Peak strength prediction of reinforced concrete columns in different failure modes based on gene expression programming

Abstract

The peak strength of reinforced concrete (RC) columns plays an important role in the appraisal of inelastic seismic performance. It depends on various parameters related to the geometry, reinforcement detail, material property, confinement effect, and loading condition. In applications, it is usually a prior condition to classify the failure modes of RC columns for predicting the peak strength accurately. Yet, classifying the failure modes of RC columns in an accurate way is a difficult task due to the complexity of the shear transfer mechanism. Thus, there is a need to develop a peak strength prediction model for RC columns failing in different modes directly. In this study, an attempt has been made by implementing the gene expression programming (GEP) method to realize this purpose. The experimental data required for the implementation of the GEP method are based on extensive results of RC columns tested in quasi-static cyclic loading. To validate the efficiency of the developed model, a detailed comparison against existing equations is conducted. The comparative results indicate that the developed model produces a rational prediction for the peak strength of RC columns in various failure modes. Based on the developed model, the peak strength can be predicted in a unified way for both ductile and non-ductile RC columns, which is beneficial for the seismic evaluation of existing structures.

Keywords

Experimental database modeling peak strength prediction reinforced concrete column

Introduction

Reinforced concrete (RC) columns are one of the most critical components of existing structures to resist gravity and lateral loads. In evaluating the seismic performance of RC structures, predicting the peak strength of RC columns is vital because it is a critical quantity of measuring the strength capacity of existing structures. However, the post-earthquake observation reveals that RC columns may suffer three different failure modes, namely, the flexural failure (FF) mode, the shear failure (SF) mode, and the flexural shear (FS) failure mode. The three failure modes exhibit different seismic damage features, especially when they enter the post-peak region of the lateral load-displacement relationship. For example, as RC columns fail in the FF mode, the post-peak behavior is controlled by the flexural deformation; its response is featured by the full development of plasticity after the yielding of longitudinal reinforcements. In such circumstance, the peak strength of RC columns is generally computed by the traditional fiber cross-sectional analysis method. However, if RC columns fail in the SF mode, the seismic response is largely contributed by the shear deformation; its response is accompanied by severe strength deterioration in its post-peak region of the lateral load-displacement relationship. In such circumstance, different semi-empirical or theoretical models, for example, the strut-and-tie model (Hwang and Lee, 2002; Zhang and Tan, 2007), the empirical regression model (ACI 318, 2008; FEMA 273, 1997; Sezen and Moehle, 2004), the modified compression field theory (Bentz et al., 2006; Vecchio and Selby, 1991), and the truss and arch model (Jin et al., 2015; Pan and Li, 2013) have been developed to predict this quantity of importance.

Given different models are used to predict the peak strength of RC columns when the specimen fails in different modes, it is obvious that there is a need to distinguish the failure modes of RC columns accurately in prior to ascertain the appropriate one. If an inaccurate hypothesis is made by researchers, a large error would be produced. However, classifying the failure modes of RC columns accurately is a challenging task (Ning and Feng, 2019; Qi et al., 2013; Zhu et al., 2007), which brings a need to predict the peak strength of RC columns failing in different modes directly. In other words, it is better to establish a peak strength model of RC columns, regardless of the failure modes. Unfortunately, the failure mode evolution of RC columns from the flexural behavior to the shear-critical behavior is so complex, leading to the difficulty of incorporating the traditional fiber cross-sectional analysis method with the semi-empirical or theoretical models. Meanwhile, empirical regression analysis method is inapplicable to develop the peak strength model of RC columns in different failure modes because the model is generally developed with relatively few data samples, and it is difficult to find an appropriate function.

To deal with this problem, an advanced prediction method is preferable. In the last two decades, several successful attempts, including the artificial neural network (ANN) and the genetic expression programming (GEP) techniques, have been made to solve complex problem in engineering applications (Abuodeh et al., 2020; İpek and Güneyisi, 2020; İpek et al., 2021; Mansouri and Kisi, 2015; Mansouri et al., 2016, 2018a, 2018b; Shafaei and Khayati, 2020). The ANN model is a data-driven technique, which searches for the most optimal predictive results by the given network topology. Due to its powerful and successful capability in the field of prediction problems, the ANN model has been widely used by researchers, for example, predicting the ultimate deformation capacity of RC columns (Inel, 2007), providing a comprehensive study to predict the compressive strength of high-performance concrete (Chou et al., 2014), predicting the flexural overstrength factor of steel beams (Güneyisi et al., 2014), identifying the failure modes of circular RC bridge columns (Mangalathu and Jeon, 2019), and predicting the involved Bouc–Wen–Baber–Noori hysteresis model parameters (Ning et al., 2019a, 2019b). However, the ANN model cannot provide practical equations. As a black-box method, the ANN model is required to identify the structure of the network, which prevents other researchers to use or benefit from or even replicate the developed model. Another way to solve complex problem in engineering applications is to employ the GEP method. The GEP method is a feature-driven technique by transforming a set of mathematical objects. In a program space, the GEP method could search for the most optimal function (Ebrahimzade et al., 2018; Mahdavi and Khayati, 2018; Mansouri and Farzampour, 2018; Mansouri et al., 2017, 2018a, 2018b). Compared to the empirical regression analysis, the GEP method is superior in finding the most optimal functional expression. Compared to the ANN model, the GEP method is also preferable in providing practical equations to facilitate users to apply in practice.

Therefore, the GEP method is advantageous in developing the peak strength model of RC columns failing in different modes. In the recent years, the GEP method has received extensive attentions among researchers. For example, Gandomi et al. (2014) developed a formulation to predict the shear strength of slender RC beams without shear reinforcement using the GEP method. Özcan (2012) developed two models to predict the splitting tensile strength of cylinder concrete by the GEP method. Güneyisi and Nour (2019) employed the GEP method to develop the axial capacity of a confined filled steel tube prediction model. Mermerdaş et al. (2013) used the GEP method to model the shrinkage behavior of metakaolin and calcined kaolin blended concretes. İpek and Mermerdaş (2020) developed an explicit formulation to predict the crushing strength of synthetic aggregates based on the GEP method. D’Aniello et al. (2015) used the GEP method to develop predictive models of flexural overstrength factor for the steel thin-walled circular hollow section beams. Güneyisi et al. (2016) and İpek and Güneyisi (2019) proposed mathematical models to estimate the ultimate axial strength of concrete-filled single and double skin steel tubular columns using the GEP method.

To the authors’ knowledge, rare efforts have been devoted to predicting the peak strength of RC columns in different failure modes based on the GEP method. Thereby, a GEP-based peak strength model is developed in this study. The main content of this article is organized as follows. First, an experimental database collected from the Pacific Earthquake Engineering Research (PEER) center is compiled and discussed. Then, the modeling approach and construction of the GEP method is presented in details, respectively. Furthermore, the formula of predicting the peak strength of RC columns in different failure modes is developed, and finally, the performance of the GEP-based peak strength model is validated comprehensively by comparing with existing equations.

Experimental database

The experimental database used to develop the peak strength model of RC columns in different failure modes is compiled from the PEER center, including 251 RC columns under quasi-static cyclic testing. Appendix enlists the detailed information of the compiled experimental database, where $B$ is the column width, $H$ is the column depth, $L$ is the equivalent cantilever length, $f_{c}$ is the concrete compressive strength, $f_{y}$ is the yield strength of longitudinal reinforcement, $f_{v}$ is the yield strength of transverse reinforcement, $ρ$ is the longitudinal reinforcement ratio, $ρ_{v}$ is the transverse reinforcement ratio, $P$ is the applied axial load, and $V_{e x p}$ is the peak strength. As observed, the experimental database includes the three failure patterns. The numbers of RC columns failing in the FF, FS, and SF mode are 197, 36, and 18, respectively. Note that the failure mode in the experimental database was identified using the criterion given by Berry et al. (2004). Specifically, RC columns are categorized into the FF mode if no shear damage is reported by experimenters. If shear damage is reported, the absolute maximum effective force (F_eff) and the strength corresponding to the maximum strain of 0.004 (F_0.004) are further computed, respectively. Then, if F_eff < 0.95F_0.004 or failure displacement ductility is less than or equal to 2.0, RC columns are categorized into the SF mode. Otherwise, RC columns are categorized into the FS mode.

Determination of input and output model parameters

Based on the compiled experimental database, typical design parameters of RC columns are defined as the input model parameter. Figure 1 shows the statistical distribution of column design parameters in the histogram. As observed, $B$ and $H$ are distributed within 80 mm and 914 mm; $L$ ranges from 80 mm to 2335 mm; $f_{c}$ covers the concrete with the low and high strength from 16 MPa to 118 MPa; $f_{y}$ has a wide distribution from 0 to 1424 MPa; and $f_{v}$ displays a narrow distribution from 0 to 587 MPa. Moreover, $ρ$ is distributed within 0.0068 and 0.0603; $ρ_{v}$ ranges from 0.00007 to 0.0295; and $P$ varies from 0 to 8000 kN. The compiled experimental database is representative of typical RC columns used in practice.

Figure 1.

Statistical distribution of column design parameters in the complied experimental database.

Accordingly, the observed peak strength is defined as the output model parameter. However, the compiled experimental database only includes the transverse force-displacement relationship, instead of the observed peak strength. Then, there is a need to extract the peak strength of RC columns from the transverse force-displacement relationship. For a consistent purpose, four types of the transverse force-displacement relationships are identified to consider the P- $Δ$ effects. In the compiled experimental database, all the transverse force-displacement relationships were processed in terms of an equivalent cantilever column to reduce the influence of the testing configuration. For a detailed description about the transverse force-displacement relationship, interested readers can direct to Berry et al. (2004). Moreover, consider the transverse force-displacement relationships are recorded along with the positive and negative directions, the enveloped curves are extracted from the transverse force-displacement relationships to represent the backbone curves. The enveloped curves define whether RC columns have reached the maximum load-carrying capacity along with the corresponding directions. Using the enveloped curves, the maximum strength of RC columns along with the positive and negative direction can then be determined, and the observed peak strength is evaluated by averaging the maximum strength.

Taken three specimens that are randomly selected from the compiled experimental database as examples, Figure 2 shows the procedure of determining the maximum strength of RC columns according to the positive and negative directions, respectively. As observed, unit No. 2 tested by Tanaka and Park (1990), No. 40.067W tested by Wight and Sozen (1973), and CUW tested by Umehara and Jirsa (1982) are selected from the compiled experimental database to represent RC columns failing in the FF, FS, and SF modes, respectively. Based on the above-proposed procedure, the corresponding maximum strength of RC columns along with the positive and negative directions are 167.76 kN and 160.19 kN, 99.37 kN and 93.92 kN, and 263.17 kN and 245.3 kN, respectively. By averaging, the peak strengths of RC columns failing in the FF, FS, and SF modes are found by 163.97 kN, 96.64 kN, and 254.24 kN, respectively.

Figure 2.

Examples to show the procedure of determining the peak strength of RC columns failing in different modes (Tanaka and Park, 1990; Umehara and Jirsa, 1982; Wight and Sozen, 1973).

Figure 3 shows the statistical distribution of peak strength and the discrepancy of maximum strength along with the positive and negative directions for all the collected RC columns. As observed, the peak strengths of RC columns display a skewed distribution, whose values range from 31 kN to 1233 kN with a mean of 211 kN and a standard deviation (SD) of 169 kN. However, there is a discrepancy of the maximum strength recorded on the positive and negative directions since some specimens are designed or tested unsymmetrically. For the compiled experimental database, the mean of the discrepancy of shear strength is 5.786 kN and the corresponding SD is 31.78 kN.

Figure 3.

Statistical distribution of peak strength and discrepancy of maximum strength along with the positive and negative directions.

Categorization of experimental database

After that, two sub-datasets are haphazardly categorized with the compiled experimental database, namely, the training sub-dataset and the testing sub-dataset. The training sub-dataset is utilized to develop the peak strength model of RC columns failing in different modes, while the testing sub-dataset is used to validate the predictability and repeatability of the developed model. By categorization, 63 specimens, about 25% of the whole compiled experimental database, are selected as the testing sub-dataset. The rest of the compiled experimental database, on the other hand, is categorized into the training sub-dataset. Table 1 enlists the statistical distribution of the whole dataset and both sub-datasets. As observed, there is a good agreement between the whole dataset and both sub-datasets. Statistically, both sub-datasets resemble the whole dataset.

Table 1.

Statistics of the experimental data utilized in the model derivation.

Statistical parameter		B (mm)	H (mm)	L (mm)	$f_{c}$ (MPa)	f_v (MPa)	f_y (MPa)	ρ _v	ρ	P (kN)	V_exp (kN)
Total data	Number of data	251	251	251	251	251	251	251	251	251	251
	Mean	289.7	314.7	1091.3	51.9	483.7	429.2	0.0082	0.0237	1234.6	211.3
	SD.	118.3	117.7	545.6	29.3	227.7	75.8	0.0051	0.0101	1379.6	168.7
	COV	0.408	0.374	0.500	0.563	0.471	0.177	0.626	0.425	1.117	0.798
	Min	80	80	80	16	0	0	0.0007	0.0068	0	30.6
	Max	914.4	914.4	2335	118	1424	587.1	0.0295	0.0603	8000	1233.2
Training dataset	Number of data	188	188	188	188	188	188	188	188	188	188
	Mean	288.8	315.0	1088.3	52.1	493.0	428.5	0.0083	0.0237	1293.6	214.4
	SD	121.1	118.9	542.7	29.9	234.2	73.5	0.0053	0.0101	1475.9	180.7
	COV	0.419	0.378	0.499	0.574	0.475	0.171	0.632	0.428	1.141	0.843
	Min	80	80	80	16	0	0	0.0007	0.0068	0	30.6
	Max	914.4	914.4	2335	118	1424	586.1	0.0295	0.0603	8000	1233.2
Testing dataset	Number of data	63	63	63	63	63	63	63	63	63	63
	Mean	292.1	313.8	1100.4	51.4	456.1	431.4	0.0076	0.0239	1058.5	202.2
	SD	110.2	115.0	558.5	27.5	206.2	83.0	0.0046	0.0101	1030.8	127.3
	COV	0.377	0.366	0.507	0.535	0.452	0.192	0.602	0.421	0.974	0.630
	Min	80	80	80	21.1	0	0	0.0007	0.0125	111.0	32.1
	Max	550	610	2335	118	1424	587.1	0.0249	0.0603	4368.0	656.0

Gene expression programming method

The GEP method is a kind of soft computing techniques. In the field of model development, the soft computing techniques can be regarded as a summation of techniques to develop robust, tractable, and low-cost solutions tolerating the presence of errors and uncertainties (Zadeh, 1994). Among the soft computing techniques, genetic algorithms and genetic programming are two of the most important ones. Functionally, the GEP method is in the same ballpark as genetic algorithms and genetic programming (GepSoft, 2021; Gen and Cheng, 1997; GepSoft, 2015; Koza, 1992). The GEP method manages the operations by employing the population of individuals that are chosen to measure the fitness, and employs genetic operators to present the genetic diversity (GepSoft, 2015; Mitchell, 1996). Following the same model mechanism, the GEP method is generally expressed as the enhanced form of genetic algorithms and genetic programming (Ferreira, 2001). In the GEP method, individuals are codified by linear strings of fixed length. The genome or chromosomes are represented by the nonlinear entities of different sizes such as expression trees or simple diagram expressions.

Flowchart of the GEP method

Ferreira (2001) is the inventor of the GEP method to conceive the computer program by utilizing the statement of learned models or discovered knowledge (Li et al., 2005). Figure 4 shows the flowchart of the gene expression algorithm as given by Ferreira (2001). As observed, the chromosomes of the preliminary population are first arbitrarily generated at the beginning of the process. Then, executing each program starts as the next step for the expression of the chromosomes. By executing each program, the fitness of individuals is estimated. After that, selecting individuals begins for the reproduction based on the fitness of individuals. Finally, the process is repeated and individuals are obtained from the new generation until exact number of generations or accurate solution is achieved. In the final solution, techniques employ almost the same genetic operators but with minor differences (D’Aniello et al., 2015; Güneyisi and Nour, 2019; Güneyisi et al., 2013; İpek and Güneyisi, 2020).

Figure 4.

Flowchart of gene expression algorithm given by Ferreira (2001).

Designated setting of GEP method

The GEP method is implemented in the software GeneXproTools 4.0. In developing the peak strength model of RC columns failing in different modes, the chromosome number of the software GeneXproTools 4.0 is assigned at 120, and the head size is designated as 15. Meanwhile, the gene number is assigned at 12 and the mathematical addition operation is chosen for linking the genes to each other. Many mathematical operations including addition (+), subtraction (−), multiplication (*), division (/), square root (√), exponential (e^), natural logarithm (ln), cubic root (³√), and arctangent (atan) are employed in the genes to improve the accuracy and reliability of the developed model. Moreover, two constants per gene are assigned, representing respectively the upper and lower limits at −30 and +30. Table 2 tabulates the mutation, the inversion, the transposition, and the recombination rates of the GEP method, as well as all the aforementioned programmatic inputs. Herein, note that the involved parameters of the GEP method have been decided by many attempts. At the beginning of the process, default settings in the software GeneXproTools 4.0 are applied to develop the peak strength model of RC columns in various failure modes. However, the model does not perform well. To improve the accuracy of the developed model, default settings are shifted step-by-step, depending on the R-squared value achieved from the trained model. To illustrate the difference before and after training, default settings, which are provided by the software, and designated settings, which produce the best prediction, are presented in Table 2.

Table 2.

Gene expression programming parameters used in proposing the model.

No	Parameter name	Default setting	Designated setting
P1	Function set	+, −, *, /, Sqrt, Exp, Ln, ^2, ^3, 3Rt, Sin, Cos, Atan	+, −, *, /, Sqrt, Exp, Ln, 3Rt, Arctan
P2	Number of generations	—	799,998
P3	Chromosomes	30	120
P4	Head size	10	15
P5	Number of genes	4	12
P6	Linking function	Addition	Addition
P7	Mutation rate	0.044	0.00206
P8	Inversion rate	0.1	0.00546
P9	1-Point recombination rate	0.3	0.00277
P10	2-Point recombination rate	0.3	0.00277
P11	Gene recombination rate	0.1	0.00277
P12	Gene transposition rate	0.1	0.00277
P13	Constants per gen	2	2
P14	Lower bound of constants	−10	−30
P15	Upper bound of constants	10	30

Development of peak strength model

Following the above construction, the peak strength model of RC columns in different failure modes is developed. Note that the GEP-based model developed in this study is carried out considering material safety factor equal to 1.0, where the model is purely derived by the experimental data. Figure 5 shows the peak strength model of RC columns failing in different modes in terms of the expression tree. As observed, the model developed is expressed with twelve sub-functions. Each sub-function is written in an expression tree. To convert the expression tree to the formulation representation, some abbreviations are undertaken thereafter. For example, 2 $\times$ d₈ is written instead of d₈ + d₈ in the transformation of Sub-ET 5 (Figure 5(e)); $d_{2}^{2}$ is written instead of d₂ $\times$ d₂ in the conversion of Sub-ET 6 (Figure 5(f)). After these transformations, the peak strength model of RC columns failing in different modes is obtained with the following expression

V_{p r e d} = V_{1} + V_{2} + V_{3} + V_{4} + V_{5} + V_{6} + V_{7} + V_{8} + V_{9} + V_{10} + V_{11} + V_{12}

(1)where V_pred is the peak strength of RC column predicted by the GEP method and V_i (i = 1, 2, …, 12) are sub-functions, following the expressions as

V_{1} = d_{6} \times (\frac{28.81}{d_{0} + d_{3} - d_{1}} \times (\sqrt{d_{4}} + d_{4} - d_{8}) - d_{4} - \ln (d_{2} + d_{5}))

(2)

V_{2} = d_{7} (\frac{d_{5} + \sqrt{d_{8}}}{\frac{2.13}{3.88 + d_{3}} \times \sqrt[3]{d_{1} + d_{6} + \ln (d_{6})}})

(3)

V_{3} = e^{- 0.42 x d_{3} \sqrt[3]{d_{7}}} \times (d_{1} + d_{8} - d_{0} - \sqrt{d_{2}} - d_{2} - d_{5} + 28.02) - d_{2}

(4)

V_{4} = (\frac{- 3.89}{d_{3} - d_{5} - d_{2} d_{8}}) e^{\sqrt[3]{d_{0} + d_{3} + d_{8} + 28.90}} + d_{2}

(5)

V_{5} = d_{7} (d_{0} + d_{6} d_{8} - 16.34 x d_{7} d_{8} \frac{d_{4}}{d_{0}}) - 8.17

(6)

V_{6} = d_{8} \frac{d_{0} d_{3}}{d_{2}^{2} + d_{8}} + d_{4} - \sqrt[3]{d_{2}} - 11.12

(7)

V_{7} = d_{7} (d_{0} + d_{5} \tan^{- 1} (d_{6} d_{7} (d_{8} - d_{1}) (d_{2} - d_{1} - d_{3}) - d_{6}))

(8)

V_{8} = d_{1} + \frac{d_{0}}{d_{2} + d_{4} + d_{6} + (1.29 + \tan^{- 1} (d_{7})) (2.13 - d_{2})}

(9)

V_{9} = d_{7} d_{8} - d_{6} d_{8} - d_{3} - d_{4} - (d_{5} - d_{3}) \sqrt{d_{7}} - 13.63

(10)

V_{10} = - \sqrt[8]{d_{8} (d_{0} + d_{4}) (d_{2} + d_{3}) (2 x d_{8} + \frac{d_{2}}{d_{6}})} - 11.92

(11)

V_{11} = d_{7} (d_{0} + d_{3} + d_{4} + 4.58 d_{0} + 2.82 x d_{0} \tan^{- 1} (d_{8}) - 2 d_{2} - 4.58)

(12)

V_{12} = d_{6} d_{0} \ln (d_{0} (d_{6} + d_{8})) - 2 d_{2} d_{7}^{2} \times \sqrt[3]{d_{5}}

(13)where the symbols d₀, d₁, d₂, d₃, d₄, d₅, d₆, d₇, and d₈ are the column width

(B in mm)

, the depth

(H in mm)

, the height

(L in mm)

, the concrete compressive strength

(f_{c} in MPa)

, the yield strengths of transverse and longitudinal reinforcements (

f_{v} in MPa

and

f_{y} in MPa

), the corresponding ratios (

ρ_{v}, unitless

and

ρ, unitless

), and the applied axial load

(P in kN)

, respectively.

Figure 5.

Expression trees of the developed model: (a) Function 1, (b) Function 2, (c) Function 3, (d) Function 4, (e) Function 5, (f) Function 6, (g) Function 7, (h) Function 8, (i) Function 9, (j) Function 10, (k) Function 11 and (l) Function 12.

Figure 6 shows the performance of the developed model for both the training sub-dataset and testing sub-dataset, respectively. As observed, the coefficient of determination (R-squared) of the training sub-dataset and testing sub-dataset are 0.9507 and 0.9509, respectively. There is a close trend between the predicted peak strength and the experimental data for both sub-datasets, demonstrating the accuracy of the developed model in predicting the peak strength of RC columns failing in different modes.

Figure 6.

Prediction performance of the developed model for (a) training sub-dataset and (b) testing sub-dataset.

Comparison with existing equations

In this section, existing equations available in design codes and literature are reviewed and analyzed to demonstrate the advantage of the developed model. These equations are proposed mainly for predicting the peak strength of shear-critical RC columns (Del Vecchio et al., 2017). According to the literature review, there are no unified equations available to predict the peak strength of both ductile and non-ductile RC columns. Table 3 enlists the expressions of existing equations, where

V_{S}

is the transverse reinforcement contribution;

V_{C}

is the concrete contribution;

V_{P}

is the axial load contribution;

A_{v}

is the shear reinforcement area;

s

is the shear reinforcement spacing;

d

is the effective depth;

A_{g}

is the gross-section area;

k

and

γ

denote the coefficients of shear strength degradation with the ductility demand;

c

is the neutral axial depth of cross-section; and

D^{'}

is the distance between centers of the peripheral hoop. As observed, existing equations used in comparison with the developed model are representative, including ACI 318 model (2008), FEMA 273 model (1997), Sezen and Moehle model (2004), and Priestley et al. model (1994), respectively.

Table 3.

Existing shear strength formulations.

Model	$V_{p r e d}$	$V_{C}$	$V_{S}$	$V_{P}$
ACI 318 (2008)	$V_{C} + V_{S}$	$0.17 (1 + (P / 13.8 A_{g})) \sqrt{f_{c}} B d$	$(A_{v} / s) d f_{v}$	0
FEMA 273 (1997)	$V_{C} + V_{S}$	$0.29 γ (1 + (P / 13.8 A_{g})) \sqrt{f_{c}} B d$	$(A_{v} / s) d f_{v}$	0
Sezen and Moehle (2004)	$V_{C} + V_{S}$	$0.8 k A_{g} (0.5 \sqrt{f_{c}} / L / d) (\sqrt{1 + (P / 0.5 A_{g} \sqrt{f_{c}})})$	$k (A_{v} / s) d f_{v}$	0
Priestley et al. (1994)	$V_{c} + V_{s} + V_{P}$	$0.8 k \sqrt{f_{c}} B H$	$(A_{v} / s) D^{'} f_{v} \cot 30$	$(H - c / 2 L) P$

Statistical evaluation for all the experimental data

Statistical evaluation is made to assess the performance of the developed model and existing equations, including several statistical measures such as the mean absolute percentage error (MAPE), the mean square error (MSE), the root means square error (RMSE), and the R-squared value. Herein, note that the above statistical measures have been widely used in the fields of structural engineering (Ang and Tang, 2007; Du et al., 2019, 2020; Li et al., 2020). The definitions of these statistical measures are presented as

MAPE = 100 \times \frac{1}{N} | \frac{o_{i} - e_{i}}{o_{i}} |

(14)

MSE = \frac{1}{N} \sum_{i = 1}^{N} {(o_{i} - e_{i})}^{2}

(15)

RMSE = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {(o_{i} - e_{i})}^{2}}

(16)

R - squared = 1 - \frac{\sum_{i = 1}^{N} {(o_{i} - e_{i})}^{2}}{\sum_{i = 1}^{N} {({\bar{o}}_{i} - e_{i})}^{2}}

(17)where

o_{i}

and

e_{i}

are the experimental data and predicted strength, respectively;

{\bar{o}}_{i}

is the mean of experimental data; and

N

is the total number of RC columns included in the compiled experimental database. Table 4 enlists the statistical performance of the proposed model and existing equations. As observed, the MAPE values of ACI 318 model (2008), FEMA 273 model (1997), Sezen and Moehle model (2004), and Priestley et al. model (1994) are 36.6, 55.9, 39.7, and 74.6, respectively. For the training sub-dataset, testing sub-dataset, and all dataset, the MAPE values of the developed model are 14.6, 13.5, and 14.3, respectively. Like the MAPE values, the developed model yields lower RMSE and MSE in comparison with existing equations. The R-squared value of the developed model is 0.95; and the R-squared value of existing equations is smaller than 0.90. Therefore, the developed GEP-based model predicts the peak strength more accurately than existing equations.

Table 4.

Statistical assessments for the proposed model and existing models.

Model		Statistical parameter
			MAPE	MSE	RMSE	R-squared
Gene expression programming–based model	Training dataset	14.6	1463	40.5	0.9507
	Testing dataset	13.5	832	28.8	0.9509
	All dataset	14.3	1440	37.9	0.9507
ACI 318 (2008)		36.6	4429	66.5	0.7561
FEMA 273 (1997)		55.9	9480	97.4	0.8184
Sezen and Moehle (2004)		39.7	4250	65.2	0.7977
Priestley et al. (1994)		74.6	12,897	113.6	0.7393

MAPE: mean absolute percentage error; MSE: mean square error, RMSE: root mean square error.

Comparison for each failure mode

Note that the most significant achievement in the present study is to develop a model that can predict the peak strength of RC columns failing in different modes, whereas existing equations such as those proposed by FEMA 273 (1997), Sezen and Moehle (2004), ACI 318 (2008), and Priestly et al. (1994) can only be used to predict the peak strength of RC columns failing in the SF mode. Figure 7 shows the variation of the average absolute errors corresponding to each failure mode. As observed, existing equations have relatively high variation and great error, with the average absolute errors of peak strength ranging from 29% to 44%. As for the developed model, only a bit of difference in average absolute errors is observed for each failure mode, where the average absolute errors of peak strength range from 13% to 15%. Therefore, the developed GEP-based model predicts the peak strength more flexibly than existing equations.

Figure 7.

Variation of average absolute errors for the peak strength predicted by the developed model and existing equations (values given on each bar show the number of experimental data for each failure mode).

Finally, Figure 8 displays the comparison of peak strength with respect to each failure mode for a better examination of the difference between the developed model and existing equations, where a normalized value equal to the unity is highlighted as the precise prediction line. As observed, the developed model has a close agreement with the experimental data. Among existing equations, Sezen and Moehle model (2004) produces close predictions as the developed model, whose normalized peak strength ranges from 0.32 to 1.37; while FEMA 273 model (1997) and Priestley et al. model (1994) yield the normalized peak strength ranging from 0.26 to 1.70 and 0.21 to 1.88, respectively. Since the peak strength of RC columns is overestimated if the normalized peak strength is less than 1.0, it is evident that the developed model performs better than existing equations, yielding the highest reliable, robust, and accurate peak strength of RC columns failing in different modes.

Figure 8.

Comparison of prediction accuracy for the proposed model against (a) ACI 318 (2008) model, (b) FEMA 273 (1997) model, (c) Sezen and Moehle (2004) model, and (d) Priestley et al. (1994) model in various failure modes.

Conclusions

A GEP-based model is developed in this study to predict the peak strength of RC columns in different failure modes. The developed formulation is examined by comparing with the experimental data under quasi-static cyclic loading. A comprehensive comparison with existing equations is conducted to illustrate the advantage of the developed model. The evidence in this study draws the following conclusions:

The GEP method established a good solution to predict the peak strength of RC columns covering all failure modes. The developed model produces the R-squared value of peak strength at 0.95 for both the training and testing sub-dataset. Different from existing equations, the developed model could predict the peak strength of RC columns failing in different modes in a unified way. Then, there is no need to classify the failure modes of RC columns in prior to use the corresponding fiber cross-sectional method or the semi-empirical models. This is beneficial for the seismic performance evaluation of existing structures.

The developed model is advantageous in yielding the highest reliable, robust, and accurate estimate for the peak strength of RC columns compared to existing equations. For a single failure mode, existing equations have the average absolute errors of peak strength ranging from 29% to 44%. However, the developed model only has a bit of difference in average absolute errors for all the failure modes, ranging from 13% to 15%. Among existing equations, Sezen and Moehle model (2004) produces close prediction as the developed model.

Note that the developed GEP-based model still has complex expression, resulting in more calculation operations than existing equations. Meanwhile, the reason why the prediction is obtained with higher accuracy cannot be well explained by the GEP method. Therefore, it is suggested to take the developed GEP-based model as a heuristic solution by incorporating the physical mechanism in the future.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The financial supports received from the National Science Foundation of China (Grant No. 51808397 and No. 51708460) the Financial Project in Shanghai (Grant No. 20130302) are gratefully appreciated.

ORCID iD

Chao-Lie Ning

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