaling laws must account for the char kinetics to produce tests with equivalent charring rates. This can be accomplished by introducing a char timescale correction factor. The char timescale correction factor ${\varphi }_{char}$ adjusts the time-scaling laws from Eq. (7) so that the reduced-scale test better replicates the full-scale equivalent char rate.

$$t_{{{\text{scale}}}} = \frac{{\chi^{2} }}{{\varphi_{{{\text{char}}}} }}t$$

(8)

The charring rate, $\beta$, is a function of the depth of char, ${d}_{char}$, and the time of exposure, $t$. Substituting in the scaling laws for thickness, Eq. (3), and the new time-scaling Eq. (8), the charring rate at a reduced-scale becomes,

$$\beta = \frac{{d_{char} }}{t}$$

(9)

$$\beta_{scale} = \frac{{d_{char,scale} }}{{t_{scale} }} = \frac{{\chi d_{char} }}{{\varphi_{char}^{ - 1} \chi^{2} t}} = \frac{{\varphi_{char} }}{\chi }\beta$$

(10)

Solving Eq. (10) for ${\varphi }_{char}$, the char timescale correction factor is now a function of the scaling factor $\chi$ and can be approximated by a power law ${\chi }^{n}$.

$$\varphi_{char} = \frac{{\chi \beta_{scale} }}{\beta } = \chi^{n}$$

(11)

The timescale and char rate equations Eqs. (6) and (7) are now modified by the char timescale exponent, $n$:

$$t_{scale} = \chi^{2 - n} t$$

(12)

$$\beta_{scale} = \chi^{n - 1} \beta$$

(13)

If the timescale and char are scaled by the Fourier number scaling laws, $=0$. Therefore, the char timescale correction factor was calculated for the GPyro model predictions and the experimental measurements of dimensional lumber samples. Fitting the dimensional lumber data, the char timescale exponent for the model predictions was $n=0.1411$ and for the experimental data was $n=0.3266$ (Gangi et al. 2023b). We will do new GPyro models and thermal-only tests to fit these factors to the plywood.

Modeling

Modeling is an important tool to allow us to predict the mechanical response of the sample during fire exposure. For example, our previous study used the reduced cross-sectional area method (RCAM) to predict the mechanical response of dimensional lumber samples. With the lumber samples, the reduced-scale cross sections are structurally similar to the larger-scale samples. Thus, the mechanical response measured in the reduced-scale tests can be scaled directly to predict the behavior of the equivalent larger-scale tests. In contrast, in many engineered wood products, such as plywood, the grain orientation of a thinner product may be vastly different from the larger-scale thicker product. In addition, composite behavior can significantly influence structural behavior, so if the stacking sequences are different, the mechanical response of the reduced-scale sample will differ from the larger-scale sample. Therefore, modeling becomes essential to extrapolating the data from the reduced-scale test to predict the behavior of the larger-scale test. This section will describe how the RCAM methodology was applied to wood products that behave like composite laminates, in this case, plywood.

Classical laminated plate theory (CLPT) to calculate deflections

The mechanical properties of the plywood composite sections were estimated using classical laminated plate theory (CLPT). The two main assumptions of classical lamination theory are that the kinematics of the entire composite plate is approximated with the Kirchhoff plate theory, and each point within the volume of the plate is in a state of plane stress (Hyer 2009; Koruche and Patil 2015). Additionally, all plies are assumed to be orthotropic and homogenous. In classical laminated plate theory, the ABD matrix, also known as the laminate stiffness matrix, defines the relationship between the stress resultants (i.e., loads) applied to a laminate, and the reference surface strains and curvatures (i.e., deformations). The laminate stiffness matrix involves everything that is used to define the laminate–layer material properties, fiber orientation, thickness, and location. A standard CLPT analysis was used to find the $\text{ABD}$ stiffness matrix of each laminate. From the $\text{ABD}$ matrix, the effective flexural rigidity of the composite beam, ${\left(EI\right)}_{eff}$, can be calculated.

The inverse of the $\text{ABD}$ matrix, called the $\text{abd}$ matrix or the laminate compliance matrix, defines the relationship between the reference surface strains and curvatures (i.e., deformations) and the stress resultants (i.e., loads) applied to a laminate. The $\text{abd}$ compliance matrix is given by

$$\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\varepsilon_{x}^{0} } \\ {\varepsilon_{y}^{0} } \\ {\gamma_{xy}^{0} } \\ \end{array} } \\ {\kappa_{x}^{0} } \\ {\kappa_{y}^{0} } \\ {\kappa_{xy}^{0} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {{\text{a}}_{11} } & {{\text{a}}_{12} } & {{\text{a}}_{16} } \\ {{\text{a}}_{12} } & {{\text{a}}_{22} } & {{\text{a}}_{26} } \\ {{\text{a}}_{16} } & {{\text{a}}_{26} } & {{\text{a}}_{66} } \\ \end{array} } & {\begin{array}{*{20}c} {{\text{b}}_{11} } & {{\text{b}}_{12} } & {{\text{b}}_{16} } \\ {{\text{b}}_{12} } & {{\text{b}}_{22} } & {{\text{b}}_{26} } \\ {{\text{b}}_{16} } & {{\text{b}}_{26} } & {{\text{b}}_{66} } \\ \end{array} } \\ {\begin{array}{*{20}c} {{\text{b}}_{11} } & {{\text{b}}_{12} } & {{\text{b}}_{16} } \\ {{\text{b}}_{12} } & {{\text{b}}_{22} } & {{\text{b}}_{26} } \\ {{\text{b}}_{16} } & {{\text{b}}_{26} } & {{\text{b}}_{66} } \\ \end{array} } & {\begin{array}{*{20}c} {{\text{d}}_{11} } & {{\text{d}}_{12} } & {{\text{d}}_{16} } \\ {{\text{d}}_{12} } & {{\text{d}}_{22} } & {{\text{d}}_{26} } \\ {{\text{d}}_{16} } & {{\text{d}}_{26} } & {{\text{d}}_{66} } \\ \end{array} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {N_{x} } \\ {N_{y} } \\ {N_{xy} } \\ \end{array} } \\ {M_{x} } \\ {M_{y} } \\ {M_{xy} } \\ \end{array} } \right\}$$

(14)

where ${\varepsilon }_{x}^{0}$, ${\varepsilon }_{y}^{0}$, and ${\gamma }_{xy}^{0}$ are the mid-surface strains, ${\kappa }_{x}^{0}$, ${\kappa }_{y}^{0}$, and ${\kappa }_{xy}^{0}$ are the mid-surface curvatures, ${N}_{x}$, ${N}_{y}$, and ${N}_{xy}$ are the force resultants, and ${M}_{x}$, ${M}_{y}$, and ${M}_{xy}$ are the moment resultants. For a cross-ply laminate where all layers are oriented at either 0° or 90°, as is the case for the plywood samples, ${\text{a}}_{16}$, ${\text{a}}_{26}$, ${\text{b}}_{16}$, ${\text{b}}_{26}$, ${\text{d}}_{16}$, and ${\text{d}}_{26}$ are zero. Equation (14) becomes

$$\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\varepsilon_{x}^{0} } \\ {\varepsilon_{y}^{0} } \\ {\gamma_{xy}^{0} } \\ \end{array} } \\ {\kappa_{x}^{0} } \\ {\kappa_{y}^{0} } \\ {\kappa_{xy}^{0} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {{\text{a}}_{11} } & {{\text{a}}_{12} } & 0 \\ {{\text{a}}_{12} } & {{\text{a}}_{22} } & 0 \\ 0 & 0 & {{\text{a}}_{66} } \\ \end{array} } & {\begin{array}{*{20}c} {{\text{b}}_{11} } & {{\text{b}}_{12} } & 0 \\ {{\text{b}}_{12} } & {{\text{b}}_{22} } & 0 \\ 0 & 0 & {{\text{b}}_{66} } \\ \end{array} } \\ {\begin{array}{*{20}c} {{\text{b}}_{11} } & {{\text{b}}_{12} } & 0 \\ {{\text{b}}_{12} } & {{\text{b}}_{22} } & 0 \\ 0 & 0 & {{\text{b}}_{66} } \\ \end{array} } & {\begin{array}{*{20}c} {{\text{d}}_{11} } & {{\text{d}}_{12} } & 0 \\ {{\text{d}}_{12} } & {{\text{d}}_{22} } & 0 \\ 0 & 0 & {{\text{d}}_{66} } \\ \end{array} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {N_{x} } \\ {N_{y} } \\ {N_{xy} } \\ \end{array} } \\ {M_{x} } \\ {M_{y} } \\ {M_{xy} } \\ \end{array} } \right\}$$

(15)

The plywood samples will be placed in three-point bending, resulting in a curvature in the sample caused by the force which induces a moment. While this loading condition will cause both bending and transverse shear. The sample will be assumed to be in pure bending in the length-wise direction since it is the dominant failure mode. Therefore, the force and moment resultants ${N}_{x}$, ${N}_{y}$, and ${M}_{y}$ can be assumed to be zero. Isolating the mid-surface curvature in the length-wise direction, ${\kappa }_{x}^{0}$, from Eq. (15),

$$\kappa_{x}^{0} = {\text{b}}_{11} N_{x} + {\text{b}}_{12} N_{y} + {\text{d}}_{11} M_{x} + {\text{d}}_{12} M_{y}$$

(16)

$$\kappa_{x}^{0} = {\text{d}}_{11} M_{x}$$

(17)

The moment resultants from CLPT are in units of moment per length so that

$$M_{x} = \frac{M}{b}$$

(18)

$$\kappa_{x}^{0} = {\text{d}}_{11} \frac{M}{b}$$

(19)

where $b$ is the width of the beam and $M$ is the total applied moment.

According to Euler–Bernoulli beam theory (Öchsner 2021), the moment–curvature relation for elastic beam bending is given in Eq. (20). Substituting Eq. (19) results in the effective flexural rigidity of the composite laminate ${\left(EI\right)}_{eff},$

$$\kappa = \frac{M}{{\left( {EI} \right)_{eff} }}$$

(20)

$${\text{d}}_{11} \frac{M}{b} = \frac{M}{{\left( {EI} \right)_{eff} }} \Rightarrow \left( {EI} \right)_{eff} = \frac{b}{{{\text{d}}_{11} }}$$

(21)

The maximum elastic deflection at the mid-span of a rectangular beam subject to symmetric three-point bending is given in Eq. (22). Substituting Eq. (19) results in the mid-span deflection of the plywood composite beam.

$${\text{Max}}\; {\text{Deflection}} = \delta_{\max } = \frac{{PL^{3} }}{{48\left( {EI} \right)_{eff} }}$$

(22)

$$\delta_{\max } = \frac{{PL^{3} }}{48b}{\text{d}}_{11}$$

(23)

Reduced cross-sectional area modeling for composite laminates

The “reduced cross-sectional area model” (RCAM) assumes that the fire resistance of a wood member or assembly depends on the extent of charring of the structural wood element. Establishing the charring rate and char level is critical to evaluating mechanical response because char has virtually no load-bearing capacity (White 2016). The RCAM for timber design in fire conditions simplifies the complex material behavior of wood exposed to fire. The effective cross section is calculated by reducing the initial cross section by the charred depth, ${d}_{char}$, and a zero-strength layer, ${d}_{z-s}$. The zero-strength layer compensates for the reduction in material properties of the wood that is heated, but not charred. By accounting for the zero-strength layer, the room-temperature material properties can be used to evaluate the section properties of the remaining effective cross-section. As the sample chars, the effective cross section will become shorter, causing a loss in stiffness. Therefore, charring leads to a deflection and stress increase as the cross section reduces.

A reduced cross-sectional area model was identified with a routine developed in MATLAB to reduce the cross section at time increments of $\Delta t$ by the linear effective charring rate, ${\beta }_{scale}$. The linear effective charring rate for a test conducted at a scale $\chi$ was calculated using Eq. (13), ${\beta }_{scale}={\chi }^{n-1} \beta$. The value for $n$, the char timescale exponent, is estimated from the GPyro analysis in Sect. “GPyro modeling to predict charring rate of plywood”. The effective charring rate compensates for the zero-strength layer of heated but uncharred timber by increasing the nominal charring rate by 20%. From guidance from the NDS Chapter 16, for solid softwood timber with a characteristic density of ≥ 290 kg/m³ the full-scale nominal charring rate is 0.64 mm/min or 39 mm/hr, therefore the effective charring rate is $\beta$=0.8 mm/min or 48 mm/hr (AWC 2018). It is important to note that similar guidelines in Eurocode 5 assume the zero-strength layer to be a constant 7 mm after 20 min of exposure (CEN 2015). It is reasonable to assume that these different approaches would produced different results in terms of scaling, thus for simplicity, this study used the NDS approach of 20% increase in char rate.

In each time increment, the total laminate thickness is reduced by $\Delta ({d}_{char}+{d}_{z-s})={\beta }_{scale}\Delta t$. The layers that have fully charred are removed, and if there is a partial layer remaining, its thickness is updated. Next, the stacking sequence is updated, and the CLPT code calculates the new ABD stiffness matrix for the reduced section. This RCAM model assumes that all uncharred wood still has room-temperature material properties. Then, given the new stacking sequence, the resulting deflections and stresses were calculated. While the linear approximation of the deflections may not be adequate for large displacements where more non-linear behavior would be seen, the model was created to be as simple as possible to approximate the behavior. Our goal was to develop and present a first-order model capturing the orientation effects. Progressive failure in tension and/or yielding in compression could be captured by changing the constitutive model: for example linear elastic-perfectly-plastic or bilinear behavior in compression and elastic-quasi-rigid behavior in tension (Saad and Lengyel 2022).

Geometry and loading conditions were specified to simulate thermo-structural tests of plywood samples subject to combined static three-point bending and fire exposure described in Sect. “Thermo-structural test procedure”. Given that the plywood tested was pinewood, the material properties of the full-stiffness uncharred wood are taken from the Wood Handbook (Forest Products Laboratory 2010) for Virginia pine, shown in Table 2. The Wood Handbook provides the properties of the three principal axes of wood: longitudinal (fiber, 1-direction), tangential (transverse, 2-direction), and radial (out-of-plane, 3-direction). The CLPT model assumes the material to be transversely isotropic in-plane, so the provided orthotropic properties were only taken in the fiber and transverse directions.

Table 2 Model parameters for pinewood in Reduced Cross-Sectional Area models (Forest Products Laboratory 2010)

Our tested samples were designed to have different stacking sequences between scales. The initial stacking sequences are shown in Fig. 1, with the ½-scale samples as 8-layer [0°/90°/90°/0°]s laminates and the ¼-scale samples as 4-layer [0°/90°]s laminates.

Fig. 1

Visual of sample stacking sequence initially and after 25% cross section char for a the ½-scale [0°/90°/90°/0°]s laminate and b the ¼-scale [0°/90°]s laminate

As the sample chars and the effective cross-section reduced, the RCAM models calculate the ABD matrix given the new stacking sequence. Visually, it can be seen that the samples at each scale will behave differently as they char. For example, after 25% char, the ½-scale sample loses 2 layers and becomes a [90°/0°/0°/90°/90°/0°] laminate that is no longer symmetrical. On the other hand, the ¼-scale sample loses only 1 layer a becomes a [90°/90°/0°] laminate after 25% char.

To demonstrate how differently the cross sections behave, initial RCAM models were run to simulate the tests with lower initial loads: the ½-scale test with 267 N applied load and the ¼-scale test with 67 N applied load. Figure 2a shows the increase in bending stresses in the top layer, while Fig. 2b shows the increase in mid-span deflection as the sections are charred. For the ¼-scale sample, the first 0° layer provides considerable bending stiffness. However, when the first layer is removed (25% of the cross section), the bending stress and mid-span deflection increase significantly. In contrast, the mechanical response remains relatively constant when the first 0° layer is removed from the ½-scale sample (12.5% of the cross section). The bending stress and mid-span deflection increase significantly when the fourth layer is removed (50% of the cross section).

Fig. 2

For the tests at lower initial load to show differences between the bending behaviors of the cross sections at ½-scale and at ¼-scale. a Bending stress vs. section removed, and b deflection vs. section removed

We note that our analysis does not consider the thermal adhesive performance of the glue, both within the as-manufactured formaldehyde plywood glue and the layer of Titebond III wood glue connecting the built-up sections. This can certainly play a role, as a loss of adhesion would reduce the composite action as the adhesive heats up. Under non-charring temperatures, Wiesner et al. (2020) found that adhesive type had a significant influence on the deflections of cross-laminated timber beams during heat. Samples bonded with polyurethane showed signs of increased shear strains along adhesive lines, interlayer slip and partial debonding, compared to samples bonded with melamine formaldehyde adhesive. The weakening of adhesives can occur at relatively low temperatures. For example, Stoeckel et al. (2013) found adhesive stiffness to deteriorate in a temperature range as low as 20–70 °C in the case of polyvinyl acetate polyurethane, and epoxies, well below charring temperatures. As noted earlier, our goal was to develop and present a first-order model capturing the orientation effects. The effect of the weakening between the wood and adhesive layers could be simulated with a more complex model accounting for the shear lag between the layers using the methodology detailed by Nairn and Mendels (2001).

GPyro modeling to predict charring rate of plywood

Heat transfer models were used to estimate the rate of charring of the plywood samples in this study, as was done in the prior analysis of the dimensional lumber experiments (Gangi et al. 2023b). GPyro, an open-source generalized pyrolysis model developed by Lautenberger and Fernandez-Pello (2009), was used to simulate the phase change that occurs with the wood sample subject to fire exposure. A sample of thermally decomposing wood is modeled as consisting of two condensed phases (virgin wood and char residue). The conversion of char to ash via a smoldering reaction was not visibly observed in any of the experimental tests due to the low oxygen environment of the furnace assembly. Thus, the smoldering reaction and the ash phase were not considered in this analysis. The gas phase outside the decomposing wood slab (the exterior ambient) is not modeled. The heat transfer through the thickness of the sample is assumed to have one-dimensional behavior.

Geometry, initial conditions, and boundary conditions were specified to simulate the three thermal-only fire exposure tests conducted on plywood samples described in Sect. ”Fire exposure test procedure”. The parameters of the three models of the full-scale, ½-scale, and ¼-scale tests are given in Table 3.

Table 3 Parameters for GPyro simulations (CEN 2002; Cheriton and Gupta 2005; Mishra et al. 2015)

The full-scale sample was exposed for 2400 s, the ½-scale sample for 600 s, and the ¼-scale sample for 150 s. With the heat transfer being one-dimensional through the sample thickness, the thickness was defined in the z-dimension (45.7 mm for full-scale, 22.9 mm for ½-scale, and 11.4 mm for ¼-scale). In each model, the thickness was divided into 101 nodes. The initial temperature, pressure, and condensed phase species are uniform throughout the thickness of the solid. The initial temperature ${T}_{0}$ is set to 293 K, the ambient pressure is 101,300 Pa, and the mass fraction is 1.0 for virgin wood and 0.0 for char residue.

Each condensed phase species has well-defined properties (bulk density, specific heat capacity, effective thermal conductivity, emissivity, permeability, and porosity), which were set to be constants. The properties of wood and char are given in Table 3. Virgin wood properties were assumed to be at the average moisture content of the tested samples. The moisture content of all tested samples was between 6 and 8%, with an average moisture content of 6.5%. For simplicity, the effect of wood dehydration and evaporation during fire exposure was not considered. The properties for wood at elevated temperatures are effective properties derived from standard fire exposure. With only two condensed phases, the one reaction is a conversion from virgin wood to char residue. The reaction parameters given in Table 3 were calculated by digitizing TGA data from Mishra et al. (Mishra et al. 2015) for the decomposition of pinewood at different heating rates. Additionally, the conversion of wood to char is an endothermic reaction, with the heat of volatilization assumed based on guidance from The Handbook of Building Materials for Fire Protection (Cheriton and Gupta 2005).

Boundary conditions were created to replicate the furnace conditions, similar to what we did in our previous study on a 1-D heat transfer analysis (Gangi et al. 2023a). The condensed phase boundary conditions occur on the front face (surface 1) and the back face (surface 2). The front face is inside the furnace, and the ambient temperature changes over time via the fire curve. The back face is outside the furnace, and the ambient temperature remains constant at the room temperature of 293 K. On both surfaces, the reradiation parameter, “Reradiation?”, was set to true, indicating that radiation was considered on both surfaces. The convection heat transfer coefficients were assigned according to the European code on fire interaction with structures, i.e., 25 W/m² K for the front face and 4 W/m² K for the back face (CEN 2002). While the heat transfer coefficient was not directly measured for this particular furnace setup, a sensitivity analysis was conducted, varying coefficient on the front face from 10 to 35 W/m² K. Resulting back-side temperatures varied between −1.7 and 1.0 °C, showing little sensitivity to the convective coefficient. Therefore, the standard value for furnace exposure of 25 W/m² K from the Eurocode was assumed. The furnace time–temperature curves in Table 1 defined the temperatures on surface 1.

Experimental methods

Thermo-structural test procedure

We conducted six tests on plywood samples in combined static three-point bending and fire exposure on our reduced-scale propane furnace. The test matrix of the thermo-structural tests is shown in Table 4. The test procedure was identical to that used in our previous study of dimensional lumber samples (Gangi et al. 2023b). Similar to that study, load capacity limits prevented full-scale samples from being tested on the small furnace-loading frame setup. Therefore, the experimental testing program included tests at two scales, ½-scale, and ¼-scale, of plywood samples of similar dimensions to previously tested lumber samples. While having full-scale validation data is preferable, having data at two different scales allows us to demonstrate the thermo-structural scaling laws if the results of the smaller ¼-scale tests can be used to predict the behavior of the larger ½-scale tests.

Table 4 Test matrix for thermo-structural scaling tests of plywood

The ¼-scale sample was chosen to be one layer of 11.4 mm thick Virginia pine plywood with four plies and a stacking sequence of [0°/90°]s. The ½-scale sample was chosen to be a built-up section of two layers of the same 11.4 mm thick Virginia pine plywood for a total of 22.9 mm thick, eight plies, and a stacking sequence of [0°/90°/90°/0°]s. The built-up section was created by gluing two of the layers with Titebond III wood glue. This served three functions: ensuring that the samples were made from the same material, creating samples exactly double the thickness, and ensuring that the stacking sequence was different to show how this composite behavior could be captured with modeling. The out-of-plane dimensions were the same at both scales, with samples being 610 mm long and 102 mm wide.

Samples were initially loaded in three-point static bending, with the loading scaled to have structural similitude. The lower loading tests (267 N at ½-scale and 67 N at ¼-scale) were conducted twice for repeatability. Then, additional tests were conducted at a 50% higher initial loading (356 N at ½-scale and 89 N at ¼-scale) to see the impact of initial stress on the test results. Once structurally loaded, the samples were exposed to the scaled E119 time–temperature fire curves. In this case, the ½-scale and ¼-scale modified fire exposure curves are shown in Table 1.

The furnace used in the tests was the same reduced-scale furnace used in the previous study (Gangi et al. 2023b), shown in Fig. 3a. The basic furnace design consists of a Unistrut frame, four propane burners, and a ceramic-fiber insulated barrier to protect the instrumentation from exposure during the testing. The frame was constructed from 41 mm (1–5/8″) Unistrut channels. Four Venturi-type 147 kW portable torch-style gas burners were mounted vertically to heat the sample directly. The burners were mounted 305 mm below the sample surface, far enough to not directly impinge on the sample surface and heat the gas layer underneath the sample instead. The thermal barrier was made from a 16 mm thick Type X gypsum wallboard insulated on both sides with a 25 mm (1″) ceramic-fiber insulation blanket of 128 kg/m³ density for a total of 51 mm of insulation. A hole was cut in the barrier to expose only the sample surface of 610 mm by 140 mm. Furnace gas temperature was measured with five Inconel-sheathed 1.5 mm diameter thermocouple probes mounted near the sample surface, shown in Fig. 3b. The average temperature was controlled by manually changing the propane flow rate in the burners using an Alicat MC-100 mass flow controller through a LabView interface. The air flow into the furnace was uncontrolled other than providing a ventilation hood. The temperature data was also collected in real-time in the LabView program. The mass flow controller allowed the propane flow to be adjusted to ensure that the test followed the proper fire curve.

Fig. 3

Thermo-structural loading frame setup a Front view with the Unistrut frame, propane burners, and sample location highlighted; b Top view indicating burner spacing and TC locations under the opening; c Top view showing the two string potentiometer measurements can be averages at the center of the span; d Side view of ½-scale loading; and (b) Side view of ¼-scale loading

As the sample is damaged from the fire exposure, the static mechanical load will cause the center of the sample to deflect. The structural response was quantified using deflection measurements. Mid-span deflection was measured using two Celesco PT1DC-10-UP-Z10-MC4 string potentiometers with a range of 250 mm and an accuracy of 0.05% of full-scale. The string potentiometers were mounted to the loading bar via hooks welded 305 mm from the center of length of the bar, shown in Fig. 3c. The center of length of the loading bar was aligned with the center of the span of the sample. Therefore, the mid-span deflection of the sample was measured by the string potentiometers. For the thermo-structural tests, the mechanical load was applied via a loading bar placed at the mid-span of the sample. The 711 mm loading bar was made from 51 mm steel square tubing and had a self-weight of approximately 22 N. Additional loading was applied via weight plates stacked on the center of length of the loading bar, shown in Fig. 3d for ½-scale tests and Fig. 3e for ¼-scale tests. In all tests, two Unistrut channels mounted above the thermal barrier and underneath the sample served as the pin supports and allowed for a 610 mm clear distance. The pin supports were mounted to allow up to 50 mm deflection before reaching the catch mechanism. Underneath the insulated barrier was the catch mechanism for the load, shown in Fig. 3d and e. Catch beams of Unistrut were mounted in the length direction of the sample and would only be engaged if the loading bar were to deflect more than 50 mm.

During the thermal exposure, the thermal response was quantified through temperature measurements. The instrumentation plan for temperature measurement in the samples is shown in Fig. 4. Temperature measurements were taken in the middle of each sample, 51 mm from the long side so that edge effects could be ignored. As shown in the top view of Fig. 4a, the surface and though-thickness measurements were taken at two redundant locations 241 mm from the short edges. Thus, we had two replicates of temperature data for each sample. Surface 24-gage, Type K glass-braided thermocouples were mounted on both the exposed and unexposed sides with stainless-steel staples. Within the thickness $d$ of each sample, Inconel-sheathed 1.5 mm diameter thermocouple probes were mounted in drilled 2 mm diameter holes. In the ½-scale and ¼-scale samples thermocouples were measured at four distances from the fire-exposed surface (on the surface, $\frac{1}{3}d$, $\frac{2}{3}d$, $d$), shown in Fig. 4b and Fig. 4c. The holes in the ¼-scale samples were offset 13 mm horizontally to avoid cracking between holes in the sample. Ceramic-fiber insulation blanket of 128 kg/m³ density was stapled on the sides of the samples to prevent flames from passing through the edge of the sample and ensure heat transfer through the thickness of the sample only.

Fig. 4

Instrumentation plan for thermocouples. a Top view of Surface Thermocouples in all samples, which is the same as the bottom view; b The side view of the location of Thermocouple Probes through the thickness of ½-scale samples; and c The offset in the ¼-scale sample to have two probes through the thickness

Tests were concluded when either of two conditions were met: the sample reached the catch mechanism indicating excessive deflection or the sample transmitted too much heat through the thickness. Both are based on failure conditions in the ASTM E119 test (ASTM 2022). For an unrestrained beam, the specimen shall be deemed as not sustaining the applied load when the maximum total deflection of ${L}^{2}/400d$ has been exceeded. For our ½-scale specimens, the clear distance $L=610 mm$ and the sample thickness $d=22.9 mm$ results in a failure condition at 40.6 mm of total deflection. Therefore, the catch mechanism was placed 50 mm below the sample to ensure the failure condition was reached. Also, for unrestrained assemblies, the transmission of heat through the test specimen during its classification period shall not raise the average temperature on its unexposed surface more than 250°F (139 °C) above its initial temperature.

Once the sample was determined to have failed, the furnace gas was turned off, thermocouples were disconnected, and the sample was promptly removed from the furnace to prevent further exposure. A spray bottle was used to thoroughly wet and cool the exposed charred surface until no visible smoldering was seen to stop any additional charring. The initial and final thickness and char depth were measured on the exposed samples. A saw was used to cut the specimens in half at the midpoint to measure the char depth. The char was scrapped off with a knife until all that remained was uncharred wood. A caliper was used to measure the remaining uncharred wood at six locations on the sample. The char depth was defined by the difference from the original sample thickness to the average depth of uncharred wood.

Fire exposure test procedure

In addition to thermo-structural tests, the thermal-only fire exposure tests previously discussed in Sect. ”GPyro modeling to predict charring rate of plywood” were conducted to validate the GPyro models and capture the plywood samples' charring rates at different exposure scales. We conducted three thermal-only fire exposure tests on our reduced-scale furnace following the procedure described in our previous paper (Gangi et al. 2023a). The test matrix is shown in Table 5.

Table 5 Test matrix for thermal-only fire expose scaling tests

Similarly to the thermo-structural tests, the average temperature was controlled in the fire exposure tests by manually adjusting the propane flow rate in the burners with a mass flow controller as needed to ensure that the test was following the proper fire curve. The primary difference is that the thermal-only tests had no mechanical loading. Heat transfer was assumed to occur primarily through the thickness of the sample, so out-of-plane dimensions were all the same, 102 mm wide by 686 mm long as with the thermo-structural test samples. Therefore, the scale was determined by the thickness of a built-up section glued together with Titebond III wood glue. The three samples were created with sections cut from the same 11.4 mm thick plywood sheet for continuity. The quarter-scale ($\chi =1/4$) sample was one single layer with a thickness of 11.4 mm, the half-scale ($\chi =1/2$) sample was two layers with a thickness of 22.9 mm, and the full-scale ($\chi =1$) sample was four layers with a thickness of 45.7 mm.

The tests were terminated when the samples experienced 40 min of equivalent fire exposure (based on full-scale duration). From Fourier number scaling laws, the times were 2400 s at full-scale, 600 s at ½-scale, and 150 s at ¼-scale. Test times were chosen to capture a similar exposure time as the thermo-structural tests. The three thermo-structural tests at ½-scale ran for an average of 602 s, while the three ¼-scale tests ran for an average of 139 s. Therefore, exposures of an equivalent of 40 min should capture a similar charring behavior as our thermo-structural tests. After the end of the desired exposure time, the furnace gas was turned off, thermocouples were disconnected, and the sample was promptly removed from the furnace to prevent further exposure. In a similar procedure to the thermo-structural tests, the initial and final thickness and char depth were measured on the exposed samples.

The samples were placed on the furnace frame over the opening to expose the middle 610 mm. Ceramic-fiber insulation blanket of 128 kg/m³ density was also placed on the sides of the samples to prevent flames from passing through the edge of the sample and ensure heat transfer through the thickness of the sample only. Once the edges were insulated, the samples were exposed to similar furnace exposures as in the thermo-structural tests: the scaled E119 time–temperature fire curves. In this case, the full-scale, ½-scale, and ¼-scale modified fire exposure curves are shown in Table 1. During the thermal exposure, the ½-scale and ¼-scale samples were instrumented identically to those in the thermo-structural tests, shown in Fig. 4. The full-scale sample instrumentation is shown in Fig. 5. The larger thickness of the full-scale samples allowed for additional measurements to be taken through the thickness. Rather than capturing the temperature profile at only 2 locations within the thickness ($\frac{1}{3}d$, $\frac{2}{3}d$) in the full-scale samples, thermocouples were placed at 5 locations within the thickness ($\frac{1}{6}d$, $\frac{1}{3}d$, $\frac{1}{2}d$, $\frac{2}{3}d$, $\frac{5}{6}d$).

Fig. 5

The side view of the location of 5 Thermocouple Probes through the thickness of full-scale samples

Results

Themo-structural test results

A summary of the thermo-structural three-point bending test results is shown in Table 6. All six tests concluded due to reaching the deflection limit of the tests, not the temperature limits. For example, tests at ½-scale ran for an average of 652 s with a lower load (267 N) and 356 s with a higher load (356 N). Similarly, at ¼-scale, the tests ran for an average of 161 s with the lower load (67 N) and 89 s with the higher load (89 N). In addition to the results of each test, the average results of the repeated tests were calculated. The repeated ½-scale tests at lower load shown in blue (Tests 1 and 2) had an average of 96.8% desired exposure, 11.51 mm of char, a char rate of 1.06 mm/min, and a full-scale equivalent char rate of 0.53 mm/min. The repeated ¼-scale tests at lower load shown in orange (Tests 4 and 5) had an average of 98.0% desired exposure, 4.57 mm of char, a char rate of 1.72 mm/min, and a full-scale equivalent char rate of 0.43 mm/min.

Table 6 Plywood thermo-structural three-point bending test with fire exposure results

Figure 6 compares the desired modified ASTM E119 fire curves and the measured temperatures at each scale. The furnace exposure was stopped once the samples reached the deflection limits. For a furnace temperature control to be acceptable, the ASTM E119 standard says that the area under the time–temperature curve, obtained by averaging the results from thermocouple readings, is within 10% for a 1 h test (ASTM 2022). For the thermo-structural tests that concluded at different times, we defined the desired exposure as the area under the standard exposure for that duration of test. For example, for Test 1 that ran for 606 s, the desired exposure was the area under the ½-scale fire curve for the first 606 s. Averaging the readings from the 5 furnace thermocouples, the ½-scale fire exposures were within 2.9%, 3.5%, and 1.2% error, well within the requirements. The corresponding values for ¼-scale tests were 2.9%, 1.1%, and 3.9% error, also within the requirements.

Fig. 6

Desired fire time–temperature curves vs. average measurements of 5 furnace thermocouples in thermo-structural tests, demonstrating that our furnace setup was able to achieve the desired exposures at a ½-scale, and b ¼-scale

Figure 7 shows the temperature evolution of tests conducted with ½-scale and ¼-scale thermal exposure. We instrumented both samples with a higher load (Tests 3 and 6) and one of the repeated lower load tests (Tests 1 and 4). Visual inspection shows that the tests were repeatable in terms of thermal exposure. The repeatability of the thermal exposure is further reinforced by comparing to the temperature evolution of the thermal-only fire tests. The temperature evolution of the ½-scale thermal-only test (Test 8 shown in Fig. 7a) and the ¼-scale thermal-only test (Test 9 shown in Fig. 7b) show good agreement with the comparable thermo-structural tests. The repeatable nature of the thermal exposure led to similar charring rates. For example, at ½-scale exposure, the measured charring rate was 1.12 mm/min for Test 1 and 1.05 mm/min for Test 3. At ¼-scale exposure, the measured charring rate was 1.62 mm/min for Test 4 and 1.71 mm/min for Test 6. Figure 7c compares the temperature evolution of the thermo-structural tests with the timescale normalized to full-scale. Scaling the time of the ½-scale tests by a factor of 4 and the ¼-scale tests by a factor of 16 provides an illustration of the Fourier number time-scaling.

Fig. 7

Comparison of temperature evolution in thermo-structural three-point bending tests, showing repeatability of thermal exposure at a ½-scale, b ¼-scale, and c all tests at a normalized equivalent full-scale time-scale

During the fire exposure, the depth of char in the sample increased, and as the effective cross-section reduced, the sample deflected more. The increase in deflection was measured with string potentiometers. Tests were concluded when the deflections reached the catch mechanism at roughly 50 mm of deflection. Figure 8 shows the measured deflections in the tests.

Fig. 8

Measured midspan deflections over time, with testing terminated at 50 mm of deflection for three-point bending tests conducted at a ½-scale and b ¼-scale

As expected, the tests with a higher initial load (Tests 3 and 6) reached the maximum deflection earlier. The repeated tests at a lower initial load behaved similarly in terms of shape of the time-displacement curves. The major difference between these is that Tests 1 and 4 were not instrumented with thermocouple probes within the cross section. As a result, the tests with the Inconel-sheathed thermocouple probes within the cross section (Tests 2 and 5) required a marginally longer time to reach deflection limits under the same conditions. This may be due to repeatability or some effects of having the thermocouple probes in the test sample providing extra stiffness to the sample as they deflected.

Fire exposure test results

A summary of the test results for the three fire exposure tests on plywood samples at full-scale (Test 7), ½-scale (Test 8), and ¼-scale (Test 9) are provided in Table 7. Ideally, at all scales, the samples would have the same equivalent charring rate as the full-scale, resulting in the same percentage of the cross section being converted to char. The measured full-scale equivalent char rate in the ½-scale test was 83.5% full-scale, and the ¼-scale test was 67.9% full-scale. The char rates were similar in the thermal-only test and the thermo-structural tests. The average charring rate in the three ½-scale thermo-structural tests was 1.06 mm/min, compared to 1.09 mm/min in the thermal-only fire exposure test. Similarly, the average charring rate in the three ¼-scale thermo-structural tests was 1.71 mm/min compared to 1.78 mm/min in the thermal-only fire exposure tests.

Table 7 Plywood thermal-only fire exposure scaling test results

Figure 9 compares the desired modified ASTM E119 fire curves and the measured temperatures at each scale. At full scale, averaging the readings from the 5 furnace thermocouples, the area under the curve of our tests was within 0.5%, well within the requirements. The corresponding values for the ½-scale and ¼-scale tests were 1.2% and 2.0%, respectively. Just as in the thermo-structural three-point bending tests, the reduced-scale furnace maintained the fire curves in all tests within the ASTM E119 requirements, with these tests all within 2%. Thermocouples were used to measure the changes in temperature during the experiment. Our thermal scaling is successful if it can capture a similar temperature profile as the full-scale sample throughout the entire exposure. Figure 10 shows the resulting temperature evolution for the ½-scale and ¼-scale tests compared to the full-scale results.

Fig. 9

Desired fire time–temperature curves vs. average measurements of 5 furnace thermocouples in thermal-only fire exposure tests at a Full-scale; b ½-scale; and c ¼-scale

Fig. 10

Thermal-only fire exposure tests temperature evolution comparison. Full-scale vs. a ½-scale; and b ¼-scale

Thermocouples were positioned on each surface and at two points in the sample thickness to sample the complete cross section. Comparing the data, the ½-scale results in blue and the ¼-scale experiments in red closely approximate the full-scale temperature profiles in black after 20 min of equivalent exposure. However, the ½-scale results showed better agreement than the ¼-scale results by the end of testing. In particular, at a depth of $1/3d$ from the exposed surface, there are larger temperature gradients. At this location, the average temperature in the full-scale sample was 856 K compared to 694 K at ½-scale and 559 K at ¼-scale. This is a relative difference at this location of 162 K (−18.9%) at ½-scale, and 297 K at ¼-scale (34.7%). This lower temperature rise in the smaller-scale samples is reflected in the full-scale equivalent charring rates, with the ½-scale sample forming 17.5% less char, and the ¼-scale sample forming 32.1% less char than the full-scale.

Discussion

The test results showed some differences from ¼-scale to ½-scale. This was hypothesized to be due to differences in stacking sequences and charring rates. To explore this, the following things were investigated: 1) how well the GPyro models predicted the behavior of the thermal-only fire exposure tests, 2) what the GPyro models predicted as the char timescale reduction factors and 3) how well the reduced cross-sectional area method (RCAM) models can predict the behavior of the thermo-structural tests. Therefore, the GPyro models of the thermal-only tests, the char time-scale reduction factors, the RCAM models of the ¼-scale thermo-structural tests, and the RCAM models of the ½-scale thermo-structural tests are presented in the discussion.

GPyro models of fire exposure tests

The GPyro models, discussed in Sect. “GPyro modeling to predict charring rate of plywood”, were created to simulate the thermal-only fire exposure tests at full-scale, ½-scale, and ¼-scale. Figure 11 shows the temperature evolution of the GPyro models in comparison to the experimental data. What is important for the model to predict are the temperature rise in the unexposed surface and the thermal gradients in the middle of the sample. By visual observation, the models predict the unexposed surface temperature well at all scales. In the middle of the sample with larger temperature gradients, there are discrepancies between model predictions and average measured temperatures. For example, we can look at measurements taken at a depth of $1/3d$ from the exposed surface (15.2 mm at full-scale, 7.6 mm at ½-scale, and 3.8 mm at ¼-scale). By the end of the test, models are under-predicting the full-scale measurements (800 K vs. 856 K). On the other hand, the ½-scale and ¼-scale models are over-predicting the temperature rise. The ½-scale model predicts a rise to 788 K vs. 694 K measured, and the ¼-scale model predicts a rise to 766 K vs. 559 K measured.

Fig. 11

Comparison of temperature evolution in GPyro models to experimental data at a Full-scale; b ½-scale; and c ¼-scale

In addition to temperature profiles, the GPyro models also calculate the amount of each reactant (virgin wood and char reside) over time, and we can compare the results at the three scales. The profiles of the mass fraction of char residue by the end of each test are shown in Fig. 12a.

Fig. 12

a Mass fraction of wood converted to char by the end of 40 min of equivalent exposure, and b char depth over time predicted using a 75% mass fraction of char to indicate the char front

The char front is typically defined by the 300 °C isotherm, a value suggested in Eurocode 5 validated for the standard fire exposure (White 2016). However, for non-standard fires, as is the case with the reduced-scale exposures, previous studies found that the char front temperature is not an explicit value as assumed by the 300 °C isotherm. For example, a study conducted by Pecenko and Hozjan (2021) found that due to the kinetics of char formation a “fast fire” had an average char front of 331 °C. Because the increased heating rate causes char to form at higher temperatures, we must choose a different parameter than the 300 °C isotherm to define the char front. Instead of using a temperature, we can then track the char formed at each test scale by assuming a mass conversion level indicating char is formed. Guidance from both Eurocode 5 and the NDS assume the full-scale nominal charring rate, which defines the movement of the char front, to occur at a rate of 0.64 mm/min or 39 mm/hr for 1 h of exposure (AWC 2018; CEN 2015). We have chosen to use 75% mass conversion to char to indicate the char front, as this produced 0.64 mm/min (39 mm/hr) of char at full-scale. The depth at which the char front, 75% mass fraction of char residue, was tracked over time in the three models, shown in Fig. 12b. Linearizing the char curve, we can predict the linear char rate in each test to be 0.64 mm/min at full-scale, 1.15 mm/min at ½-scale, and 2.02 mm/min at ¼-scale.

We can compare the char rate $\beta$ in the models, found using the parameter of 75% char residue, to the char measured levels from the experimental tests, shown in Fig. 13a. We can also compare the full-scale equivalent char rate, $\chi \beta$, shown in Fig. 13b, which should ideally be the same as full-scale.

Fig. 13

Comparison of model prediction to experimental measurements of a Linear char rate $\beta$, and b Full-scale equivalent char rate $\chi \beta$

The experimentally measured char values are presented as the average of the measurements at six locations, with the standard deviation shown displayed with error bars. Experimental measurements of char rate were a mean of 0.65 mm/min (SD 0.02) at full-scale, 1.09 mm/min (SD 0.09) at ½-scale, and 1.78 mm/min (SD 0.12) at ¼-scale. Model predictions of char rate were 2.9% lower than average measurements in full-scale test, 5.5% higher than in the half-scale test, and 13.5% higher than in the quarter-scale test. The predictions at full-scale and ½-scale are within the standard deviation of the measurements with just this simple two-phase reaction model. This difference in charring rate reflects the previously identified differences in temperature rise, where the full-scale model was under-predicting, while the ½-scale and ¼-scale models were over-predicting temperature rise.

Char time-scale reduction factors from GPyro models

The experimental data and the models show a similar trend of more char in the full-scale test than in the ½-scale and ¼-scale tests. The GPyro models consider the effect of heating rate on the conversion of wood to char. Even though the models predict a similar temperature rise in the reduced-scale samples, the full-scale equivalent charring rate is lower. The charring rate is lower because, at higher heating rates, wood is converted to char at a higher temperature due to char kinetics. As explored in our previous study, when applied to wood-based materials, the scaling laws must account for the char kinetics to produce tests with equivalent charring rates (Gangi et al. 2023b). For this set of plywood samples, the char timescale correction factor was calculated for both the GPyro model predictions and the experimental measurements, shown in Fig. 14.

Fig. 14

Curve fitting the normalized equivalent char rate χβ_scale/β to calculate the char timescale correction factors using model data, φ_char = χ^0.1656, and using test data, φ_char = χ^0.2785

The char timescale correction factor was calculated from Eq. (11), where ${\varphi }_{char}= \frac{{\chi \beta }_{scale}}{\beta }={\chi }^{n}$. The char timescale correction factor ${\varphi }_{char}$ adjusts the time-scaling laws so that the reduced-scale test better replicates the full-scale equivalent char rate. The factors are calculated by plotting the normalized equivalent char rate $\frac{{\chi \beta }_{scale}}{\beta }$ vs. scaling factor $\chi$ and fitting the curve to a power series${\chi }^{n}$. Fitting the data, the char timescale exponent for the model predictions is $n=0.1656$ and for the experimental data is$n=0.2785$.

Validation of reduced cross-sectional area method models to ¼-scale tests

The reduced cross-sectional area method (RCAM) models, discussed in Sect. “Reduced cross-sectional area modeling for composite laminates”, were created to simulate the thermo-structural tests at ¼-scale. The purpose of modeling the ¼-scale tests first was to demonstrate the ability of the reduced-scale tests to be used for model validation. To validate the ¼-scale RCAM models, we will assess the ability of the models to predict the increase in center deflection of the sample over time. The models predict the time-deflection response as the sample is damaged due to char and the time to reach the maximum center deflection of 50 mm. Fire resistance tests traditionally assess the duration a structure can hold a structural load during fire exposure. The time to reach the maximum center deflection indicates the time the sample can hold the particular static structural load under fire exposure before reaching an excessive deflection. Thus, we will compare the time-deflection predicted by the model to the experimental measurement, as well as the time to reach maximum deflection of 50 mm.

In the RCAM models, the amount that the cross section was damaged over time is controlled by the charring rate, which is impacted by the char timescale correction factor. The reduction of the effective cross-section in the RCAM models is now defined by scaling the full-scale effective charring rate $\beta$ using Eq. (13), ${\beta }_{scale}= {\chi }^{n-1}\beta$, which accounts for the char timescale correction factor. From guidance from the NDS and Eurocode 5, the full-scale effective charring rate was assumed to be $\beta$=0.8 mm/min (AWC 2018; CEN 2015). This effective char rate is an increase of 20% from the linear char rate of 0.64 mm/min to account for reduced strength and stiffness properties and accelerated charring at the corners. Thus, the model is able to account for the thermally impacted timber beneath the char layer. To show the effect of the char timescale correction factor, we considered three scenarios of the char correction factor:

• (1) The correction factor that was calculated based on GPyro models ($n=0.17$), making ${\beta }_{scale}= {\chi }^{-0.83}\beta ={\chi }^{-0.83}(0.8\frac{mm}{\text{min}})$. For ¼-scale exposure, ${\beta }_{scale}=2.54 \frac{mm}{\text{min}}$

• (2) The correction factor that was calculated based on thermal-only fire exposure tests $(n=0.28)$, making ${\beta }_{scale}= {\chi }^{-0.72}\beta ={\chi }^{-0.72}(0.8\frac{mm}{\text{min}})$. For ¼-scale exposure, ${\beta }_{scale}=2.18 \frac{mm}{\text{min}}$

• (3) Having no correction factor to show the initial Fourier number scaling laws ($n=0$), making ${\beta }_{scale}= {\chi }^{-1}\beta ={\chi }^{-1}(0.8\frac{mm}{\text{min}})$. For ¼-scale exposure, ${\beta }_{scale}=3.20 \frac{mm}{\text{min}}$

Figure 15 shows the RCAM model prediction of ¼-scale tests with a lower static load (Tests 4 and 5) and a higher static load (Test 6). In all instances, the RCAM models had a similar shape to the time-deflection curves as the tests.

Fig. 15

RCAM model predictions of ¼-scale tests using char timescale correction factors φ_char = χⁿ for n = 0.17 (GPyro predictions), n = 0.28 (measured char) and n = 0 (Fourier number scaling) vs. measured deflections of tests with a lower load; and b higher load

The differences in the models were the rates of damage, input as the effective charring rates. The RCAM models with no correction factor, scaled by the Fourier time scaling laws ($n=0$) predicted Tests 4 and 5 to reach the maximum mid-span deflection at 107 s, while the tests averaged 161 s. Similarly, Fourier scaling models predicted Test 6 with a larger static load to reach the maximum mid-span deflection at 66 s, while the test concluded at 95 s. These predictions are overly conservative, demonstrating the need to use the char timescale reduction factors to have more accurate predictions. On the other hand, the RCAM models based on experimentally measured charring rates ($n=0.28$) were very accurate, with predictions of 156 s for Tests 1 and 2, and 97 s for Test 3. Because these models use experimental inputs, the model was validated to predict the structural response well of three-point bending tests of these multi-directional wood materials.

Table 8 compares the three sets of ¼-scale RCAM models to thermo-structural test data. The RCAM models were validated to show good agreement with the measured time-deflection curves when the charring rate is scaled by experimental results ($n=0.28$) with an error between -2.4% to 2.1%.

Table 8 RCAM model predictions ¼-scale tests with different char timescale correction factors

It is not always feasible to run thermal-only fire exposure tests to predict the char exponent, so this exponent can either be assumed to be zero or be estimated with modeling in a program such as GPyro. With no char timescale correction factor, the ¼-scale test behavior is conservatively predicted by 30.7% to 33.3%. The RCAM models incorporating the results of the GPyro models ($n=0.17$) predicted the time of maximum mid-span deflection to be 135 s for Tests 1 and 2, and 83 s for Test 3. The results of these models based purely on prediction were also conservative in predicting test end time, but that prediction improved to an error between 12.8 and 16.0% using GPyro modeling.

Using validated RCAM models to predict the behavior of ½-scale tests

Since the ¼-scale tests cannot directly be used to predict the behavior of the ½-scale tests because of the dissimilar stacking sequences, modeling has to be used to predict the behavior. Therefore, the reduced cross-sectional area method models validated to show good agreement with the ¼-scale tests were modified to predict the ½-scale test thermo-structural behavior. Again, we will compare the time-deflection prediction to the experimental measurement because the time at failure is an important performance indicator for fire resistance tests. The ½-scale RCAM models will predict the time-deflection response as the sample is damaged due to char and the time to reach the maximum center deflection of 50 mm. The time to reach the maximum center deflection indicates the time the sample can hold the particular static structural load under fire exposure before reaching an excessive deflection.

For the ½-scale tests, we wanted to use a purely analytical predictive model that could be created without the benefit of testing. Therefore, the reduction of the effective cross-section was assumed to occur based on the results of the GPyro model, which accounts for the reduction in charring rate in a reduced-scale test due to heating rate effects. From those GPyro simulations, a char time-scale correction factor of $n=0.17$ was calculated, making ${\beta }_{scale}= {\chi }^{-0.83}\beta ={\chi }^{-0.83}(0.8\frac{mm}{\text{min}})$. For the ½-scale fire exposure, $\chi =1/2$, and the effective charring rate assumed in the RCAM models was 1.43 mm/min. Figure 16 shows the RCAM model prediction of the ½-scale tests with a lower static load (Tests 1 and 2) and a higher static load (Test 3).

Fig. 16

RCAM model predictions of ½-scale tests using a purely predictive model with char timescale correction factors calculated from GPyro simulations (φ_char = χⁿ with n = 0.17) vs. measured deflections with a lower load; and b higher load

The RCAM models predict Tests 1 and 2 to reach the maximum mid-span deflection at 575 s, while the tests averaged 652 s. Similarly, the RCAM models predicted Test 3 with a higher static load to reach the maximum mid-span deflection at 477 s, while the test concluded at 502 s. In all instances, the RCAM models had a similar shape to the time-deflection curves as the tests and were conservative in predicting test end time by between 4.9 and 11.6%. Therefore, these purely analytical models of the test behavior with the time-scale reduction factor calculated from the GPyro simulations ($n=0.17$) show good agreement with the experimental results and can be a useful method to get conservative predictions of the larger-scale test without having to do experiments or computationally expensive thermo-mechanical models.

To further demonstrate the approach, models were also run with the other two char time-scale reduction factor scenarios: calculated from experimental char measurements ($n=0.28$) and ignoring the heating rate effects ($n=0$). Table 9 compares the three sets of RCAM models to thermo-structural test data. As expected, the predictions based on experimentally measured charring rates match very well (error −4.4% to 2.8%). More simple initial predictions can be created using the Fourier number time-scaling law which are much less computationally expensive because they do not require any GPyro analysis. Unfortunately, in the models with $n=0$, the predicted behavior will be overly conservative (error −15.2% to −21.1%) because it does not account for the heating rate effect on char kinetics.

Table 9 RCAM model predictions of ½-scale tests with different char timescale correction factors

While the results agree well going from ¼-scale to ½-scale, as discussed in Sect. “Thermo-structural test procedure” load capacity limitations of the experimental setup meant that we were unable to demonstrate the scaling for a full-scale sample. Our previous studies (Gangi et al. 2023a, b) demonstrated well the correlation between ¼-scale and full-scale with the heat transfer, and we are confident based on the results going from ¼-scale to ½-scale that the scaling methodology can account for material orientation. Prediction of the full-scale test, which will be saved for future work, would include modifying the reduced cross-sectional area method models validated to show good agreement with the ¼-scale tests. The full-scale RCAM model would have the geometry, loading, and composite stacking sequence of the full-scale sample with the reduction in the cross-section assumed to occur at the full-scale effective charring rate of $\beta$=0.8 mm/min.

Conclusion

This study develops and demonstrates an alternative methodology for using analytical modeling to predict the behavior of tests with combined thermo-structural loading on wood-based composite materials. This new alternative approach is applicable to a scenario when the laminate stacking sequence in the reduced-scale and the larger-scale test sample differ in their response to a progressing char front. The first step was to calibrate an analytical model to the smaller ¼-scale three-point bending tests. Reduced cross-sectional area models (RCAM) incorporating classical laminated plate theory (CLPT) were used to predict the mechanical response of the composite samples as the char front increased. Three alternate methods were proposed for calibrating the char timescale correction factor: (1) Fourier number scaling, (2) from predictive GPyro models, and (3) fitting to measured char rates. To validate the ¼-scale RCAM models, we assessed the ability of the models to predict the increase in center deflection of the sample over time. With Fourier number scaling ($n=0$), the ¼-scale test behavior is conservatively predicted by 30.7–33.3%, demonstrating the need to consider using a char timescale reduction factor. The RCAM models with the reduction of the effective cross-section fit to the results of the GPyro models ($n=0.17$) improved the predictions (error 12.8–16.0%), however the RCAM models fit to the measured char rates ($n=0.28$) were validated to show the best agreement with the ¼-scale tests (error −2.4–2.1%).

The final step was to create predictive models of the larger ½-scale tests by modifying the reduced cross-sectional area method models validated to show good agreement with the ¼-scale tests. The ½-scale RCAM models have the geometry, loading, and composite stacking sequence of the ½-scale sample. As expected, the predictions based on experimentally measured charring rates ($n=0.28$) match well with the measured time-deflection curves of the ½-scale tests (error −4.4–2.8%). Thus, the models validated to show good agreement with the ¼-scale tests were able to predict the behavior of the ½-scale tests, despite the differing stacking sequences. For the most accurate predictions, we recommend calibrating the char timescale reduction factor to char measurements from reduced-scale thermal-only tests. GPyro simulations ($n=0.17$) provide a useful method to get conservative predictions of the larger-scale test (error 4.9–11.6%) without more computationally expensive thermo-mechanical models. While improvements can be made to further refine the mechanical model, future research opportunities exist for improvements on the char-temperature progression predictions.