Scale modeling of thermo‐structural fire tests

Standard methods for fire resistance testing require large‐scale assemblies and are typically conducted on specialized furnaces at considerable cost. This research focused on developing a scaling methodology for a reduced‐scale fire resistance test that reduces the size of the test article while maintaining the same thermal and structural response exhibited in the large‐scale test. The developed scaling methodology incorporates uniform geometric scaling, Fourier number time scaling, and furnace boundary condition matching. The scaling laws were experimentally validated with fire exposure tests on gypsum wallboard samples at three scales (full‐scale, 1/2‐scale, and 1/6‐scale). In the tests, samples were exposed to a full‐scale equivalent of 60‐min of ASTM E119 fire curve exposure on a reduced‐scale horizontal furnace, and the temperature rise through the thickness profile was measured. Models were created to calculate the modified fire curves for the smaller‐scale tests. Experimental results show that on the exposed surface, the 1/2‐scale absolute temperature was within 1.7% of full‐scale, while the 1/6‐scale temperature was within 2.5%. While the time‐dependent properties of burning and cracking caused visual differences in these gypsum tests, modeling and temperature measurements demonstrated that the test results were thermally similar. The good similarity of temperatures is achievable in fire exposure tests of non‐combustible gypsum wallboard down to 1/6‐scale.

reducing the size of the test article while maintaining the same thermal and structural response exhibited in the large-scale test.
][7][8][9][10][11] Usmani et al. 6 documented an overview of the structural response of beam assemblies with no externally applied loads subjected to fire.Under single sided thermal exposure, unloaded beams undergo thermal bowing due to through-thickness temperature gradients.Thermal bowing can act in conjunction with the applied structural loads to initiate structural failures.McGuire et al. 7 investigated the similitude of the thermal response of the reduced-scale ASTM E119 fire exposure test for geometries reduced in every dimension (length, width, and thickness) by the same scaling factor, χ.Their work aimed to maintain nondimensional thermal gradients through the thickness of an assembly.
Through analysis of Fourier's law, to scale the heat conduction, the timescale in the reduced-scale test must be reduced by a factor of χ 2 .Thus, if the thickness of an assembly were reduced by a factor of two, the timescale of the exposure would be reduced by a factor of four to maintain non-dimensional thermal gradients.However, the heat transfer on the exposed surface of the sample during furnace loading is more complex, and this method alone did not account for surface heat transfer.A series of subsequent studies conducted by O'Connor et al. [9][10][11] addressed the scaling of ASTM E119 exposure through a modified furnace temperature curve based on comparative energy to account for surface heat transfer.In their work, the furnace temperature was increased to maintain the energy balance between the fullscale and an equivalent reduced-scale assembly.In addition, they assumed that the surface heat transfer was radiation-dominated to simplify the calculation of the modified furnace temperature. 11The modified furnace temperature resulted in similar thermal gradients in the assembly down to half-scale thickness.However, at less than ½-scale, this assumed radiation-dominated exposure methodology resulted in unrealistically high modified furnace temperatures for the scaled thicknesses.
3][14][15][16][17][18] Looking at different types of fire exposure, a series of compartment fire studies that experimentally validated scaling compartment fire conditions at a reduced scale were conducted by Perricone et al., 15 Perricone, 12 Wang et al., 16 and Wang. 13Their test results on wood cribs at 1/8, 1/4, and 3/8-scale compared well with each other in terms of heat transmission, but experimental limitations did not allow for comparison testing at full-scale.Regarding standard fire resistance testing, an experimental study conducted by Sorathia 19 reinforced the observation that the heat transfer in an ASTM E119 furnace test remains approximately one dimensional (1D) through the thickness of the assembly.In this study, the fire exposure tests reduced only the length and width of the test sample and exposed samples of the same thickness to the same ASTM E119 fire curve.Results showed that samples with surface areas reduced to 1/3-scale had similar unexposed surface temperatures as full-scale samples.However, not scaling the sample thickness when adding structural loading would result in different structural behavior.
Modeling has been an important tool in demonstrating the similarity between thermo-structural tests at different scales.A common approach has been to conduct small-scale experiments that inform models of large-scale tests.For example, Mehaffey et al. 20 conducted small-scale fire exposure tests on gypsum-board/wood-stud wall assemblies without loading to validate analytical models.Subsequent full-scale structurally loaded fire resistance tests were conducted, and the analytical model predicted the heat transfer through these walls rather well.Modeling has also been directly used to simulate furnace exposure in thermo-structural fire tests.Vallee developed numerical models with Abaqus to validate experimental furnace fire exposure tests on partition walls. 21Results showed that the analytical models could predict the fire resistance based on temperature rise with 6% to 28% error.Analytical modeling can be used to put experimental data into better context.This study builds on previous research by developing scaling laws for fire resistance tests conducted in a furnace.The geometry of the test samples is scaled down, including the sample thickness, to account for the 1D heat transmission through the thickness and to help maintain structural similarity.The fire exposure is applied at a reduced timescale to maintain the Fourier number.A modified fire curve is also calculated, assuming both radiation and convection inside the furnace.Fire exposure tests are performed on gypsum drywall samples to confirm that the scaling laws produce a similar thermal response at the reduced scale.3][24][25][26][27][28] Heat transfer finite element models are used to verify that the scaling laws produce thermal similarity.The tests and models are developed to demonstrate that the new scaling laws can yield small-scale tests that behave similarly to their full-scale equivalents in terms of thermal behavior.

| SCALING LAW PROCEDURE
Our scaling methodology for thermo-structural test scaling combines three approaches: uniform geometric scaling, Fourier number time scaling, and boundary condition matching.This methodology reduces the original full-scale test that requires specialized facilities to a test that can be performed in a typical lab setting, which we will refer to as reduced-scale.

| Uniform geometric scaling
In the reduced-scale test, uniform geometric scaling can be applied to calculate the dimensions of the reduced-sized test sample.Uniform scaling, or isotropic scaling, shrinks the full-scale test samples by a scale factor that is the same in all directions.The result of the uniform scaling factor is a reduced-size sample that is similar (in the geometric sense) to the original.The uniform scaling factor, χ, is the ratio of the scaled dimensions (reduced-size) to the original dimensions (full-size).For example, for a half-scale test, χ = 0.5.As shown in Figure 1, the original assembly has a width, a, length, b, and thickness, d.If the same distributed load per unit area, p, is applied to both assemblies, to maintain structural similitude, the dimensions of the scaled assembly are: Uniform scaling of the assembly in a thermo-structural test has implications for both the structural and thermal behavior.The main goal of many thermo-structural tests is to capture the failure behavior of the given sample.Structural failure is often determined by exceeding a stress limit, whether a normal bending stress failure, a planar shear stress failure, or a stress concentration.Uniform geometric scaling combined with scaling of the total mechanical load allows the reduced-scale test to produce similar stresses.For example, in Figure 1, the sample is a homogeneous, isotropic, rectangular plate with a uniformly distributed load per unit area, p and simply supported on all four edges.From small-deflection theory, the maximum stress at the center of the sample due to cylindrical bending is given by 29 : If the materials are the same at both scales, the same maximum stress will be seen in the reduced-scale test if the same distributed load is applied as in the full-scale test.

| Fourier number time scaling
Unfortunately, reducing the through-thickness scale of the assembly is expected to affect the temperature profile.Our interest is in thermo-structural tests with one-sided exposure where the exposure is predominately one-dimensional through the thickness of the sample.If two samples of different thicknesses are exposed to furnace temperatures for the same duration, the similar thinner sample will heat up much quicker than its thicker counterpart.As a result, the exposure time needs to be reduced when testing a thinner sample.
Fourier-number time scaling will reduce the fire exposure time in the reduced-scale test.Once the thickness of the sample is reduced by a factor χ, time scaling is required to have similarity between the 1D heat transfer through the thickness of the sample.During the heat transfer, the conduction through the thickness of the sample is governed by the transient heat diffusion.In this study,"$" will be used to define the dimensionless variables.Dimensionless variables will be shown as the ratio of the variables, for example, x ¼ x=d, t ¼ t=τ, and . The Fourier number controls the solution to the 1D heat equation.This non-dimensional equation is given by: where d is the thickness of the part, α is its thermal diffusivity, and τ is an appropriate characteristic time (e.g., the total test time).Consequently, Fourier number scaling is required to have thermal similarity at both scales Thus, the characteristic time for the scaled test differs by a factor of χ 2 from that of the original test.For our example half-scale test, χ 2 = 0.25; therefore, 15 min of fire exposure at half-scale is equivalent to 60 min of exposure on the full-scale test.Therefore, we can define a ½-scale test with 15 min of fire exposure as having a full-scale equivalent test time of 60 min.

| Boundary condition matching
Maintaining the thermal gradients through the thickness of the sample also requires that the surface temperatures be maintained at the same full-scale equivalent time in the reduced-scale sample as in the fullscale.The 1D conduction of heat through the sample thickness is caused by a thermal flux incident on the surface.The net thermal flux Comparison of dimensions from original to scaled test.
transmitted through a sample, q 00 net , is a combination of convective heat transfer, q 00 conv , and radiative heat transfer, q 00 rad . 30 The first term on the right-hand side of the equation represents convective heat transfer between furnace gasses T f and the sample surface temperature T s .Although the details of the convective heat transfer mechanism are complex, the magnitude is generally described as a simple linear relationship controlled by the film coefficient h. ) to develop the boundary conditions (BC) on the exposed (x ¼ 0) and unexposed (x ¼ 1) surfaces, given as: Unexposed BC : q00 where Tf , Te , Tu , and T∞ are the non-dimensional furnace, exposed surface, unexposed surface, and far-field ambient temperatures, respectively, h e and h u are the convection heat transfer coefficients on the exposed and unexposed sides, and T max f is the maximum (absolute) furnace temperature.
The other common terms in these equations, shown in Figure 2A, are the sample thickness, d, the thermal conductivity, k, radiative emissivity, ε, and the Stefan-Boltzmann constant, σ.To properly scale the heat transfer process, the non-dimensional heat flux must be maintained between the original (full-scale) and scaled (reduced-scale) tests.
On the exposed surface, defined by Equation (8), we can set the boundary condition of the original assembly (Figure 2A) equal to that of the scaled assembly (Figure 2B).In the resulting Equation (10), it is assumed that the surface temperature, thermal conductivity, emissivity, and convective heat transfer coefficients remain constant at both scales.
On the exposed surface, the furnace temperature of the scaled test furnace can be adjusted to give the same non-dimensional heat flux.
Isolating the non-dimensional modified-scale furnace temperature, Tf,scale results in the following equation: with: On the unexposed surface, defined by Equation ( 9), we can set the boundary condition of the original assembly (Figure 2A) equal to that of the scaled assembly (Figure 2B).The process for boundary condition matching is the same as that on the exposed surface, though it is much more difficult to change the ambient temperature in a real testing scenario.Therefore, in the resulting Equation ( 12), the surface temperature, thermal conductivity, emissivity, and ambient temperature are assumed to remain constant at both scales.
On the unexposed surface, the non-dimensional heat flux was matched by adjusting the heat transfer coefficient of the unexposed side: As with the exposed surface, Equation ( 13) alone is not sufficient to determine the required heat transfer coefficient of the unexposed side.Knowledge of the unexposed surface temperature-either from measurements or simulations-is required.

| HEAT TRANSFER MODELS
Numerical simulations help demonstrate that the thermo-structural scaling laws will produce similar behavior in scaled samples in realistic testing scenarios.Initially, we focused on modeling thermal-only fire exposure tests.The fire exposure tests simulate the heat transfer in a thermo-structural test, which is crucial for maintaining similar thermal and structural behavior.Maintaining the thermal gradients through the thickness of a sample is essential to the structural behavior, as this drives thermal bowing. 6There are also benefits to maintaining a thermal gradient regarding material degradation at elevated temperatures.
Notably, temperature-dependent properties are preserved if the same temperatures are achieved at similar positions in the scaled and original samples.Therefore, if the temperature-dependent structural stiffness or failure stress remains similar, this leads to similar structural behavior.
Fire exposure tests were conducted on samples at different scales made from lightweight Type X gypsum wallboard with a density of ρ ¼ 582 kg m 3 .First, the full-scale test was a 96 mm (3.75 00 ) thick gypsum sample built up from six layers of lightweight Type X wallboard exposed to a 60 min ASTM E119 standard time-temperature fire curve. 3Subsequent tests were a ½-scale test of a 15 min modified time-temperature fire exposure on a 48 mm (1.875 00 ) thick sample that contains three layers and a 1/6-scale test of a 100 s modified exposure on a 16 mm (0.625 00 ) thick sample with only one layer of wallboard.The modified time-temperature fire curves for the ½-scale and 1/6-scale tests were calculated from Equation (10), using the exposed surface temperature from the full-scale test model as an input.
Two-dimensional (2D) finite element models were created in Abaqus CAE 32 to simulate the heat transfer in the fire exposure tests.The heat transfer occurs primarily through the thickness of the sample.
Therefore, the 2D model geometry only considered the sample width and thickness.Material properties used in these models come from the SFPE Handbook of Fire Protection Engineering. 33This handbook suggests values for the thermal density, specific heat capacity, and thermal conductivity of the Type X gypsum wallboard as a function of temperature, shown in Figure 3.
The initial temperature of the samples was assumed to be uniform and equal to 20 C (293 K).The boundary conditions shown in Figure 4A were radiation and convection on the exposed surface, with the furnace temperature Tf changing over the time of the exposure via the fire curves.On the unexposed surface, boundary conditions are also radiation and convection, though the far-field temperature T∞ remains constant at the ambient lab temperature of 20 C. The boundary conditions were set using "Interactions."The interaction types, used for convection and radiation boundary conditions, were "Surface film condition" and "Surface radiation to ambient," respectively.The effective emissivity of the radiation was set to 0.8. 34The convection heat transfer coefficients were assigned according to the European code on fire interaction with structures, 35 that is, 4 W/m 2 K for the unexposed surface and 25 W/m 2 K for the exposed surface.The modified fire curve temperature is a function of the exposed surface temperature of the sample, which is not explicitly known.The full-scale heat transfer model was used to determine the exposed surface temperature Te as shown in Figure 5A.From this exposed surface temperature curve and given the full-scale fire curve, the modified fire curves at both ½-scale and 1/6-scale can be calculated using Equation (2.11).The full-scale ASTM E119 fire curve was explicitly defined as a linear approximation between five points in time (300, 600, 1800, 3600, and 7200 s).Therefore, the modified fire curves were calculated at five similar points on the reduced timescale.
For example, the ½-scale modified fire curve, with a Fourier number time scale factor of χ 2 = 1/4 was calculated at 75, 150, 450, 900, and 1800 s.The results of the calculations are summarized in Table 1 and shown visually in Figure 5B.The newly calculated modified fire curves were used as boundary conditions in the ½-scale and 1/6-scale finite element models.As the scaling factor decreases, the furnace temperature needs to increase more quickly to maintain heat flux.

| EXPERIMENTAL METHODS
We conducted a series of thermal-only fire exposure tests at different scales on our reduced-scale furnace to determine if our thermal scaling laws work in practice.The objective is to demonstrate the thermal aspect of the scaling laws.In addition, we investigated the feasibility of achieving fire exposure on a reduced-scale furnace.

| Small horizontal furnace setup
The goal of the small horizontal propane furnace design was to create an inexpensive furnace that could provide the modified exposure of the ATSM E119 time-temperature fire curves at smaller scales.However, for this reduced-scale furnace design to remain inexpensive, it must be small enough to be constructed in a typical lab setting.Therefore, the dimensions of the reduced-scale furnace, shown in

| Material tested
Gypsum wallboard was chosen as a material for testing due to its wide application as a building material and its fire resistance properties.
Locally sourced Type X lightweight gypsum wallboard sheets were cut down into test samples.Lightweight Type X gypsum wallboard, 16 mm (5/8 00 ) in thickness of 582 kg/m 3 density, must be manufactured according to established ASTM standards to provide not less than one-hour fire resistance. 36Thus, a 16 mm section of gypsum wallboard should pass an ASTM E119 fire exposure of 1 hour.Gypsum, particularly Type X fire-resistant gypsum, is non-combustible, which means it contributes no fuel to a fire.Therefore, during fire exposure, the cross-section should remain intact.The 16 mm crosssection was chosen as the smallest scale (1/6-scale) to have a fully intact section with both face sheets and allow enough depth for two thermocouple probes to be placed through the sample thickness.

| Test procedure
We tested the gypsum samples in our small horizontal propane furnace.Six fire exposure tests were conducted, and two gypsum drywall samples were tested at full-scale (χ ¼ 1), half-scale (χ ¼ 1 2 ), and sixthscale (χ ¼ 1 6 ).The test matrix is shown in Table 2.For the purposes of preparing test samples, it was assumed that the heat transfer occurred primarily through the thickness of the sample.Therefore, all samples had the same out-of-plane dimensions, 152 mm wide by 686 mm long.
The scale was determined by the sample thickness of the built-up section.Built-up sections were glued together with Titebond III wood glue.Therefore, all sections came from the same sheets of material.
The gypsum samples were 96 mm thick at full-scale, 48 mm thick at half scale, and 16 mm thick at sixth scale.
The instrumentation plan for the samples is shown in Figure 9.
Temperature measurements were taken in the middle of each sample, 76 mm from the long side, so that edge effects can be ignored.As shown in the top view of Figure 9A, both the surface and throughthickness 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.The holes were drilled 76 mm into the side so that measurements were taken in the center, in line with the surface measurements.In the ½-scale and 1/6-scale samples, thermocouples were placed at four distances from the fire-exposed surface (on the surface, 1 3 d, 2 3 d, d), shown in Figure 9C,D.The holes in the 1/6-scale samples were offset 13 mm horizontally to avoid cracking between holes in the sample.While the thickness of the 1/6-scale sample was limiting, in the full-scale samples the thickness of 96 mm allowed for more thermocouples to be placed to get a more complete profile.Therefore, in the full-scale samples, thermocouples were placed at seven distances from the fireexposed surface (on the surface, 1 6 d,  9B.
The samples were placed on the furnace frame over the opening so that the middle 610 mm was exposed.The remaining surface area of the furnace that the samples did not cover was blocked with barriers made from Type X gypsum wallboard and insulated with a 51 mm ceramic-fiber insulation blanket.A ceramic-fiber insulation blanket 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.
The tests were terminated when the slabs experienced 1 hour of equivalent fire exposure (based on full-scale duration).From Fourier number scaling laws, the times were 3600 s at full-scale, 900 s at ½-scale, and 100 s at 1/6-scale.Upon termination time, the furnace gas was turned off, the thermocouples were disconnected, and the sample was promptly removed from the furnace to prevent further exposure.The initial and final thickness and sample dehydration depth were measured on the slabs.A knife was used to penetrate the surface near the centerline to measure the sample dehydration depth.
The knife could only penetrate the volume of gypsum that had been dehydrated.A caliper measured the knife penetration depth at six locations and averaged it, which was related to the original sample thickness.

| RESULTS
A summary of the test results for the fire exposure tests on gypsum drywall samples at different scales is provided in Table 3   Thermocouples were used to measure the changes in temperature during the experiment.Our scaling is successful if it can capture a similar temperature profile as the full-scale sample throughout the entire exposure.Figure 11 shows the resulting thermal distributions for the typical half (1/2) and sixth (1/6) scale tests compared to the full-scale results.Thermocouples were positioned on each surface and at two points in the sample thickness to sample the complete crosssection.Inspection shows a good agreement between the full-scale and reduced-scale temperatures.However, the ½-scale results showed better agreement than the 1/6-scale results when compared with the full-scale.On the exposed surface, the ½-scale temperature was within 1.7% of full-scale by the end, while the 1/6-scale temperature was within 2.5%.By the end of the test, this non-combustible material had achieved reasonable comparability of temperatures near the fire-exposed faces down to 1/6-scale, but there was some disagreement around the full-scale equivalent 30 minute point.This early test discrepancy may be due to the exposure in Figure 10C lagging slightly behind the desired fire curve to that point.
Another way to visualize the temperature change is with thermal gradients.The thermal gradient is the temperature difference from the exposed surface to the unexposed surface.Figure 12 shows the comparison temperature profiles of the full-scale, ½-scale, and 1/6-scale tests at snapshots of what is equivalent to 30 and 60 min of exposure.On the y-axis, the location of the thermocouple within the cross-section depth is normalized, where 0 is the exposed surface and 1 is the unexposed surface.Comparing the data, the ½-scale results in blue and the 1/6-scale experiments in yellow closely approximate the full-scale temperature profiles in black.However, the ½-scale results showed better agreement than the 1/6-scale results.On the unexposed surface in particular, the ½-scale temperature was within 1.08% of full-scale by the end, while the 1/6-scale temperature was within 11.24%.
Figure 13 shows the gypsum samples both before and after the fire exposure.During fire exposure, the non-combustible gypsum drywall cross-section should remain intact.This is because gypsum contains 21% by weight of chemically bound water, also called crystalline water, as part of the gypsum crystal itself. 37When gypsum drywall panels are exposed to fire, the heat of the fire converts the crystalline water to steam.The heat energy that converts water to steam is thus absorbed, keeping the opposite side of the gypsum panel cool as long as there is water left in the gypsum.Therefore, a measure of the level of damage is the amount of dehydrated section.From measurements based on knife insertions, the samples had a similar level of dehydration, with a full-scale of 31.8% of the cross-section dehydrated, a half- The results of the models for the other two scales followed a similar trend.The ½-scale model, for example, had surface temperature rises within 2.4% and 5.8% on the exposed and unexposed sides, respectively.Similarly, the 1/6-scale model was within 4.9% exposed and 20.2% unexposed.Thus, the finite element models were validated to  show good agreement with the experimental data.A summary of the test model predictions compared to the experimental measurements is shown in Table 4.
Table 5 shows the summary of the ability of the reduced-scaled tests to capture the surface temperature rises of the full-scale test.
The ½-scale test showed good agreement with the full-scale, with surface temperature rises within 5.0% (43.7 K) and 12.1% (2.7 K) on the exposed and unexposed sides, respectively.On the other hand, the 1/6-scale tests showed more discrepancy, with 9.6% (83.1 K) exposed and 123.4% (28 K) unexposed.
One of the of the experimental results at 1/6-scale was that after the full-scale equivalent 60 min exposure, the unexposed surface temperatures were 28 K higher than at the full-scale.The modeling investigates two possible explanations for this difference: the higher initial temperatures at 1/6-scale and the disregard for boundary condition matching on the unexposed surface.From Figure 14, the test models accounted for the higher ambient temperature in the 1/6-scale tests (301 K vs. 293 K) in the boundary condition and the initial temperature profile used as initial conditions.Additionally, while the film coefficient was not explicitly measured during the tests, no external fans were used to control the air speed inside the lab, so the film coefficient was assumed to be 4 W/m 2 K in all models.The effect of using the same heat transfer coefficient at 1/6-scale rather than the calculated 49 W/m 2 K from boundary condition matching was seen in the previous theoretical model results.From Figure 7, when assuming 4 W/m 2 K at 1/6-scale, the unexposed surface temperatures were 8 K higher than in the full-scale models.Therefore, the combination of the higher initial temperature and lower heat transfer coefficient compared with the recommended value through scaling resulted in the test models predicting within 10 K of the measured results at 1/6-scale.
Previously, it was observed that the time-temperature curve at 1/6-scale was more challenging to achieve because of the short timescale.The area under the curve, averaging the readings from the five furnace thermocouples, was only within 3.4% of the desired exposure, compared to 0.2% at ½-scale and 0.5% at full-scale.The discrepancy mainly showed up before the full-scale equivalent 30 min point in that test.By an equivalent 30 min of exposure, the exposed surface temperature at 1/6-scale was measured to be 121 K less than the measured temperature at full-scale.The 1/6-scale test model with measured furnace temperatures predicted the exposed temperature to be 104 K less than the full-scale models.Therefore, modeling reinforces that the discrepancy was caused by not being able to match the desired exposure within the first 50 s of the 1/6-scale test.By the end of the 60 min test exposure, though, the discrepancy had decreased, and the 1/6-scale exposed surface was measured to be within 1.90% of full-scale.Taking this into account, the scaling appears appropriate down to a 1/6-scale if the faster exposure temperature rise can be met.The lagging behind desired temperature at 1/6-scale was an issue with the reduced-scale furnace setup.Future testing will include a reconfiguration of the propane burners so that they impinge directly on the sample surface.that the paper facing remains on the exposed lining for between 5 and 10 min of severe furnace exposure. 22The complete char oxidation process occurred in the full-scale and ½-scale samples, exposed to 60 and 15-min, respectively.However, as seen in Figure 16A, the 100 s exposure time at a 1/6-scale does not allow the complete char oxidation process.While the face sheet burned and charred, not enough time was allowed for char oxidation.

| Visual observations
In addition, cracking was seen in the full-scale samples, shown in Figure 16B.Once the paper had oxidized, exposing the gypsum, continued exposure led to the formation of small horizontal cracks.A previous study from Manzello et al. saw similar cracking in Type X gypsum board samples after the paper on the exposed face had oxidized. 38It is commonly known that when gypsum panels are heated, they contract due mainly to dehydration. 39Cracking was observed in the dehydrated gypsum of the full-scale samples.In both the ½-scale and 1/6-scale samples, with exposure times of 900 and 100 s, respectively, no significant horizontal cracking was observed in the dehydrated gypsum.Therefore, the large cracking in the full-scale samples occurred because of the longer exposure time of 3600 s.These visual observations support the idea that certain time-dependent processes, such as char oxidation, or dehydration cracking, may not be fully accounted for using our test scaling laws in small-scale tests.

| CONCLUSION
This study develops and demonstrates a methodology for reducing the scale of a test with combined thermo-structural loading to provide similar results at all scales.The scaling methods incorporate three approaches: geometric scaling, Fourier number time scaling, and boundary condition matching.Modeling demonstrated that boundary condition matching on the unexposed surface was not necessary for this test geometry.First, we built a small horizontal furnace for conducting fire exposure tests at a reduced scale.Next, the thermal portion of the scaling laws was demonstrated with fire exposure tests on non-combustible (gypsum drywall) samples.
Good similarity of temperature profiles and extent of dehydration in the cross-section was demonstrated in samples down to 1/6-scale.
The similar temperature profiles indicate that scaling of the boundary condition on the unexposed surface is not necessary for this testing scenario.The heat transfer models using experimentally measured conditions were validated to show good agreement with the experimental data.The scaling agrees with the modeling as well.However, 1/6-scale may be the practical limit for scaling gypsum drywall samples due to many factors: difficulty of achieving the exposure in the 100-s tests, the higher initial temperature gradients, and the measurement uncertainty of the thermocouples within the thin cross-section.In a sample so thin, to fit the two thermocouples at the correct depth within the 16 mm cross-section required precise drilling.Additionally, while thermal performance was similar, certain timedependent phenomena seen at the full-scale like char oxidation of the face sheet and cracking, were not captured at the 1/6-scale.
Future work will include a series of thermo-structural tests to investigate if achieving similar thermal behavior will lead to similar structural behavior.Additionally, these tests will investigate the effect of material type on the scaling laws, including wood materials that have time-dependent properties of pyrolysis and charring.
Heat transfer boundary conditions on the (A) Original full-scale sample and (B) Reduced-scale sample.

orig À T4 e
Standard fire time-temperature curves define full-scale furnace gas temperatures, such as ASTM E119 or ISO 834.31The first step in calculating the modified fire curves for the reduced-scale tests is reducing the time of the exposure by the Fourier number scaling factor χ 2 .Then, at each new time step, the modified furnace temperature must be calculated to give the same non-dimensional heat flux as in the full-scale test.This approach alone cannot calculate the modified scale test furnace temperature because the modified scale test furnace temperature is a function of the exposed surface temperature of the sample, which is not explicitly known.Determining the non-dimensional exposed surface temperature, Te , requires either experimental data from tests conducted at the original scale or validated computer simulations.Prior to conducting the original full-scale experiments, the modified furnace time-temperature curve of a reduced-scale test can be predicted with a detailed heat transfer model of the full-scale test.In this study, the exposed surface temperature from the simulations of a full-scale test was used to calculate the fire curves at ½-scale and 1/6-scale.

A
finite element mesh was created to allow for 30 DC2D4 fournode linear heat transfer quadrilateral elements through the thickness of the sample.The temperature profiles through the thickness of the samples in the models are shown in Figure 4B.This figure shows the model mesh density, the boundary conditions on both the unexposed and exposed sides, and a contour plot of the thermal gradient.The transient heat transfer in the model occurs for a full-scale equivalent of 60 min.

F
I G U R E 3 Properties of lightweight Type X gypsum board as a function of temperature.(A) Density, (B) Specific Heat Capacity, and (C) Thermal Conductivity.F I G U R E 4 Gypsum models (A) Summary of boundary conditions for full-scale Abaqus finite element model, and (B) Mesh and temperature contours.Similarly, the modified heat transfer film coefficient is a function of the samples' unexposed surface temperature on the unexposed boundary, which is not explicitly known.The full-scale heat transfer model was used to determine the unexposed surface temperature Tu , as shown in Figure 6A.From this unexposed surface temperature curve and given the constant ambient temperature, the modified film F I G U R E 5 (A) Full-scale ASTM E119 fire curve and resulting exposed surface temperatures, and (B) Resulting modified furnace timetemperature fire curves from Boundary Condition Matching.T A B L E 1 Summary of modified furnace time-temperature fire curves at different scales.

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I G U R E 6 (A) Full-scale ambient temperature and resulting unexposed surface temperatures, and (B) Resulting modified film coefficients from Boundary Condition Matching.coefficients at both ½-scale and 1/6-scale can be calculated using Equation(13).The modified film coefficients were calculated at five similar points in time as the modified fire curve.Because the estimated values had a minimal variance across time, an average value was chosen, which can be used as a constant.The results of the calculations are shown in Figure6B.Models were run with the newly calculated modified film coefficients used as boundary conditions in the ½-scale and 1/6-scale finite element models.As the scaling factor decreases, the film coefficient needs to increase significantly to maintain heat flux.Models run with these boundary conditions produced the exact unexposed surface temperatures as the full-scale.However, in practice, it is difficult to adjust the heat transfer coefficient on the unexposed surface.The required heat transfer coefficient would be challenging to control without using large fans or blowers to move the ambient air and difficult to measure.For example, to increase the film coefficient from 4 W/m 2 K at full-scale to 49 W/m 2 K at 1/6-scale, the ambient lab air velocity would have to increase by roughly 10 times.Increasing the film coefficient that much would be unrealistic without the aid of a wind tunnel.Therefore, the impact of not scaling the unexposed boundary needs to be assessed.Analyses were conducted with samples at ½-scale and 1/6-scale, with the unexposed boundary condition being a film coefficient of 4 W/m 2 K.The transient heat transfer in the model occurs for a fullscale equivalent 60 min, which is 900 s at ½-scale and 100 s at 1/6-scale.Figure7A,B compare the model temperature profiles at snapshots of equivalently 30 and 60 min of exposure, respectively.On the y-axis, the location within the depth of the cross-section is normalized, where 0 is the exposed side and 1 is the unexposed side.The temperature profiles of the models are shown at full-scale (black), ½-scale (blue), and 1/6-scale (yellow).Results from the finite element models show that without boundary condition matching on the unexposed surface, the scaled tests have very similar thermal behavior to the original full-scale test.The heat transfer models matched behavior very well, considering the heat transfer boundary condition was scaled only on the exposed surface.A constant heat transfer coefficient on the unexposed boundary condition made a minimal impact on the model results.The unexposed surface temperatures are predicted to be 316 K at full-scale, 320 K at half-scale, and 324 K at sixth-scale.The 1/6-scale model is within a margin of 10 K, thus the experimental testing program will not consider boundary condition matching on the unexposed surface.

Figure 8A ,
Figure8A, with an overall length of 838 mm, a width of 686 mm, and a depth of 305 mm.The opening at the top allows for samples up to

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I G U R E 8 Reduced-scale horizontal furnace (A) Furnace setup (B) Thermocouple diagram.

F I G U R E 9
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 5 Thermocouple Probes through the thickness of full-scale samples; (C) The reduction of Thermocouple Probes to 2 through the thickness of the Half-scale sample; and (D) The offset in the 1/6-scale sample to have 2 probes through the thickness.arepresented as an average of the two tests performed on each scale, Test Groups 1, 2, and 3.In addition, this table includes the test termination time, the comparison of the actual furnace exposure to the desired exposure, the surface temperatures at the end of the exposure, and the measured level of damage of the cross-section.

Figure 10 4 F I G U R E 1 0
Figure 10 compares the desired modified ASTM E119 fire curves and

F I G U R E 1 2 | DISCUSSION 6 . 1 |F I G U R E 1 5
Figure 10 were used as the sink temperature on the exposed surface boundaries.The temperature profiles of the test models are shown in comparison to the experimental data at an equivalent exposure time of 30 min in Figure 15A and 60 min in Figure 15B.The test model results at full-scale, ½-scale, and 1/6-scale are shown in the dashed lines, compared to the measured data in the solid lines.By visual

F I G U R E 1 6
Visual observations of gypsum samples (A) Char oxidation of the face sheet was seen in the full-scale and ½-scale samples, but not at 1/6-scale, and (B) Horizontal cracks developed in the full-scale sample due to dehydration, but not in the other samples.

Figure 16 shows
Figure 16 shows two primary visual observations from fire exposure tests of gypsum samples: char oxidation of the paper face sheet and cracking of the panels.Paper-faced gypsum plasterboard linings are most commonly used, particularly when fire resistance is required.The paper facings, which contain the core material and provide tensile strength to the plasterboard linings, are burned away after temperatures reach approximately 300 C. A previous study from Jones found . The results T A B L E 2 Test matrix for fire exposure scaling tests.