An Integrated Thermomechanical Method for Modeling Fiber Reinforced Polymer Composite Structures in Fire Ziqing Yu¹ and Aixi Zhou² ¹Department of Engineering Technology, University of North Carolina at Charlotte, 921 University City Blvd, Charlotte, NC 28223-1; PH (74) 687-3273; email: zyu1@uncc.edu ²Department of Engineering Technology, University of North Carolina at Charlotte, 921 University City Blvd, Charlotte, NC 28223-1; PH (74) 687-3727; email: aixi.zhou@uncc.edu ABSTRACT This paper presents an integrated thermomechanical method for modeling the behavior of fiber reinforced polymer (FRP) composite structures subject to simultaneous fire and mechanical load. The model includes heat transfer modeling to calculate temperature history of the structure and structural modeling to predict the mechanical performance of the structure. Heat transfer modeling assumes that heat and mass transfer of volatile by decomposition are one dimensional through thickness. With the temperature profile calculated by heat transfer modeling, structural modeling analyzes the deformation and failure. Both thermal and mechanical properties in modeling are temperature dependent. Arrhenius equation is used to describe decomposition reaction of resin and a concept of shift temperature is introduced to account for heating rate s effect on the decomposition temperature. The integrated model is validated with experimental data from structural fire tests. INTRODUCTION Fiber reinforced polymer (FRP) composites have been widely used in various structural applications. FRP composites offer many advantageous physical and mechanical properties, such as high specific strength, lightweight, and good fatigue and corrosion resistance. However, since FRP composites contain polymer matrixes, they are highly combustible. FRP composites will degrade, decompose, and sometimes yield toxic gases at high temperature (Pering 198). Due to its combustible nature, fire safety and fire protection of FRP composites are of great concern. Evaluation of the performance of FRP composites in fire includes reaction to fire and fire resistance studies. Reaction-to-fire study examines fire growth and fire effluents. Fire resistance study examines how a structure resists fire and usually measures three parameters: insulation, integrity, and load bearing capacity (for loadbearing elements). The intention of this paper is to address the fire resistance of FRP composite structures with a focus on developing and validating an integrated thermomechanical method for predicting the response of FRP structures in fire.
Analyzing the structural fire resistance of FRP structures requires at least two analyses: (1) a heat transfer analysis including effects of decomposition to predict the temperature profile in the structures as a function of time and location (Hendeson et.al. 1985 and 1987; Florio et.al., 1991); and (2) a structural analysis to examine the mechanical and structural response of FRP structures in fire (Bausano, et.al., 26; Feih et.al, 26; Gu 25; Keller et.al., 26; Zhou and Keller, 25). The heat transfer analysis is critical since estimating the response of a structure in fire requires temperature-dependent thermal properties and mechanical properties of the material (Mouritz and Gibson 26). The effect of insulation can also be estimated based on heat transfer analysis. The decomposition analysis is important since many physical and mechanical properties are affects by the decomposition process. The structural analysis will give fire protection engineers or structural engineers critical information related to structural fire resistance analysis and design (Mouritz and Gibson 26), such as deflection and strain as a function of time, residual stiffness and strength of the structure, stability and integrity of the structure, time-to-failure estimation of the structure. This paper presents an integrated method for modeling the response of FRP composite structures in fire. The integrated model consists of heat transfer modeling and a structural modeling. In the heat transfer section, both heat and mass transfer are assumed to be one dimensional. In the structural analysis section, decomposition s effects on mechanical properties are considered. Temperature and deformation predictions will be compared with experimental data from FRP laminate under onesided constant heat flux and intermediate scale FRP laminate panels under one-sided furnace fire. HEAT TRANSFER MODELING One Dimensional Heat Transfer The following one-dimensional heat transfer model has been developed by Henderson et. al. (1985, 1987) for degrading material with instantaneous, unidirectional flow of gases toward the heated surface. = h h = =
where three terms on the right hand side of the first equation above relate to heat conduction, heat of resin decomposition and volatile convection.,,,,,h are instantaneous density, virgin density, decomposed density, specific heat, thermal conductivity, and enthalpy of the composite;,h, are mass flux, enthalpy and specific heat of gases generated from decomposition of resin; Arrhenius equation is used to describe decomposition reaction where the change in density with time is defined as a function of temperature. A, E, n are rate constant, activation energy, and the order of resin decomposition reaction respectively. The resin decomposition term is negative when the decomposition process is endothermic and positive when exothermic. Arrhenius equation is accurate only for a small range of heating rates because it is only a function of temperature provided that the instantaneous density is also only temperature dependent. A concept of shift temperature is introduced here to take into account effects of heating rate. The modified Arrhenius equation is given by = exp where the shift temperature is a function of heating rate and can be determined from TGA tests. To determine the shift temperature, series of TGA tests need to be conducted at a practically wide range of heating rates. Since higher heating rates can increase the temperature of decomposition and therefore each TGA test at a specific heating rate will give its own temperature of decomposition, the shift temperature can be determined by fitting a function of heating rate for the decomposition temperature. Decomposition degree is measured by residual weight fraction of resin to original resin, which is given by = The decomposition fraction F runs from 1 to where 1 indicates no decomposition and implies no resin remaining. Decomposition s effects on thermal and mechanical properties are accounted for using the decomposition fraction as in the following thermal and mechanical property equations. Thermal Properties at Different Material Sates Thermal conductivity and specific heat capacity depend on temperature and the decomposition state of the material. Under three material states--virgin material, decomposing material, and decomposed material, material has different thermal properties. While thermal properties at virgin and decomposed states can be determined by thermal tests such as DSC and Hot Disk, rule of mixture can be used to compute thermal properties at decomposing state = +1
= +1 where,,,, are thermal conductivities and specific heat capacities at virgin and decomposed states. Thermal Boundary Conditions If fire is defined by temperature, the thermal boundary at the exposed surface is then given by = +h If fire is define by heat flux, the thermal boundary at the exposed surface is = h The thermal boundary condition at the unexposed surface is given by = +h In boundary equations above, is emissivity of the exposed surface;,, are temperatures of fire, the exposed surface, and ambience, respectively. STRUCTURAL MODELING In the study, only elasticity is considered, no viscoelasticity included. Most FRP structures, such as those pultruded and VARTMed can be considered as orthotropic in structural analysis, and are assumed to be orthotropic during the whole duration of fire exposure. Assumption of Plane Strain The first analytical mechanical model by McManus and Springer (1992) used the following governing equation to include influence of thermal expansion, internal gas pressure, moisture as well as charring expansion. = + +Λ + + Δ where,λ,, are the thermal, pressure, moisture, and charring expansion coefficients, and,,,δ are the temperature, pressure, moisture content and char volume differences, respectively. The model is remarkable but expensive or less practical since all the coefficients must be experimentally determined before stress and strain can be calculated. In this study, only the material compliance and thermal expansion will be considered. The equation above reduces to
= + where i,j,k,l=1,2,3. are the strains, is elastic compliance and = for isotropic and orthotropic materials, are stresses, are the thermal coefficients of expansion, and = is the temperature difference. The above equation assumes = initially. Because in this study compressive load is applied only along longitudinal direction (x-direction) with a uniform distribution over width direction (y-direction) and the panel is very thin, we can assume that strain oriented in y-direction is negligible and panels only experience plane strain in x-z plane, then = = = and the problem is reduced to be two dimensional. The compliance matrix D for orthotropic material becomes, [ ]=[ ]= 1 1 1 2 where = and Poisson s ratios are assumed independent of temperature. Young s modulus and shear modulus,, are functions of temperature. A temperature-displacement analysis step in ABAQUS can be used to solve the constitutive equation above numerically for deformation history using plane strain element CPE4RT. Temperature-Dependent Mechanical Properties Validated by Mouritz et al. (26), the following hyperbolic tangent function is used to describe the relationship between mechanical properties and temperature before decomposition: ={ + 2 2 tanh } where P is the particular mechanical property, P U and P R are values of that property at the room and at high temperatures respectively. T g is the mechanically determined glass transition temperature, at which mechanical properties are half reduced compared with those at room temperature. In general, T g is not the same for all properties; is a constant describing the breadth of the distribution; A power law factor,, is used to take into account effects of decomposition on mechanical
properties where the decomposition fraction F is calculated in the heat transfer model and the exponent n can be determined by tests. Load Bearing Capacity In structural fire testing, according to ISO 834-1(1999) (Patterson 23), failure to support the load is deemed to have occurred for axially loaded structure when both of the following limits have been exceeded: = 1 and =3 1 where C in mm and in mm/min are limits of axial contraction and axial contraction rate. H is the initial height in millimeters VALIDATION OF MODEL Validation Problem I: FRP Laminate under One-Sided Heat Flux In the first validation study, a one-sided heating test (Feih et. al., 26) is modeled. As in the test, a glass/vinyl ester laminate of size 56x5x9 is exposed directly to a constant heat flux of 5 kw/m² at one side and thermally insulated at another side. Top end of the laminate is clamped but can move in vertical direction and bottom end is completely clamped. Constant compressive loading of 8% of the Euler buckling load is applied at top of the laminate. In FEA implementations, onedimensional thermal analysis is conducted in COMSOL. The same thermal analysis can be conducted in ABAQUS, but in COMSOL is easier to implement. 3 elements of size.3 mm are uniformly through thickness direction. Time Dependent Solver is used with time step of 1 second. Temperatures at hot face, middle, and cold face from modeling were compared with experimental results. With temperature profile from thermal analysis, two-dimensional structural analysis is carried out in ABAQUS. 12 elements of size.75mm are distributed uniformly along thickness. ABAQUS/Standard with initial time step of 1 seconds is used to solve constitutive equation. Temperature comparisons of modeling with tests in figure 1 show that the agreement is excellent, revealing that the model can accurately predict the thermal effects of heat conduction, endothermic decomposition of vinyl ester and convection flow of the volatiles. Included in figure 2 are deflection comparisons at center point. Both results from modeling and tests indicate that the laminate bends initially toward heat source due to thermal moment caused by uneven thermal expansion over thickness, then the eccentric moment reverses the laminate away from heat source because the neutral axis moves away from heat source due to larger stiffness loss at the exposed surface.
Temperature ( C) 7 6 5 4 3 2 1 Hot face Middle Cold face Temperature by Modeling Temperature by Test 5 1 15 2 25 3 Time (s) Figure 1 Temperature at Center Point Deflection (mm) 1.8.6.4.2 -.2 -.4 -.6 -.8-1 Modeling Measurement 5 1 15 2 25 Time (s) Figure 2 Deflection at Center Point Included in figures 3 and 4 are axial contraction and axial contraction rate from modeling, as there is no test result available for comparisons. According to load bearing capacity, time to failure is 8 minutes when both limits of axial contraction and axial contraction rate are met. It is noted that buckling failure cannot be determined based on deformation history, including axial contraction and deflection at center point because no sudden deformation change is detected.
Aixial Contraction (mm) 1 8 6 4 2-2 Axial Contraction Axial Contraction Limit 2 4 6 8 1 12 Figure 3 Axial Contraction Aixial Contraction Rate (mm/min) 6 4 2-2 -4 Axial Contraction Rate Axial Contraction Rate Limit 2 4 6 8 1 12 Figure 4 Axial Contraction Rate Validation Problem II: FRP Laminate Panel under Furnace Fire In second validation study, a structural test (Lattimer et. al., 27) is modeled. In the test, a glass/vinyl ester laminate of size 91x71x12 mm with 25.4 mm thick superwool as thermal insulation is exposed to IMO A. 754 furnace fire. Top end of the laminate is clamped but can move in vertical direction, while bottom end is simply supported.a constant force of 9.7 kn is applied on top of the laminate. Thermocouple TC1 is located at the exposed surface of superwool, while
thermocouples TC2-TC5 are positioned 4 mm apart along thickness of the laminate for temperature measurement. Deflections at 16, 31, and 62 mm away from the bottom surface and in-plane deformation of top surface were measured in the test. One-dimensional thermal analysis is conducted in COMSOL. 6 elements of size.62 mm are uniformly through thickness direction. Time Dependent Solver is used with time step of 1 seconds. In the 2D structural analysis in ABAQUS, element size was 2x2 mm with 19 elements through the thickness (including superwool). The solver was ABAQUS/Standard and the initial time step is 1 seconds. Figures 5-8 compare temperature results from modeling from TC2 to TC5, which are locations of thermocouples in tests, with experimental data for fire IMO A. 754. Overall, the agreement is good. Heat transfer through the interface between superwool and laminate is complicated by accumulation of volatile gases and how good superwool and laminate contact each other, which makes temperature measurement at the interface very difficult. Temperature ( C) Temperature ( C) 4 3 2 1 Test Modeling 1 2 3 4 5 Figure 5 Temperature at TC2 25 2 15 1 5 Modeling Test 1 2 3 4 5 Figure 7 Temperature at TC4 Temperature ( C) Temperature ( C) 3 25 2 15 1 5 Modeling Test 1 2 3 4 5 Figure 6 Temperature at TC3 2 15 1 5 Modeling Test 1 2 3 4 5 Figure 8 Temperature at TC5 Figures 9-12 show deflection results by modeling compared to test data. Modeling successfully predicts global buckling failure, which is indicated by sudden change of deflection.
Both deflection history and time to failure from modeling are in good agreement with Deflection (mm) Deflection (mm) 16 12 8 4 16mm_FEA 16mm_Test 1 2 3 4 5 Figure 9 Deflection at 16 mm 12 8 4 62mm_FEA 62mm_Test 1 2 3 4 5 Figure 11 Deflection at 62 mm Deflection (mm) Deflection (mm) 16 12 8 4 31mm_FEA 31mm_Test 1 2 3 4 5 Figure 1 Deflection at 31 mm 6 4 2-2 Inplane_Test Inplane_FEA 1 2 3 4 5 Figure 12 In-plane Displacement test results, indicating that the mechanical property model works well and plain strain assumption is reasonable. Based on in-plane deformation, time to failure given by criterion of load bearing capacity is 42 minutes, which is very close to test result and time to buckling failure. CONCLUSION In the paper, an integrated method is developed to predict temperature and deformation of FRP laminates under one sided heating and compression. The model includes one dimensional heat transfer modeling and plain-strain based structural modeling. Both mechanical and thermal properties are temperature dependent. Effects of decomposition on both temperature and deformation are included in the model. A concept of shift temperature is introduced to account for heating rate s effect on the decomposition temperature and therefore behavior of laminates in fire. Criterion of load bearing capacity works well for FRP panels under one sided heating and compression.
ACKNOWLEDGEMENT The authors would like to thank the Office of Naval Research (Award No. N14-7-1-514) and UNC Charlotte Faculty Research Grant (Grant No. FRG-111181) for supporting this research. Special thanks to Prof. Brian Y Lattimer at Virgin Tech (USA) and Dr. Stefanie Feih at Royal Melbourne Institute of Technology (Australia) for sharing experimental data. REFERENCES Bausano, J.V., Lesko, J.J. and Case, S.W. (25). Composite life under sustained compression and one sided simulated fire exposure: Characterization and prediction. Composites: Part A 37,192. Feih, S., Mathys, Z., Gibson, A.G. and Mouritz, A.P. (26). Modeling the tension and compression strengths of polymer laminates in fire. Composites Science and Technology 67, 551 564 Florio, J., Henderson, J.B., Test, F.L. and Hariharan, R.. (1991). A study of the effects of the assumption of local thermal equilibrium on the overall thermally- induced response of a decomposition, glass-filled polymer composite. International Journal of Heat & Mass Transfer; 34:135-147 Gu, P. and Asaro, R.J. (25). Structural buckling of polymer matrix composites due to reduced stiffness from fire damage. Composite Structures 69, 65. Henderson, J.B., Wiebelt, J.A. and Tant, M.R. (1985). A model for the thermal response of polyer composite materials with experimental verification. Journal of Composite Materials; 19:579-595 Henderson, J.B. and Wiecek T.E. (1987). A mathematical model to predict the thermal response of decomposing, expanding polymer composites. Journal of Composite Materials; 21:373-393 Keller, T., Tracy, C, and Zhou, A. (26). Structural response of liquid-cooled GFRP slabs subjected to fire: Part II. Thermo-chemical and thermomechanical. Composites Part A, 37, (9), 1296. Lattimer, B.Y., Asaro, R. et. al. (27). Structural response of fiber reinforced plastic composites during fires. The 11 th Interflam Conference, London, UK, Sept., 27 pp.653 Mouritz, A.P. and Mathys, Z. (1999). Post-fire mechanical properties of marine polymer composites. Composite Structures, 47, 643-653 Mouritz, A.P., Feih, S., Mathys, Z. and Gibson, G. (26). Mechanical property degradation of naval composite materials in fire. Modeling of Naval Composite Structures in Fire, Office of Naval Research, 51-16. Mouritz, A.P. and Gibson, A.G. (26). Fire properties of polymer composite materials. Springer. McManus, H.L. and Springer, G.S. (1992). High temperature behavior of thermomechanical behavior of carbon-phenolic and carbon-carbon composites, II Results. Journal of Composite Materials, 26:23-225
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