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Emirati Journal of Civil Engineering and Applications
Vol 1 Issue 1 (2023)
Pages (45 –69)
Available at www.emiratesscholar.com
© Emirates Scholar Research Center
45
Emirati Journal of Civil Engineering and Applications
© Emirates Scholar Research Center
Simple Visual Aids for Predicting Fire Response of RC Columns:
Nomograms via Machine Learning
M.Z, Naser1,2, Arash Teymori Gharah Tapeh1, Haley Hostetter1, Mohammad Khaled al-Bashiti1, William Qin1,
mznaser@clemson.edu
School of Civil & Environmental Engineering and Earth Sciences (SCEEES), Clemson University, USA1
Artificial Intelligence Research Institute for Science and Engineering (AIRISE), Clemson University, USA2
Abstract:
Assessing the ability of reinforced concrete (RC) columns to withstand the effects of fire is a multifaceted and intricate
problem due to the various factors that influence their fire response. As such, engineers may find it challenging to precisel y
predict such fire resistance. While some codal provisions exist and fire testing/advanced modeling can be adopted, the
same methods may suffer from poor predictivity and can be costly and/or complex. In this paper, we shift focus toward
machine learning techniques (by means of Nomograms) that can produce simple visual aids to assess the fire resistance of
RC columns. Our analysis shows that Nomograms can be accurate, account for a series of factors currently absent from our
domain knowledge and provisions, and outperform existing methods adopted in building codes. Our analysis also infers
that such Nomograms could be possible candidates for adoption in standardized settings, given their simplicity, ease
of
use, and lack of multi-stepped procedures.
Keywords:
Nomograms, Fire resistance, Fire rating, Concrete columns, Machine learni
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Emirati Journal of Civil Engineering and Applications
Vol 1 Issue 1 (2023)
Pages (45 –69)
Available at www.emiratesscholar.com
© Emirates Scholar Research Center
46
Emirati Journal of Civil Engineering and Applications
© Emirates Scholar Research Center
1.0 Introduction
Due to its superior properties, concrete (and hence
reinforced concrete) has become one of the most widely
used construction materials, especially in environments
where fire or extreme temperatures are expected [1].
However, despite its resilience, elevated temperatures cause
concrete to undergo a series of chemical and physical
changes that result in damage. This damage can include loss
of mechanical and bond strength and may trigger the
structure to collapse [2]. The study of reinforced concrete
(RC) under elevated temperatures has thus become
significant in understanding the type and magnitude of
damage fire can cause and predicting thermal and structural
response [3]. However, an examination of existing literature
reveals a continued challenge in predicting the wide variety
of fire effects on RC members at elevated temperatures [4],
such as fire resistance and fire rating.
One of the key factors that can influence the fire resistance
of RC columns is the compressive strength of concrete.
Several studies have concluded that columns made from
higher strength (i.e., high strength and ultra-high-
performance concrete) tend to have a lower fire resistance
than normal strength concrete [5, 6]. For example, Dwaikat
and Kodur [7] attributed the loss in fire resistance to the
occurrence of spalling. Bolina et al. [8] fabricated 16
concrete columns of equal dimensions with four different
mix designs for strength. In each group of four, the concrete
cover was varied. Each element was then tested in a
standard furnace using the ISO 834 standard fire for 240
minutes. Results showed a correlation between the
propensity of spalling and fire resistance and noted that the
concrete cover thickness and diameter of reinforcement had
a greater influence on fire resistance than mix design.
Other key factors, i.e., the geometry, loading, and restraint
conditions, can affect the fire response of RC columns. Such
properties include the cross-sectional dimensions, section
shape and length (square, circular, etc.), loading amount and
eccentricity, and axial or bending restraint. Kodur and Phan
[9] noted an increase in cross-sectional dimensions
decreases heat flux (and therefore increases fire resistance);
such an increase results in a more significant thermal
gradient [10]. This effect can lead to a loss in confinement
and fire resistance. Martins and Rodrigues [11] studied the
effect of column length on fire performance. They found
that an increase in column length tends to increase the
member's slenderness ratio, leading to P-δ effects that
decrease the fire resistance.
Studies [12–14] each concluded that increasing load level
decreases the fire performance of concrete columns.
Similarly, increasing eccentricity also results in a decrease
in fire resistance [15, 16]. Finally, the type and location of
restraints also play a crucial role in the fire resistance of RC
members. Recently, Yang et al. [17] found that increasing
the restraint ratio for axially restrained square tube RC
columns resulted in increased fire resistance. The previous
studies have demonstrated the complexity of studying RC
members in fire scenarios and the difficulty in predicting
their behavior [18].
At a different front, O'Meagher and Bennetts [19] developed
a mathematical model to analyze RC walls under fire
exposure, and this method can be extended to other building
elements such as walls, floors, beams, and columns. Further,
Harmathy [20] created another method to tie real fires to
standard fires and proposed its use for structural design.
More recently, researchers have attempted to tackle the lack
of available fire tests in the literature by using machine
learning (ML) to predict fire resistance. Studies [21, 22]
used various algorithms and model approaches to accurately
predict the fire resistance of RC members with a variety of
geometric and practical scenarios. For example, Naser and
Kodur [23, 24] used the random forest, extreme gradient
boosted tree, and deep learning algorithms trained with a
database of 494 RC columns to predict fire resistance. When
deployed, such a model can accurately predict the fire
resistance of over 5,000 RC columns in under 60 seconds.
In addition to its use for predicting RC behavior in fire, ML
has also been shown to benefit the field of structural fire
engineering greatly. For example, much work has been done
to model fire dynamics in an effort to reduce the current
limitations of equations and methods. Similar studies,
including [25–27], show the need for more realistic fire
modeling capabilities. Other ML studies have explored a
variety of structural materials and member types, each
showing that ML can provide improved accuracy and
understanding of such complicated phenomena. For
example, Khan et al. [33] used a hybrid simulation approach
to analyze restrained composite beams exposed to fire. Such
a method makes studying realistic boundary conditions and
the behavior of an entire structural system exposed to fire
possible, and their study demonstrated good agreement with
the test results of actual composite beams. Research on
concrete slabs and floor systems [28–30], timber structures
[31], and steel structures [32] using ML further
demonstrates the versatility of the approach for evaluating
structural performance in fire scenarios.
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Pages (51 –69)
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In this paper, we continue the notion of adopting ML, but rather than creating blackbox models, we utilize ML to generate
simple and visual aids to assess the fire resistance of RC columns. Such methods fall under Nomograms. We create a series of
Nomograms to cover regression (i.e., fire resistance prediction) and classification (fire rating) problems. These Nomograms
can have similar accuracy to traditional ML models while accounting for a series of factors currently absent from our domain
knowledge and provisions [33]. Our analysis also shows that the proposed Nomograms can also outperform currently adopted
prediction methods in building codes.
2.0
Statistical description of the dataset
We gathered 248 full scale fire tests of RC columns for our investigation from diverse literature sources [34, 35]. This dataset
consists of ten independent variables that describe the geometry, loading, and mechanical properties of reinforced concrete
columns, namely, column width (b), steel reinforcement ratio (r), column effective length (L), effective length factor (k),
concrete compressive strength (f′c), steel yield strength (fy), concrete cover (C), eccentricity in the x-direction (ex) and y-
direction (ey), and applied load (P), and one dependent variable, the column fire resistance (R). Table 1 lists key statistical
insights about these collected factors. Overall, this dataset satisfies the recommendations of data useability and health as
noted by:
•
Van Smeden et al. [36] – having a minimum set of 10 observations per feature.
•
Riley et al. [37] – having a minimum of 23 observations per feature.
•
Frank and Todeschini [38] – maintaining a ratio of 3 and 5 between the number of observations to the number of
features.
Table 1 Statistical insights of the dataset
Factors
Mean
Standard Deviation
Kurtosis
Skewness
Minimum
Maximum
b (mm)
306.90
56.27
0.40
-0.06
200.00
406.00
r (%)
2.12
0.64
2.39
0.53
0.89
4.39
Le (m)
3.99
0.73
1.45
0.25
2.10
5.70
f′c (MPa)
45.94
25.94
2.84
1.90
24.00
138.00
fy (MPa)
461.55
60.70
-0.44
0.38
354.00
591.00
k
0.83
0.22
-1.47
-0.64
0.50
1.00
C (mm)
38.67
8.24
-0.86
-0.24
25.00
64.00
ex (mm)
18.78
31.73
7.83
2.70
0.00
150.00
ey (mm)
2.10
10.42
28.21
5.28
0.00
75.00
P (kN)
1103.75
991.22
5.26
2.16
0.00
5373.00
R (min)
150.48
101.05
2.00
1.16
22.00
636.00
2.1
Data distribution
The histograms in Fig. 1 show the distribution of each variable in the dataset graphically. These graphic depictions of the data
are a helpful tool for comprehending the underlying patterns and trends, which can guide additional analysis and result
interpretation.
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Fig. 1 Histograms for all variables in the dataset
2.2
Correlation investigation
Correlation is frequently used to uncover general patterns
between distinct variables to help engineers visualize how
changes to one variable may affect the other. Simply,
correlation is a statistical method for describing the direction
and intensity of the link between the listed variables herein.
The most common correlation methods, Pearson and
Spearman, gauge the linear and monotonic relationships
between two dataset variables. Other correlation techniques,
like Chatterjee and Mutual Information, can also be used to
detect dependence and nonlinear correlations, respectively.
Figure 2 presents a heatmap matrix that visually represents
the results of the four correlation methods listed above. The
results from the various correlation methods reveal several
key findings. For example, concrete cover (C) exhibits the
strongest positive correlation with fire resistance (R).
Additionally, these methods reveal a strong negative
correlation between effective column length (K) and the fire
resistance of reinforced concrete columns, with negative
values of -0.71 for Spearman, -0.70 for Pearson, and -0.57
for Chatterjee. In contrast, the mutual information model
predicts a strong correlation between applied load (P) and
fire resistance (R), with a coefficient of 4. At the same time,
eccentricity in the y-direction (ey) and steel ratio (r) exhibit
the weakest correlations in most methods.
One can think of these four heatmaps as guiding tools to
visualize the type and strength of relationships between the
factors from the different lenses of each method. In other
words, the intention of showing these maps is to educate the
readers on the existence of possible relations between the
factors and not to identify the best relation type – as, in
reality, all of these relations are calculated through the
principles of each method.
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Emirati Journal of Civil Engineering and Applications
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DOI: 10.54878//EJCEA.109
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Fig. 2 Correlation matrices for Pearson, Spearman,
Chatterjee, and Mutual Information
3.0
Detail of machine learning algorithms
The detailed procedures and techniques used in this study
are described in this section for the reader's convenience. All
models, with their full settings, can be found in their cited
works. The analysis of the model was carried out using
RapidMiner's [39] automated platform. We opted to use this
automated platform to examine a different approach to that
practiced in an earlier study by our group wherein
traditional (i.e., coding-based) ML was used, as cited in the
introduction section [23, 24].
3.1
Linear regression
The goal of linear regression is to find a linear relationship
between a set of parameters. This relationship is fitted via
coefficients and mapped using a linear equation that
connects the dependent and independent variables. Setting
the intercept to zero in linear regression models is a
common practice to ensure that the model does not predict
any response when all input variables have values of zero –
which is valuable for our analysis of fire resistance
prediction. This is because the model would not be
physically realistic to forecast fire resistance when all the
parameters are set to zero. In addition, the Lrm (Logistic
regression modeling) is an extension of the linear model and
aims to reduce the squared sum of differences between the
predicted and actual values. The Lrm can be adopted to
transform a linear relation between a dataset into a
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Emirati Journal of Civil Engineering and Applications
Vol 1 Issue 1 (2023)
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DOI: 10.54878//EJCEA.109
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Emirati Journal of Civil Engineering and Applications
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classification problem (such as fire rating, as will be shown
in a later section) [40], [41].
3.2
Deep learning
A deep learning model is a subclass of ML that uses 2+
hidden layers [42]. This model seeks to replicate human
cognition and uses a comparable methodology to
comprehend the underlying pattern in various fields of
study. We opted for a model with Multi-Layer Perceptron
(MLP) with Rectified Linear Unit (ReLU) activation
function and Adam optimizer. The minimum batch size was
1, the learning rate was set at 0.001, and the number of
layers and neurons was set to 3 and 113, respectively.
3.3
Decision tree
The decision tree is a simple ML algorithm [35].
Recursively partitioning the data allows the model to run
until the conditions are met and terminated. The simplicity
of presentation and interpretation is this model's key benefit.
However, decision trees have a high risk of overfitting –
especially when working with noisy data. Based on the
optimal parameter analysis, the decision tree model had a
depth equal to seven.
3.4
Random forest
Like a decision tree, a random forest is a tree-based method
in which several trees are simultaneously formed [43]. The
results from all the trees are combined to make the final
forecast. This model performs well regarding overfitting and
their ability to handle high-dimensional data. The analysis's
best parameters, which comprised 20 trees, a maximum
depth of 7, and an error rate of 22.4%, were applied to the
random forest model.
3.5
Gradient boosting trees
Another tree-based model attempts to improve model
performance by reducing the impact of the erroneous feature
in each modeling iteration [43]. This model finds the error
in the first tree and builds a new tree model with the error as
a new feature. Although the model performs well in
classification and regression, it may be less appealing to use
with big data because of the high volume of computations.
The gradient boosting approach performs best with the
number of trees set to 30, the maximum depth they can
grow, and the learning rate of 0.1.
3.6
Support vector machine
While the support vector machine is frequently used for
classification problems, it may also be utilized for
regression [44]. The model operates by fitting a hyperplane
that best distinguishes between a dataset's various clusters or
classes. As already said, the technique can handle high-
dimensional datasets, but it can be challenging to work with
because the ideal tuning parameters require careful research.
To minimize overfitting, the kernel gamma and complexity
constant for the gradient-boosting tree model were found to
be 0.05 and 1000, respectively.
3.7
Model performance evaluation
Finding the appropriate metrics' evaluation process for the
research datasets is a crucial step that must be completed to
ensure the quality of analysis in a machine learning analysis.
In this study, we employed two sets of metrics that belong to
regression and classification. All these metrics are
commonly used in ML studies [45].
On the regression front, we used the squared error (SE),
which measures the discrepancy between projected values
and their corresponding actual values, is used. In addition,
the relative error (RE) is used. This metric evaluates the
accuracy of a prediction by the ratio of the absolute error to
the ground truth. A relative error has the benefit of being a
dimensionless quantity, allowing comparison of prediction
accuracy across many scales and units of measurement. The
mean absolute error (MAE), which measures the average
size of the errors between the anticipated and actual values,
is also used. One benefit of using MAE is that it offers a
more perceptible measure of prediction error than other
metrics like mean squared error. Also, we used the R-
squared (R2), which shows how much of the variance in the
dependent variable can be predicted based on the
independent variable (s). Higher R2 values indicate a better
fit between the expected and actual values. Then, The last
regression metric used is the Root Mean Squared Error
(RMSE), which calculates the average of the squared
differences between predicted and actual values. As
predicted and actual values are squared before being
averaged, employing RMSE has the advantage of penalizing
large errors more severely than small errors. Table 2 lists the
above metrics.
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Table 2 List of used regression and classification metrics
Metric
Formula
E (Error)*
𝐸 = 𝐴 − 𝑃
SE (Squared Error)
𝑛
𝑆𝐸 = ∑ 𝐸
2
𝑖=1
RE (Relative Error)
|𝑦
𝑖
−
𝑦|
RE = 𝑦
MAE (Mean Absolute Error)
∑𝑛 1|𝐸𝑖|
MAE = 𝑖= 𝑛
R2 (Coefficient of Determination)
𝑛 𝑛
𝑅
2
= 1 − ∑(𝑃
𝑖
− 𝐴
𝑖
)
2
/ ∑(𝐴
𝑖
− 𝐴
𝑚𝑒𝑎𝑛
)
2
𝑖=1 𝑖=1
RMSE (Root Mean Absolute Error)
∑
𝑛
𝐸 2
𝑅𝑀𝑆𝐸 = √
𝑖=1 𝑖
𝑛
TPR (True Positive Rate) or Sensitivity
𝑇𝑃
𝑇𝑃𝑅 = 𝑇𝑃 + 𝐹𝑁
TNR (True Negative Rate) or Specificity
𝑇𝑁
𝑇𝑁𝑅 = 𝑇𝑁 + 𝐹𝑃
ACC (Accuracy)
𝑇𝑃 + 𝑇𝑁
𝐴𝐶𝐶 = 𝑇𝑃 + 𝑇𝑁 + 𝐹𝑃_𝐹𝑁
The A and P letters stand for Actual and Predicted values, and n is for the number of points. On the classification front, we
used the confusion matrix to visualize the performance of the classification quantity models. This matrix presents the
accuracy of classifiers by providing statistics on actual and predicted classifications. This matrix's rows indicate real cases,
whereas its columns reflect predicted events. A given set of predictions corresponding to true positives (TP), false positives
(FP), true negatives (TN), and false negatives (FN) numbers are shown in the matrix. Calculations based on the values in the
confusion matrix can be made for three standard performance metrics: sensitivity, specificity, and accuracy. The percentage
of real positives the model correctly classifies is measured by sensitivity. While accuracy gauges the general accuracy of the
model's predictions, specificity quantifies the percentage of actual negatives that the algorithm accurately detects (see Table
2).
4.0 Results and discussion
4.1
Predicting the fire resistance time of RC columns
We start our discussion by presenting a comparison among the various ML models examined, including linear regression,
gradient boosted trees, decision trees, support vector machines, random forests, and deep learning. In this analysis, our goal
is to predict the fire resistance (time to failure) of the collected RC columns in this study.
We note that the gradient boosted trees and linear regression model yielded the first and second best performance – as can be
seen in Table 3. However, the gradient boosted trees, as well as the other used ML models, are considered blackboxes and
cannot be used in an intuitive manner or without the use of coding. Unlike the gradient boosted trees, linear regression can be
used to create Nomograms; hence, this model is used, as will be discussed in a later section.
While we acknowledge the modest performance of the linear regression model (e.g., R2 = 0.64), we will show that this model
still outperforms existing codal provisions in predicting the fire resistance of RC columns. First, we compared predictions
from the linear model to predictions
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obtained from Eurocode 2 (Eq. 1) and AS3600 (Eq. 2) – see Table 3 and Fig. 3. As one can see, the accuracy of the latter two
models fall short than that of the linear model.
However, the same two models are not applicable to eccentrically loaded RC columns. Thus, our analysis listed in Table 3
and Fig. 3 also shows the predictivity of these two codal provisions when only applied to concentrically loaded columns, and
to columns primarily made from normal strength concrete. Here, we also show that the predictivity of these provisions is
lesser than that of the linear model.
Table 3 Evaluation of different models performance in terms of fire resistance time
Model
R2
SE
MAE
RMSE
RE
Linear regression
0.64
148507
37.34
54.49
0.39
Gradient boosting
0.77
102257.04
31.07
45.22
0.32
Decision tree
0.51
215923.84
43.74
65.71
0.47
Support vector
0.47
232278.89
52.93
68.15
0.49
Random forest
0.63
160781.63
34.25
56.70
0.41
Deep learning
0.23
342179.31
66.60
82.72
0.59
Codal provisions
AS3600
-9.70
108859.28
209.83
329.93
2.38
AS3600 (without eccentricity)
-7.36
77262.85
172.83
277.96
0.99
Eurocode 2
0.42
5803.80
55.96
76.18
0.57
Eurocode 2 (without eccentricity)
0.23
7048.67
63.05
83.95
0.37
𝑅 = 120 (𝑅𝑓𝑖+𝑅𝑎+𝑅𝑙+𝑅𝑏+𝑅𝑛 1.8 , and 𝑅 = 83 (1 − 𝜇 1+𝜔
𝐴
𝑠
𝑓
𝑦𝑑
where,
)
120
𝑓𝑖
𝛼𝑐𝑐
+𝜔
)
,
𝜔 =
𝐴
𝑐
𝑓
𝑐𝑑
(1)
R = fire resistance of column (min),
αcc = coefficient for compressive strength,
Ra = 1.6(a-30); a is the axis distance to the longitudinal steel bars (mm); 25 mm ≤ a ≤ 80 mm,
Rl = 9.6(5-lo,fi); lo,fi is the effective length of the column under fire conditions; 2 m ≤lo,fi ≤ 6 m; values corresponding to lo,fi = 2
m give safe results for columns with lo,fi < 2 m,
Rb = 0.09b'; b' = Ac/(b+h) for rectangular cross-sections or the diameter of circular cross sections,
Rn = 0, if 4 rebars are used, and 12 for more than 4 rebars.
𝑘 × 𝑓1.3×𝐵3.3×𝐷1.8
𝑅 = 𝑐
0.9 (2)
10
5
×𝑁
1.5
× 𝐿
𝑒
where,
R = fire resistance of column (min),
k = a constant dependent on cover and steel reinforcement ratio (equals to 1.47 and 1.48 for a cover less than 35 mm and
greater than or equal to 35 mm, respectively),
fc = 28-day compressive strength of concrete (MPa),
B = least dimension of column (mm),
D = greatest dimension of column (mm),
N = axial load during fire (kN),
Le = effective length (mm).
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bottom) [Note that the axes of the AS3600 model had to be
extended beyond 600 min as this equation yields relatively
larger fire resistance times]
4.2
Predicting the fire resistance rating of RC columns
In this stage of analysis, our goal is to predict the fire
resistance rating of RC columns. Our goal is to create visual
aids that continue to account for a wide range of features
that are absent from codal existing provisions. Such ratings
can become handy for quick evaluation of RC columns in
design and analysis scenarios. Since these ratings are
primarily given in 60 min, this analysis turns into a
classification problem. As such, all fire resistance times in
the collected dataset were converted into classes, namely, 0-
60 min [Class 1], 60-120 min [Class 2], 120-180 min [Class
3], and 180-240 min [Class 4]. It should be noted that some
columns were reported to fail beyond 240 min, and hence
these were labeled as 240 min.
We used the same approach followed in the previous section
and applied the same ML models to predict all four classes
at once. In this process, we noticed difficulty in training the
machine learning models on all classes (see Table 4). After
several attempts, it became clear that this approach does not
yield acceptable results. Yet, we note that the logistic model
(with a sigmoid function) ranked within 1.3% of the highest
two models.
Table 4 Evaluation of models’ performance on all classes in
terms of fire resistance rating
Model
Accuracy
Logistic regression
63.4%
Gradient boosting trees
49.3%
Decision tree
46.6%
Support vector
40.0%
Random forest
64.3%
Deep learning
65.7%
To further improve the accuracy of the models, a sensitivity
analysis was conducted to realize that combining the
database into two alternate classes (Class 1 and Class 3) and
(Class 2 and Class 4) would lead to arriving at the best
classification metrics. Thus, we repeated the and also noted
that the logistic model continued to rank top three1 (in terms
of Class 1 or 3) and as the leading model (for Class 2 or 4).
Fig. 3 Actual versus predicted fire resistance regarding three
models (Linear model, Eurocode 2, and AS3600 from top to
1 While some ML models outperform the logistic model,
such models can not be easily converted into a simple visual
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According to our analysis, the logistic regression model
performs on par with sophisticated ML models. The logistic
regression model demonstrated strong performance in
predicting the dataset's two classes (Classes 1 and 3) with an
overall accuracy of 0.88, sensitivity of 0.93, and specificity
of 0.80. Furthermore, the logistic model achieved an
accuracy of 0.95 on the other two classes. This model
showed a high sensitivity of 0.91 and a specificity of 0.97
(see Table 5).
Table 5 Evaluation of different models’ performance on two
alternate classes in terms of fire resistance rating
5.0
Nomograms
Engineers, practitioners, and medical professionals
historically used Nomograms, also known as alignment
charts, to solve mathematical equations or to depict complex
interactions within phenomena [46, 47]. Typically, a
Nomogram has one output variable and several input
variables and can vary between simple parallel line scales or
more intricate parabolic shapes. In comparison to other
informational aids like equations or tables, Nomograms can
be more user-friendly and do not require calculations
beyond simple addition.
Nomograms are flexible tools that may be constructed using
various techniques, including software packages like
Pynomo in Python and Lrm in R. The standard process for
creating a Nomogram entails analyzing the relations
between independent and dependent variables, choosing a
layout, deciding on the right scales for the variables, and
drawing the Nomogram. From the perspective of this work,
this process is tied to the use of linear regression and logistic
regression2.
5.1
Nomogram development
In this paper, we utilized Nomograms to predict the fire
response of RC columns to fire via regression and
classification. In the first case, a regression-based
Nomogram utilizing linear regression was developed, and in
the second case, logistic regression-based Nomograms
utilizing the total point approach to determine the
probability of different classes for RC columns were
created.
To create the first Nomogram, the linear regression model
was selected. The coefficients corresponding to each
variable in this model work as a function to scale the axis
associated with each parameter. The Nomogram was
produced by plotting the resulting alignment chart with
ratios and dimensions according to the functions and
coefficients. The isopleth () function, also known as a
contour map, was used to predict a fire test from the dataset
in the Nomogram. Our codes for creating the Nomograms
are provided in the Appendix.
In order to create a target variable for the second Nomogram
type, we divided it into four classes: the first class represents
columns with fire resistance of lesser than 60 minutes, the
second class represents columns with fire resistance of
between 60 and 120 minutes, the third class represents
columns with fire resistance of between 120 and 180
minutes, and the fourth class represents columns with fire
2 The use of other forms of regression (i.e., nonlinear
regression) is possible but is likely to require a more
complex approach to that presented in this work.
Model for
fire Classes
1 and 3
Accuracy
Sensitivity
Specificity
Logistic
regression
88.7%
93.3%
80.0%
Gradient
boosting
trees
93.3%
93.3%
93.3%
Decision
tree
93.3%
86.7%
100.0%
Support
vector
48.0%
0.0%
100.0%
Random
forest
93.3%
93.3%
93.3%
Deep
learning
model
85.3%
93.3%
73.3%
Model for
fire Classes
2 and 4
Accuracy
Sensitivity
Specificity
Logistic
regression
95.8%
91.7%
97.7%
Gradient
boosting
trees
91.1%
89.3%
95.0%
Decision
tree
86.7%
100.0%
63.3%
Support
vector
68.1%
53.3%
88.3%
Random
forest
93.3%
96.7%
88.3%
Deep
learning
86.4%
96.7%
70.0%
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resistance of between 180 minutes and 240 minutes. As
mentioned above, we opted to filter Classes 1 and 3 together, followed by Classes 2 and 4. After training and testing the
dataset, we fitted a linear regression that adheres to the Eq. 3 to determine the relationship between parameters. Then, using a
logistic regression prime, we convert the values into binary classes Eq. 4.
Where, 𝛽0, 𝛽1are coefficients derived during the training process, and 𝑥1, 𝑥2, 𝑥3, etc., are the features identified in the study
(those are listed in Table 1). Now, the model can predict the expected classes using the above equations. We again used the
Lrm packages to make the model explainable for the engineer and practitioner by creating a Nomogram model via the
Nomogram () function and plotting the derived result.
5.2
The Nomograms in use: step-by-step examples
In this section, we illustrate the practical applicability of the derived Nomograms by applying them to a specific RC column.
Through this demonstration, we provide step-by-step instructions for effectively using the Nomograms to predict reinforced
concrete columns' fire resistance. Utilizing a fire test with the following features:
•
b (column width) = 406 mm,
•
r (reinforcement ratio) = 2%,
•
L (effective length) = 4 m,
•
f′c (compressive strength of concrete) = 100 MPa,
•
fy (steel yield strength) = 400 MPa,
•
K (effective length factor) = 1,
•
C (column concrete cover) = 48 mm,
•
ex (eccentricity in the x direction) = 0 mm,
•
ey (eccentricity in the y direction) = 0 mm,
•
P (applied load) = 1410 kN
Using the Nomogram shown in Fig. 4, a user can:
1.
Locate the two first parameters (b and r) on their respective axes, then draw a straight line to intersect with the R1 line. In
this example, we find the values 406 and 2 and connect them to their corresponding R1 line.
2.
Locate the third value, L (find the value four on the L axis), then draw a straight line from the previous point found in step
1, intersecting with the R2 line.
3.
Locate the fourth value f′c (find the value 100 on the f'c axis), then match the last line to intersect with the R3 line.
4.
In the fourth step, search for fy values (find the value 400 on the fy axis) and match this quantity with the value of line 3 to
intersect with the corresponding value on line R4.
5.
In the fifth step, search for the effective column length K (in this case, K=1) and follow the same process as step 4 to locate
the value on the R5 line.
6.
In this step, find the concrete cover C on the C axis (search for the value 48), then draw a straight line that passes the C axis
and intersects with the R6 line.
7.
Locate the eccentricity in the x direction ex (search for the value 0) and match R6 with R7.
8.
Find the eccentricity in the y direction ey (search for the value 0) and match R7 with R8.
9.
The final step involves finding the R value (fire resistance). In this step, locate the applied load on the P axis, draw a
straight line to intersect with R8, then follow the line to intersect with the R axis, which gives the predicted (R) value.
As a result, as it can be read from Fig. 4, the value for the fire resistance of the column is 231 minutes, which is equal to what
we have from the test experiments.
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Fig. 4 Nomogram for predicting fire resistance of concrete columns in minutes. [Note: Numbers in circles refer to steps].
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We followed a similar procedure for the same RC column to demonstrate the practical application of the classification
Nomograms. First, we locate the appropriate axes related to the parameters we were looking for and draw a perpendicular line
from each axis to the point axis to determine the value of each parameter using the Nomogram. Next, we sum all the values
we found and use the total point line to determine the probability of fire resistance classes.
Here, one must consider the two generated classification-based Nomograms simultaneously. In this example, the RC column
will be classified through Nomogram A and Nomogram B. If the column falls under the 50% probability line in Nomogram
A, then this column is unlikely to fall under Class 3 (effectively implying that the column is likely to be labeled as Class 1
and vice versa if the column falls above the 50% probability line to be labeled as Class 3). The same column is also examined
in Nomogram B to evaluate its probability of falling under Class 2 or Class 4. Then, the arrived at probabilities from the two
Nomograms are compared, and the larger probability is used to identify the fire rating class for the column at hand3.
For example, applying the above procedure and utilizing the Nomograms in Fig. 5 for the same RC column used in the
regression example, we obtain the following scaled values (which can be obtained visually from the Nomograms or via the
complimentary Table 6):
·
Nomogram A: b = 14.1, r = 39, L = 21, f′c = 5, fy = 19, K = 0, C = 28, ex = 0, ey = 0. The sum of these values equals 138.1
with a probability of 0.84 (interpolated linearly).
·
Nomogram B: b = 74, r = 8, L = 17, f′c = 43, fy = 6, K = 0, C = 33, ex = 54, ey = 17.5. The sum of these values equals 327. 5
with a probability larger than 0.99.
Since the probability of this column falling under Class 4 is larger than the other classes (i.e., 0.99 > 0.84.3 [for Class 3 ]),
then we can conclude that this column could be considered under Class 4 (indicating a fire resistance rating between 180-240
min). Considering the actual observed fire resistance is 231 min, then our analysis is valid.
3 In the rare event that the two Nomograms return identical probabilities, then we advise selecting the more
conservative class.
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(a)
Nomogram A for predicting Classes 1 and 3 [Note: only b and rare shown for eligibility]
(b)
Nomogram B for predicting Classes 2 and 4 Fig. 5 Nomograms for fire rating
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Table 6 Companion tabulated values for Nomograms A and B
5.3
Further remarks
As one can see, the created Nomograms are user-friendly
and easy to use. These require basic information about RC
columns, including the eccentricity of the applied load and
boundary constraints, as well as account for some features
absent from currently available codal provisions.
Nomograms, like all other methods and procedures, can be
utilized for various issues and fields of study, including the
classification of fire severity, fire spread, and fire risk. Once
fully validated, Nomograms have the potential to be
incorporated into upcoming building regulations and
standards, since they are visual tools engineers can
efficiently utilize.
In addition, we would like to point out some limitations that
need to be acknowledged. For example, additional
diversified datasets, fire tests, and larger ranges of features
can be better used to validate this approach further. It would
also be advantageous to consider different Nomogram types
that accommodate a broader range of model types. Further,
while the generated linear regression model has a decent
accuracy that exceeds those from Eurocode 2 and AS3600,
we invite interested readers to further improve upon the
developed model's accuracy and explore the use of other
forms of regression, especially those for multi-classification
analysis.
6.0
Conclusions
This study investigates the fire resistance of reinforced
concrete (RC) columns using a dataset of 248 laboratory fire
tests from various literature sources. The study's objectives
were to create user-friendly aids to predict fire resistance
and fire rating of RC columns and develop Nomograms
capable of handling regression and classification problems.
The following conclusions can also be drawn from the
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findings of this study:
•
The fire resistance of RC columns could be
classified and predicted using the independent
variables' geometry and mechanical characteristics.
The correlation analysis showed that the boundary
conditions and applied load significantly affected
fire resistance, while the reinforcement ratio and
yielding steel strength had less impact.
•
This study shows that traditional linear regression
can yield comparable accuracy to traditional ML
and may outperform codal provisions.
•
This analysis provides valuable insights into the
fire resistance and fire rating of RC columns and
lays the foundation for future research to improve
the understanding and prediction of fire resistance
in reinforced concrete structures.
Acknowledgment
The authors would like to thank the steering and organizing
committees of the 12th International Conference on
Structures in Fire (SiF'22), Hong Kong Polytechnic
University. The authors also thank the Fire Technology
Journal for sponsoring this special issue.
Conflict of interest statement
None.
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Appendix
Codes for running this analysis will be provided upon
publication.
Nomograms (for blank nomograms) will be provided upon
publication.