Finite element analysis

Finite element analysis is the use of the finite element method (FEM), a numerical method for solving the equations of heat flow or structural mechanics in a solid body. The method divides the body or structure into small elements joined at nodes, and it assembles the equations of all elements into one set of matrix equations. Within each element, the unknown quantity, such as the temperature, is described by a simple polynomial. In fire safety engineering, the method is used to calculate how heat penetrates into structural members and how the heated structure deforms and carries its loads.

Introduction to Fire Safety Engineering describes a finite element model for heat penetration as one that divides a construction into many computational cells, much as computational fluid dynamics divides the gas in a room. Finite element models can follow temperature gradients across a cross-section, restrained thermal expansion and large deflections, which simple hand methods cannot. The global model used to study the collapse of World Trade Center Building 7 contained about 3 million elements, and 25 seconds of its simulated collapse took about 8 weeks to compute. The results depend on the mesh, the material data and the boundary conditions, and they have to be checked against fire tests.

How the method works

The finite element method shares its first step with the older finite difference method. Both divide the object into small segments, using nodes connected by lines that form a grid or mesh, so both approximate the geometry. A finer mesh gives a more accurate result but takes more computing time. The spacing may therefore vary, with the smallest elements where the temperature gradient is steepest, for example near the heated surface or in insulation.

A finite element mesh for a quarter of an insulated steel column: nodes joined into elements, with smaller elements where the temperature gradient is steepest. Based on Rockett and Milke (2002), pp. 1-38 and 1-41, and Iqbal and Salley (2004), p. B-81.

The two methods differ in how they treat the governing equation. The finite difference method replaces it by algebraic equations through a Taylor series expansion, while the finite element method keeps the exact equation and fits a polynomial temperature profile within each element. Because only the geometry is approximated, the finite element method typically gives a more accurate result on coarse grids. The equation of each element is written in matrix form, and the element matrices are assembled into one system for the whole body: $$[C]\{\dot{T}\}+[K]\{T\}=\{Q\}$$

Symbol	Description	Unit
$[C]$	capacitance matrix, built from density times specific heat of each element	J/K (per unit area in one dimension: J/(m²·K))
$[K]$	conductivity matrix, built from the thermal conductivity of each element	W/K (per unit area in one dimension: W/(m²·K))
$\{T\}$	vector of temperatures at the nodes	°C
$\{\dot{T}\}$	vector of the rates of temperature change at the nodes	°C/s
$\{Q\}$	vector of heat input at the nodes	W (per unit area: W/m²)

Solving this system is the main computational task. Standard methods such as Gaussian elimination are not efficient, because most entries of the conductivity matrix are zero, so much of the progress in finite element modelling has come from better solution techniques. As with finite differences, the time steps can be taken explicitly or implicitly. For the explicit finite element formulation, however, no simple stability criterion is available.

Thermal analysis of structural members

The most common use of finite elements in fire safety engineering is the calculation of temperatures inside structural members. Such structural fire resistance models solve the conservation of energy within the solid, which is divided into elements in much the same way as a field model divides a room. They need the thermal conductivity, specific heat and density of each material, and data on phase changes such as those of intumescent coatings. The fire exposure enters as a time-temperature history, with heat transfer by convection and radiation at the exposed surface and conduction inside the solid. Steel and concrete members with and without fire protection are the most common cases.

Several programs were developed for this purpose. FIRES-T3, TASEF-2, SUPER-TEMPCALC and STABA-F all handle two-dimensional problems, and FIRES-T3 also handles three-dimensional ones. Some programs can only simulate the standard fire exposure of a fire resistance test, while FIRES-T3 and TASEF accept time-varying exposures and temperature-dependent material properties. For concrete members, Fleischmann and Buchanan note that FIRES-T3, TASEF-2 and SAFIR can accurately predict the temperature distribution. None of them models the movement of moisture, so the thermal properties must be adjusted to account for the heat absorbed by evaporating water. All three use a finite element technique and therefore require a skilled operator.

Simple methods are often adequate, and finite elements become necessary where they fail. The International Fire Engineering Guidelines state that simple lumped-mass methods for steel are not suitable when there are significant temperature gradients across a section. A beam whose top flange supports a concrete slab is an example, and finite element analysis is then appropriate. The Australian data sheets for the Fire Safety Verification Method likewise list methods for the thermal response of an element that range from lumped thermal mass correlations to finite element analysis.

A thermal model is only as good as the heat transfer at its boundary. In 2008, the SP Technical Research Institute of Sweden ran three compartment fire experiments with a steel beam. Their aim was to show how plate thermometer measurements near the beam can supply the boundary conditions for a multidimensional heat conduction calculation of the beam. They are known as the SP adiabatic surface temperature experiments, and the report on them deals with the use of that temperature for predicting heat transfer. A second series in 2011 used plate thermometers around a 6 m steel column standing in pool fires.

Worked example: heating of a wall

Rockett and Milke illustrate the finite difference method with a simple wall. The wall is 10 cm thick and initially at 20 °C, with a thermal conductivity of 25 W/(m·K), a density of 6000 kg/m³ and a specific heat of 500 J/(kg·K). One face is exposed to surroundings at 500 °C with a convective heat transfer coefficient of 10 W/(m²·K), while the other face is held at 20 °C. With a grid of 2.5 cm and a time step of 30 s, their explicit finite difference calculation gives 28.6 °C at the exposed face after 5 minutes.

The same problem can be solved with four linear finite elements of 2.5 cm, so that the nodes coincide with those of the handbook example. Each element contributes a conductivity term of k/L_e and a heat capacity term of ρcL_e, and the convective boundary adds the heat transfer coefficient to the last node. With implicit time steps of 30 s, the exposed face reaches 30.4 °C after 5 minutes. At depths of 2.5 cm, 5 cm and 7.5 cm, the temperatures are 26.3 °C, 23.5 °C and 21.5 °C. Distributing the heat capacity directly to the nodes instead of over each element changes the exposed face only to 30.3 °C.

A much finer model, with 40 elements and a time step of 1 s, gives 30.6 °C at the exposed face. The coarse finite element model is thus within 0.2 K of the refined result, while the coarse finite difference result is 2.0 K lower. This agrees with the handbook's statement that the finite element method typically gives more accurate results on coarse grids. The heated zone is also easy to estimate. With a thermal diffusivity of 25 / (6000 × 500) = 8.3 × 10⁻⁶ m²/s, the characteristic penetration depth after 300 s is about √(αt) = 50 mm, half the wall thickness.

Structural analysis

Finite element models are also used to calculate how heated structures deform and fail. Milke notes that several such models exist for fire-exposed members and frames, and that most were first developed for research. An example from the United States is FASBUS-II, which models floor assemblies made of a two-way concrete slab, steel decking and steel beams. Its input consists of the temperature distribution, temperature-dependent mechanical properties, geometry, end conditions and loads. The output gives deflections, rotations and stresses, which are then compared with performance limits.

The main reason for using such models is that a frame behaves differently from its members. In a frame, load transfer or membrane action may let a steel member keep its integrity at temperatures above those usually associated with failure, and capturing this requires a frame analysis. The International Fire Engineering Guidelines call this the general method, needed for steel frames where deformations are imposed on the structure. It calculates stresses, strains and internal forces in each member with material properties at elevated temperatures, using specialist programs operated by structural engineers. The guidelines name SAFIR and VULCAN as finite element programs used extensively for steel and concrete structures in fire.

The models rest on simplifying assumptions. A review cited by Milke found that the models then available all assumed that plane sections remain plane and that steel and concrete act together without slip. They also disregarded torsion and moisture, and they did not model large displacements accurately. The Australian data sheets describe the structural stage as a range from hand calculations with critical temperatures or charring rates to finite element analysis with temperature-dependent properties. Where the interaction of elements matters, detailed modelling of the whole structure or major parts of it may be required.

Linking fire models and structural models

A full analysis of a structure in fire combines several stages. The Australian data sheets distinguish the fire exposure, the thermal response of an element and its structural response, and finite element analysis can serve in the last two. In the analysis of the twin towers, the thermal loads on the structural models came from thermal analyses, which in turn came from fire dynamics simulations. Introduction to Fire Safety Engineering describes the same division of tasks in fire modelling: computational fluid dynamics can determine the thermal load on a construction, and a finite element model the heat penetration into it.

For the towers, NIST developed a method called the Fire Structure Interface to couple the thermal response of the structure to the fire simulations. It calculated heat transfer to the structural members with a simple radiation model that divides a room into a hot, sooty upper layer and a cooler, clearer lower layer. The properties of the layers were taken from time averages of the fire simulations. Finite element models of the steel members, their fireproofing and the concrete slabs then produced three-dimensional temperature histories in a format suited to the structural models.

The transfer between models is not trivial. NIST described the mapping from fire simulations to thermal models, and from thermal models to structural models, as complex and challenging. The reason was the very different length scales, time steps, element types and software of the three kinds of analysis. The fire simulations used cells of 50 cm and time steps of 10⁻³ s, the thermal analyses elements of 1 to 2 cm and steps of 1 s. The structural analyses used elements of 1 to 60 in. and steps of 600 s, so that the analyses together spanned five orders of magnitude in length and ten in time. The coupled analyses also showed which input mattered most: the thickness of the fireproofing and its damage by the aircraft impact had the largest effect on steel temperatures and structural response.

Large models in the World Trade Center investigations

The investigation of World Trade Center Building 7 shows how large such analyses can become. The analysts first used a model of a few floors to test modelling methods and hypotheses about the initiating event, and then analysed a model of the entire building. A separate model of the north-east floor corner was used to explore failure mechanisms, which were then included in a 16-storey ANSYS model and the 47-storey LS-DYNA model. The building model consisted mainly of shell elements, with beam elements for diagonal members, nonlinear discrete elements for connection details and brick elements for some rigid masses. In the model of the north-east floor corner, steel members had elements of about 1.8 in. (46 mm), while the slab and metal deck had elements of about 7.5 in. (190 mm).

The size of the global model set its limits. To keep the model at about 3 million elements, or about 10 million degrees of freedom, the elements had to be a few inches long on average. Failure was modelled by deleting elements whose failure criterion was exceeded, which let failure spread through the structure. At this resolution the gradients around a fracture could not be resolved, so the damage criteria had to be adjusted to give the right strength and ductility. Processing 25 seconds of simulated time took about 8 weeks on 12 processors working in parallel.

The analyses of the twin towers met a related problem. Modelling all components and connections of the towers with refined meshes would have been prohibitive, so NIST worked from individual components to major subsystems and then to the whole structure. Each simplification made for the larger models was validated against the detailed component models. Even so, local failures such as connection failures, large deflections, inelastic buckling and creep caused convergence problems that stopped many simulations early. Analyses of this kind show both the reach and the practical limits of finite element methods in structural fire engineering.

Validation and limitations

A finite element model is only credible if it reproduces tests. Milke shows a full-scale composite steel frame tested in a standard ISO fire, with a 6 m beam under a distributed load of 94.2 kN/m and a column load of 972 kN. The numerical model followed the measured sideways displacement of the column closely for most of the test and predicted failure after 149 minutes, against 150 minutes in the test. Another frame model showed reasonable agreement between predicted and measured results. For a large building frame, a further model gave deflections and forces in good agreement whether the frame was analysed as a whole or divided into subframes to reduce the computing effort.

NIST's Fire Structure Interface also compared favourably with a series of large-scale experiments. NIST used exploratory studies as well, to estimate how sensitive the results were to the input parameters. The SP experiments became part of the data set that NUREG-1824 Supplement 1 uses for the verification and validation of fire models.

The limitations are as important as the successes. Assumptions about composite action, torsion, moisture and large displacements can matter in a real fire. Thermal programs that ignore moisture movement need adjusted material properties, and all of them need a skilled operator. The International Fire Engineering Guidelines state that fire engineers, unless suitably trained, generally limit themselves to predicting the thermal effects of fire, while structural engineers analyse the behaviour of the structure. Fleischmann and Buchanan expected the use of such models to grow with performance-based design and more realistic time-temperature curves.

See also

• Computational fluid dynamics
• Fire Dynamics Simulator
• Mesh resolution
• Sensitivity analysis
• Verification and validation

References

• Australian Building Codes Board. (2020a). Data sheets: Fire Safety Verification Method (Handbook annex, Print version 1.1).
• Fleischmann, C., & Buchanan, A. (2002). Analytical methods for determining fire resistance of concrete members. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 4-239–4-256). National Fire Protection Association; Society of Fire Protection Engineers.
• Gross, J. L., & McAllister, T. P. (2005). Structural fire response and probable collapse sequence of the World Trade Center towers (NIST NCSTAR 1-6). National Institute of Standards and Technology.
• Iqbal, N., & Salley, M. H. (2004). Fire Dynamics Tools (FDTs): Quantitative fire hazard analysis methods for the U.S. Nuclear Regulatory Commission Fire Protection Inspection Program (NUREG-1805). U.S. Nuclear Regulatory Commission, Office of Nuclear Reactor Regulation.
• MacNeill, R., Kirkpatrick, S., Peterson, B., & Bocchieri, R. (2008). Global structural analysis of the response of World Trade Center Building 7 to fires and debris impact damage (NIST NCSTAR 1-9A). National Institute of Standards and Technology.
• McAllister, T. P., Gann, R. G., Averill, J. D., Gross, J. L., Grosshandler, W. L., Lawson, J. R., McGrattan, K. B., Pitts, W. M., Prasad, K. R., Sadek, F. H., & Nelson, H. E. (2008). Structural fire response and probable collapse sequence of World Trade Center Building 7 (NIST NCSTAR 1-9, Vols. 1–2). National Institute of Standards and Technology.
• Milke, J. A. (2002a). Analytical methods for determining fire resistance of steel members. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 4-209–4-238). National Fire Protection Association; Society of Fire Protection Engineers.
• National Research Council of Canada, International Code Council, Department of Building and Housing, & Australian Building Codes Board. (2005). International fire engineering guidelines (2005 ed.). Australian Building Codes Board.
• Prasad, K., & Baum, H. (2005). Fire Structure Interface and thermal response of the World Trade Center towers (NIST NCSTAR 1-5G). National Institute of Standards and Technology.
• Rockett, J. A., & Milke, J. A. (2002). Conduction of heat in solids. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 1-27–1-43). National Fire Protection Association; Society of Fire Protection Engineers.
• Stroup, D., Lindeman, A., McGrattan, K., Peacock, R., Overholt, K., Joglar, F., LeStrange, S., & Montanez, S. (2016). Verification and validation of selected fire models for nuclear power plant applications: Supplement 1 (NUREG-1824, Supplement 1; EPRI 3002002182). U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research; Electric Power Research Institute.
• Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.