Uncertainties in structural fire design

Decisions made at different phases of a design process are affected by uncertainties. This is due to the impossibility of gaining complete and accurate analytic or design information needed at each stage. Undoubtedly, the presence of uncertainties is acknowledged in various design standards with the provision of safety factors for many design conditions as code drafters/writers are aware of design uncertainties and risks either by intuition or perception. Therefore, structural fire design entails guaranteeing that in post-flashover fires there is a very low probability of failure of the building structure (Wong, 1999). Notably, the assessment of reliability (which is the converse of risk or probability of failure) in a limit state design is mostly a method for assessing and mitigating design uncertainties (Lemaire, 2013). Reliability-based design has also influenced the ultimate limit state design principle in many design codes including the Eurocode, whereby failure probability is measured from the

difference between structural resistance Rc and load demand Rdas shown in Figure 2.7. The

shaded area in Figure 2.7 is considered as the failure region in which the probability of

failure, Pf in a normal distribution is less than or equal to zero, i.e. P (Rc – Rd ≤ 0). Figure 2.7

also shows that reliability index β is another way of expressing the failure of a system based

on the failure distance. Importantly, the prediction of Pf and β does not inevitably imply

actual rate of structural failure given that real failure rate is significantly dependent on human error (BSI, 2002b). The basic theory including the mathematical assumptions and derivations

for reliability analysis can be found in Zhang et al. (2013) and Guo et al. (2014). In the

management of design uncertainties, Eurocode 0 (BSI, 2002b) mentions that historic and probabilistic calibrations may be combined to determine limit state partial factors for loadings and materials. And such that structural reliability levels should be as close as possible to a target reliability index. Reliability index takes into account stochastic uncertainties in the effects of loading actions, resistances and model uncertainties, thereby allowing for comparison of reliability levels of representative structures of buildings. This also entails that a reliability analysis can be carried out to investigate the suitability of a structural system given expected loading actions and uncertainties.

Figure 2.7. Reliability design concept adapted from Lemaire (2013).

Many types of uncertainties exist in different engineering designs; in the design of steel structures in fire conditions, the uncertainties can range from variable parameters (including basic variables, e.g. unit weight, steel density, yield strength etc.) to structural fire model uncertainties (e.g. assumed/adapted calculation values etc.). Melchers (1987) discussed key

uncertainties considered in the evaluation of reliability and their interdependencies. In this review, a summary of the relevant uncertainties and brief explanations are given in Table 2.2.

Table 2.2. Summary of relevant steel structural fire design uncertainties (Melchers, 1987; Wong, 1999)

Type of uncertainty

Explanation

Design fire Refers to the remarkable dynamics of fire phenomenon, which give room for

many assumptions and possibilities in structural fire design.

Model Steel structural models are developed for expected outcomes and are mainly

deterministic. Hence, models are susceptible to produce inaccurate results especially in scenarios of different assumptions.

Stochastic This refers to the uncertainty in the form of the statistical distributions of the considered variables.

Loading action The variation of loading conditions in fires is inherent uncertainties that may

affect structural fire design adequacy. Examples of loading actions, in this case include variance of gravity, live and fire loading actions on steel.

Parameter/property Material properties of steel are extracted from empirical data which may not represent the exact statistical distributions due to experimental limitations and lack of accurate information, e.g. yield stress and elastic modulus of steel in fires, thicknesses and thermal properties of insulation materials.

Completeness Inability to fully describe the complexities of the given design problem, or the

deliberate omission of some branches of a given design problem.

Estimation Estimation of structural steel response in fires does not depend only on steel

material properties; it also involves designers’ knowledge, which may be improved in time or not.

Human error This is adjudged in the literature as the greatest source of design uncertainty and

is divided into gross error (e.g. oversight) and error due to human variability (e.g. human incapability, poor skill, and performance).

Decision Suitable design decisions largely depend on the mitigation of other design

uncertainties. Other kinds of decision uncertainty are within the appropriate use of engineering judgement and cost-benefit analysis outcomes.

In evaluating design uncertainties, different approaches can be considered. An ideal uncertainty measure in engineering design would typically combine, model, statistical, and parameter uncertainties in one system. Ang and Tang (1975) opine that the probability distribution function (PDF) adjudged as the best combination of uncertainty may be

insufficient in solving practical problems. Random variables are currently used in the evaluation of uncertainty whereby statistical moments are employed to measure the dispersion of random variables. In this case, normalised calculations using standard deviation and mean of various distributions of random variables give the coefficient of variation (COV) suitable for uncertainty combinations.

The different methods of analysing uncertainties include, among others: Rosenblueth’s method (point estimate method); estimation of the first-order variance; and Monte Carlo simulation. Of relevance to this research is the Monte Carlo simulation. This is because Monte Carlo simulation is more preferable to the other methods in terms of the generation of a large number of random variables and the production of model and statistical outputs by iteration. In other words, a typical Monte Carlo simulation simply generates lots of data, computes the uncertainty model a thousand or ten thousand times using randomly selected values from probabilistic functions. In using Monte Carlo simulation, multiple probabilistic outcomes can be compared and the model is customised/combined with respect to different uncertain scenarios. The application of Monte Carlo simulation is also more desirable than other methods due to its ease of sensitivity analysis (i.e. ease of observing the inputs having the most effect on key results) and input correlation (i.e. possibility of modelling interdependent input variables). Notably, the Rosenblueth’s method has been shown to have a weaker computational capability albeit it is also adjudged as a method that suitably accounts

for skewness of uncertain parameters (Chang et al. 1995).

Monte Carlo methods are referred to as stochastic simulations. Halton (1970) defined a

Monte Carlo approach as “representing the solution of a problem as a parameter of a

hypothetical population, and using a random sequence of numbers to construct a sample of the population, from which statistical estimates of the parameter can be obtained”. Historically, the Monte-Carlo approach was an expensive uncertainty evaluation technique and therefore a “difficult” method in terms of practical implementation (Rubinstein, 1981). However, the advent of digital computers mainly increased its usage in quantitative risk analysis. Another challenge with Monte Carlo simulation is with its sampling method for the propagation of uncertainty, which typically requires millions of simulations. There are other sampling methods used for uncertainty propagation including random, stratified and Latin hypercube sampling. Latin hypercube sampling has been studied and compared to others. In recent times, the method is most preferred due to its efficiency in stratifying and reducing the number of samples required to perform Monte-Carlo simulations (Helton and Davis, 2003).

The other sampling methods are considerably challenging to implement as they will require the development of empirical and/or alternate models and derivatives which is beyond the scope of this research. In structural fire design, it is possible to use Monte Carlo simulation with reference to a pair of random variables. The random variables represent steel structural demand and capacity used to obtain an output regarding a probability estimation of the structural failure, i.e. when and to what extent demand exceeds the structural member’s capacity. Elms (2004) highlighted this capability of Monte Carlo by estimating the

probability of failure (pf) from failure ratios in an iterative process based on randomly chosen

samples. The study suggested that this approach could be implemented in limit state structural analysis if probability distributions are defined for chosen variables.

There have been several applications of Monte Carlo techniques in evaluating uncertainties in structural fire design. Wong (1999) studied the reliability of structural fire design through deterministic and probabilistic analyses of case study building structures, which included the

evaluation of parameter uncertainties using Monte Carlo simulation. Guo et al. (2013)

investigated the probabilistic structural fire resistance of a steel beam by combining Monte

Carlo and finite element simulation. Recently, Zhang et al. (2014) carried out probabilistic

structural fire analysis of protected steel columns. In their study, various parameters were defined as stochastic distributions to understand the reliability of intumescent coatings. The commonest feature in these studies is the characterisation of uncertain parameters as

stochastic variables for the probabilistic structural fire analysis, albeit Zhang et al. (2014)

considered ‘professional factor’ as a model uncertainty for maximum steel temperature and steel buckling temperature in a parametric fire. The studies also relied on characterisations of parameter uncertainties in the literature. For instance, the statistical characterisation of variable load, fire load density, thermal absorptivity and material properties of applied fire

protection on steel can be found in Iqbal and Harichandran (2010). Thermal conductivity, ki

of sprays and gypsum materials are characterised as lognormal distributions; while thermal absorptivity for normal and light-weight concrete is characterised as normal distributions. These associated statistical distributions were obtained from the analysis of raw experimental data. Iqbal and Harichandran (2010) also mentioned that the reported distributions for live, dead and fire loads were taken from the literature. The availability of statistical data has remained a challenge in evaluating uncertainties in structural fire design as exemplified by

the many unknown distributions presented in the uncertainty parameter table in Guo et al.

defined as statistical variables in the probabilistic analysis. This may be adjudged as the case

in the statistic characteristics table presented in Zhang et al. (2014). In other cases, statistical

characteristics of uncertain variables are assumed if rational reasons for the assumptions are provided (e.g. Wong, 1999). Nevertheless, statistical approaches in structural fire design can provide rational outcomes; hence, the unavailability of statistical data may not be a reasonable excuse to avoid probabilistic evaluations. In scenarios of assumed uncertainty characterisation, sensitivities can also be tested for informed decision-making.

There are different software programs used for defining statistical distributions of uncertain variables and evaluating them in a probabilistic analysis. The @Risk add-on macro to

Microsoft Excel (Palisade, 2012) and B-RISK fire risk simulation model (Wade et al., 2013)

are potential software as they include the Monte Carlo function, which can be used for uncertainty evaluation given a deterministic structural fire design. With any of these tools, the uncertain parameters or input values, and deterministic model outputs will be specified as probability distributions.

Following a structural fire analysis based on the calculation method set out on a spreadsheet, @Risk can be used for the propagation of uncertainties in the input parameters to predict failure probabilities. In the @Risk software, probabilistic distributions must be defined for the parameters and model outputs from any of 30 probability distributions ranging from Beta to Weibull distributions on the @Risk menu. Also, recent versions of the @Risk software include the Latin hypercube sampling option in Monte Carlo simulation. The laudable prospects of evaluating several design uncertainties are ideal for design scenarios involving multi-disciplinary stakeholders having different/conflicting inputs in the design decision- making process.