# Shear Strength Model for Reinforced Concrete Corner Joints Based on Soft Computing Techniques

Authors: Moiz Tariq; Azam Khan; Asad Ullah; Muhammad Waseem; Hassan Nasir; Irfan Jamil
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

The shear strength of cyclically loaded RC corner joints, resulting in opening and closing moments, has not been extensively studied. In addition, experimental studies of the joint shear strength are time-consuming and expensive. Therefore, to overcome this challenge, two separate gene expression programming (GEP) based empirical models are developed for the shear strength of the corner joints, one under the opening moment and the other under the closing moments. One of the key parameters overlooked in previous studies is the joint shear reinforcement, which has been incorporated in the GEP models. These models are developed by compiling an experimental database of 59 specimens in terms of the concrete compressive strength, the joint aspect ratio, the reinforcement tensile strength, and the reinforcement compressive strength. A detailed statistical study is undertaken that indicates superior accuracy of the proposed models and a high potential for their application in the design practice.


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## Body

## 1. Introduction

Beam-column joints made from reinforced concrete (RC) have attracted considerable attention in the last few decades [1–10] because of the difficulty in predicting their behavior. Although extensive effort has been put forward to studying various conventional RC joints, a reliable prediction method for the beam-column corner joints is rare. Corner joint response is more complex in nature, which can be because of the joint opening; resulting in compression outside the joint, or the joint closing; resulting in compression inside the joint. In addition, the corner joints’ behavior is uncertain when supported by a column that is lightly loaded. Despite this complexity and uncertainty, the seismic building codes place little emphasis on the corner joint design.When the corner joint opens and closes, different load-resisting mechanisms are triggered, thereby further complicating the joint’s behavior. Therefore, various models are developed for the shear strength of these joints. For example, Priestley et al. [11] proposed a shear strength model in terms of the principal tensile 0.29fc′ and the principal compressive stresses 0.3fc′ MPa. Later, Megget’s [12] revised the corner joint shear strength to 0.1fc′, which was also adopted by New Zealand Standard [13]. However, several experimental investigations have shown that the corner joint behaves differently in the opening and the closing action, so a single expression of the shear capacity is not sufficient [14–17]. A weaker diagonal concrete strut during the corner joint opening almost halves the shear capacity as compared to the capacity during the joint closing action [18, 19]. Thus, their seismic efficiency is more dependent on the opening than the closing shear stress.Corner joints are not included in any of the major design codes, such as the New Zealand design standard [13], EC 8 [20], Chinese Seismic Code [21], or Architectural Institute of Japan [22]. As a result, the exterior joint provisions are applied to the corner joint design. ACI318-19 [23] and ACI352R-02 [24] recommend only one expression for the corner joint’s nominal shear strength. Besides these actions, the corner joint has normally lower axial load than joints in other locations because of a discontinuous column. Due to all these differences, Moiz et al. [25] proposed two regression models, one for corner joint opening and the other for the corner joint closing. The model showed satisfactory predictive ability, however, it tends to omit the joint shear reinforcement ratio, which has a significant influence on the shear strength of the RC corner joint.Various experts have put forth analytical and computational models for determining the shear capacity of RC joints. In this respect, the compression field theory has been successfully applied by Hwang et al. [26]. Similarly, the rotating-angle softened truss model (RA-STM) [27] and the modified rotating angle softened truss theory (MRA-STM) [28] have been implemented for predicting RC joint shear capacities. Other models include the softened truss theory (FA-STM) [29] and an improved strut and tie-based shear strength model [30]. However, these models are not able to accurately predict the behavior of corner joints. For the corner joints' shear capacity, very few analytical models exist that can determine the capacity under the opening and closing actions [31, 32]. Attempts have also been made to simulate corner joints using a computationally complex finite element analysis approach [33].Due to the requirement of expertise and involvement of the computational complexity of the existing models, the design capacity of corner joints needs to be reexamined using a technique that can predict the response based on the existing experimental evidence. As a result, the current research aims to develop two reliable shear capacity models using the Gene Expression Programming by consulting experimental results for 59 specimens. The proposed soft computing models are validated and compared statistically with the existing models.


## 2. Research Significance

The transfer of the shear and the moment forces from one structural element to another is ensured by the use of beam-column joints. RC structures in earthquake-prone areas frequently collapse due to the shear failure of joints [34–41]. One of the failure causes is the treatment of joints as a rigid element in structural design, which ignores the shear strength of joints, thereby challenging its design and detail for better seismic performance. Only a few reliable models exist that can predict the shear capacity of joints based on the ductility requirements [41–53]. In most of the existing models, the concrete compressive strength is the only influencing parameter, but other factors, such as the joint geometric properties, joint shear reinforcement, and member longitudinal reinforcement, are often overlooked. Attempts have been made, without success, to apply these shear capacity models on the corner joints because of the lack of corner joint-specific models. Hence, the current research aims to develop two soft computing models, incorporating the influence of horizontal and vertical joint shear reinforcement, for assessing the complex behavior of corner joints under the joint opening and closing actions, respectively.


## 3. Parameters Affecting the Shear Strength of RC Corner Joint

Essentially, all of the variables influencing the shear capacity of RC corner joints so that an improved model is developed. Research [4, 7, 52–68] shows that the shear strength of corner joints depends on a variety of factors including the concrete compressive strength, the longitudinal tension reinforcement ratios, the joint horizontal and vertical shear reinforcement, and the joint aspect ratio. These studies have confirmed that joint shear strength is positively correlated with the concrete compressive strength when the joint is tending to close, and the shear strength is negatively correlated when the joint is tending to open.Another factor influencing corner joint shear strength is the joint aspect ratio (beam to column depth ratio), which is positively correlated with the shear strength, regardless of whether the joint is opening or closing [12, 14, 15, 18, 31, 32, 69–71, 72]. Surprisingly, the design code of practices neglects the influence of joint aspect ratio on the strength of the corner joints. It has also been shown [69] that adequate anchorage can increase the joint shear capacity of a longitudinal reinforcement (Figure 1). In addition, the shear capacity can be improved if a corner joint column is wider than the beam, because of the indirect confinement offered by the column. It is concluded from this finding that, under the joint closing, the flexural moment capacity reduces if the opening to the closing moment of the joint is higher. Contrarily, under the joint opening, the capacity reduces if the opening to the closing capacity of the joint is lower.Figure 1
Corner joint longitudinal reinforcement. (a) Closing moment (b) Opening moment.
(a)(b)


## 4. Experimental Investigation and Existing Shear Capacity Models

When it comes to studying corner joints, the Loma Prieta Earthquake of 1989 may be seen as a turning point. Since then, numerous experimental studies have been undertaken, especially involving cyclic loading. Before 1989, most of the experimental studies involved the monotonic loading of corner joints. This section examines some of the most significant studies into cyclically loaded corner joints.


### 4.1. Current Experimental Database

A database of 59 experiments is compiled based on the key influencing parameters discussed in Section3. Out of these 59 specimens, 40 are used to build the GEP model, and the remaining 19 samples are used to test the model. Few experiments are excluded because of inconsistencies in the specimen geometry, material and reinforcement properties, etc. Table 1 summarizes the available specimens in the database.Table 1
RC specimens dimensions and properties.
Specimen IDFc′ (MPa)fyb (MPa)bj (mm)hj (mm)ρbTρjTMogili and Kuang [73]KJ-NO37.60526.00300.00300.000.0200.024KJ-N136.80526.00300.00300.000.0240.024KJ-F034.50560.00300.00420.000.0200.026KJ-F129.20560.00300.00420.000.0240.026KJ-S027.80560.00300.00300.000.0110.026KJ-S126.80560.00300.00300.000.0130.026Zhang [31]KJ-NS38.40500.00300.00300.000.0240.000KJ-F36.50500.00300.00300.000.0240.011KJ2-H12V1029.30520.00300.00300.000.0240.013KJ3-H10V1232.20500.00300.00300.000.0240.013KJ-H8V1035.40500.00300.00300.000.0240.009KJ-BD50030.90500.00300.00300.000.0140.008KJ-CW43030.80500.00365.00300.000.0240.011KJ-BD70032.50500.00300.00300.000.0100.008KJ-CD50032.30500.00300.00500.000.0240.008KJ-CW60033.20500.00450.00300.000.0240.013Mazzoni et al. [15]KJ#142.10503.00254.30304.800.0240.004KJ#242.10503.00254.30304.800.0240.008KJ#332.85503.00254.30304.800.0240.008Cote and Wallace [16]KJ#145.70448.00317.50406.400.0140.011KJ#249.80448.00317.50406.400.0140.011KJ#345.00448.00317.50406.400.0140.005KJ#445.60448.00317.50406.400.0140.011McConnell and Wallace [17]KJ#531.50448.00342.50406.400.0150.006KJ#633.00448.00342.50406.400.0150.005KJ#732.90448.00342.50406.400.0150.010KJ#836.30448.00342.50406.400.0150.006KJ#938.50448.00342.50406.400.0150.005KJ#1037.90448.00342.50406.400.0150.010KJ#1135.00448.00342.50406.400.0150.006KJ#1232.90448.00342.50406.400.0150.005KJ#1331.70448.00342.50406.400.0150.010Megget [12, 14]KJ-127.80358.00225.00250.000.0200.006KJ-227.80358.00225.00250.000.0180.006KJ-334.00328.00225.00250.000.0260.011KJ-434.00328.00225.00250.000.0220.010KJ-533.60355.00225.00250.000.0220.001KJ-633.60325.00225.00250.000.0220.011KJ-750.00333.00225.00250.000.0410.017KJ-840.40340.00225.00250.000.0220.001KJ-939.80333.00225.00250.000.0220.006KJ-1039.70333.00225.00250.000.0220.006KJ-1126.80333.00225.00250.000.0220.006Angelakos [33]KJ-145.70448.00340.00400.000.0140.011KJ-249.70448.00340.00400.000.0140.011KJ-345.00448.00340.00400.000.0140.005KJ-445.60448.00340.00400.000.0140.011KJ-531.50461.00340.00400.000.0220.006KJ-633.00461.00340.00400.000.0220.005KJ-732.90461.00340.00400.000.0220.010KJ-836.30461.00340.00400.000.0160.006KJ-938.50461.00340.00400.000.0160.005KJ-1037.90461.00340.00400.000.0160.010KJ-1135.00461.00340.00400.000.0190.006KJ-1232.90461.00340.00400.000.0190.005KJ-1331.70461.00340.00400.000.0190.010KJ-1433.60448.00340.00400.000.0190.006KJ-1536.90434.00340.00400.000.0150.010KJ-1637.20487.00340.00400.000.0150.010KJ-145.70448.00340.00400.000.0140.011KJ-249.70448.00340.00400.000.0140.011KJ-345.00448.00340.00400.000.0140.005KJ-445.60448.00340.00400.000.0140.011KJ-531.50461.00340.00400.000.0220.006KJ-633.00461.00340.00400.000.0220.005KJ-732.90461.00340.00400.000.0220.010KJ-836.30461.00340.00400.000.0160.006KJ-938.50461.00340.00400.000.0160.005KJ-1037.90461.00340.00400.000.0160.010KJ-1135.00461.00340.00400.000.0190.006KJ-1232.90461.00340.00400.000.0190.005KJ-1331.70461.00340.00400.000.0190.010KJ-1433.60448.00340.00400.000.0190.006KJ-1536.90434.00340.00400.000.0150.010KJ-1637.20487.00340.00400.000.0150.010


### 4.2. Previous Experimental Investigations

Several experimental studies [15–17] have noted anomalous results for corner joint shear capacities when using the empirical expression proposed by ACI 352-02 [24]. A similar difference [14] is noted in the New Zealand Code of Practice (NZS 3101-2006). This disparity is because of leaving out various key influencing factors from the code proposed expressions [30, 32, 72]. Material, geometry, and reinforcement properties have a significant influence on the joint shear strength [30, 72]. As a result of this precedent, it is necessary to develop an expression that incorporates all these factors as input parameters.


### 4.3. Review of Existing Shear Capacity Models

Extensive variation exists in the proposed empirical expressions of corner joint shear strength. However, many of these expressions produce results that are invalid and impractical. There are many proposed corner joint shear capacity equations; for instance,0.58fc′ [17] and 0.58fc′ [33, 72, 74]. Similarly, other researchers [12, 14] have proposed 0.12fc′ for the joint closing and 0.10fc′ for the opening shear capacity. Some investigators [11] have bounded the principal tensile stress in the corner joint to 0.29fc′ , rather than predicting the joint shear capacity. An identical variable trend in the shear capacity equations is also observed in the expressions of building codes. Some of the empirical models used in the subsequent comparative study are discussed in Table 2Table 2
Previously proposed empirical models.
NameEmperical formulaLimitationsNew Zealand Code of practice (3101-2006) [13]Vjh≤min0.2fc′bjhc;10bjhcExtending the provisions of exterior joint on corner jointChinese Standards 50011:2010) [21]Vjh≤1/γRE0.3ηjβcfcbjhjExtending the provisions of interior joint on corner jointγRE=0.85ηj = 1Architectural Institute of Japan [22]Vju=kϕFjbjDjReduction factor is same for both closing and opening behaviourFj=0.8fc′0.7Vju=0.272fc′0.7bjDjk = 0.4American Concrete Institute [24]Vn=0.083γcfc′AjReduction factor is same for both closing and opening behaviourVn=γcfc′AjEurocode 8 [20]Vjhd=ληfcd1−vd/ηbjhjcExtending the provisions of exterior joint on corner jointη=0.61−fck/250Tariq et al. [25]vjhc=−0.438/−1.23hb/hc2−−1.23−hb/bb+fc′−hb/hc−13.73hb/hc/2.31fc′/fy1/4+ρbfyb/fc′hb2/bjhcMPaDoes not incorporate the effect joint vertical and horizontal joint transverse reinforcementvjho=1/ρbfyfc′−4.9671/2+1.33+hb/hc+hb/bb−0.37bj/hc+1.54fc′/fyb−hb/bbhb/bb1/4+hb/bb4ρbfyb/fc′3MPa


## 4.1. Current Experimental Database

A database of 59 experiments is compiled based on the key influencing parameters discussed in Section3. Out of these 59 specimens, 40 are used to build the GEP model, and the remaining 19 samples are used to test the model. Few experiments are excluded because of inconsistencies in the specimen geometry, material and reinforcement properties, etc. Table 1 summarizes the available specimens in the database.Table 1
RC specimens dimensions and properties.
Specimen IDFc′ (MPa)fyb (MPa)bj (mm)hj (mm)ρbTρjTMogili and Kuang [73]KJ-NO37.60526.00300.00300.000.0200.024KJ-N136.80526.00300.00300.000.0240.024KJ-F034.50560.00300.00420.000.0200.026KJ-F129.20560.00300.00420.000.0240.026KJ-S027.80560.00300.00300.000.0110.026KJ-S126.80560.00300.00300.000.0130.026Zhang [31]KJ-NS38.40500.00300.00300.000.0240.000KJ-F36.50500.00300.00300.000.0240.011KJ2-H12V1029.30520.00300.00300.000.0240.013KJ3-H10V1232.20500.00300.00300.000.0240.013KJ-H8V1035.40500.00300.00300.000.0240.009KJ-BD50030.90500.00300.00300.000.0140.008KJ-CW43030.80500.00365.00300.000.0240.011KJ-BD70032.50500.00300.00300.000.0100.008KJ-CD50032.30500.00300.00500.000.0240.008KJ-CW60033.20500.00450.00300.000.0240.013Mazzoni et al. [15]KJ#142.10503.00254.30304.800.0240.004KJ#242.10503.00254.30304.800.0240.008KJ#332.85503.00254.30304.800.0240.008Cote and Wallace [16]KJ#145.70448.00317.50406.400.0140.011KJ#249.80448.00317.50406.400.0140.011KJ#345.00448.00317.50406.400.0140.005KJ#445.60448.00317.50406.400.0140.011McConnell and Wallace [17]KJ#531.50448.00342.50406.400.0150.006KJ#633.00448.00342.50406.400.0150.005KJ#732.90448.00342.50406.400.0150.010KJ#836.30448.00342.50406.400.0150.006KJ#938.50448.00342.50406.400.0150.005KJ#1037.90448.00342.50406.400.0150.010KJ#1135.00448.00342.50406.400.0150.006KJ#1232.90448.00342.50406.400.0150.005KJ#1331.70448.00342.50406.400.0150.010Megget [12, 14]KJ-127.80358.00225.00250.000.0200.006KJ-227.80358.00225.00250.000.0180.006KJ-334.00328.00225.00250.000.0260.011KJ-434.00328.00225.00250.000.0220.010KJ-533.60355.00225.00250.000.0220.001KJ-633.60325.00225.00250.000.0220.011KJ-750.00333.00225.00250.000.0410.017KJ-840.40340.00225.00250.000.0220.001KJ-939.80333.00225.00250.000.0220.006KJ-1039.70333.00225.00250.000.0220.006KJ-1126.80333.00225.00250.000.0220.006Angelakos [33]KJ-145.70448.00340.00400.000.0140.011KJ-249.70448.00340.00400.000.0140.011KJ-345.00448.00340.00400.000.0140.005KJ-445.60448.00340.00400.000.0140.011KJ-531.50461.00340.00400.000.0220.006KJ-633.00461.00340.00400.000.0220.005KJ-732.90461.00340.00400.000.0220.010KJ-836.30461.00340.00400.000.0160.006KJ-938.50461.00340.00400.000.0160.005KJ-1037.90461.00340.00400.000.0160.010KJ-1135.00461.00340.00400.000.0190.006KJ-1232.90461.00340.00400.000.0190.005KJ-1331.70461.00340.00400.000.0190.010KJ-1433.60448.00340.00400.000.0190.006KJ-1536.90434.00340.00400.000.0150.010KJ-1637.20487.00340.00400.000.0150.010KJ-145.70448.00340.00400.000.0140.011KJ-249.70448.00340.00400.000.0140.011KJ-345.00448.00340.00400.000.0140.005KJ-445.60448.00340.00400.000.0140.011KJ-531.50461.00340.00400.000.0220.006KJ-633.00461.00340.00400.000.0220.005KJ-732.90461.00340.00400.000.0220.010KJ-836.30461.00340.00400.000.0160.006KJ-938.50461.00340.00400.000.0160.005KJ-1037.90461.00340.00400.000.0160.010KJ-1135.00461.00340.00400.000.0190.006KJ-1232.90461.00340.00400.000.0190.005KJ-1331.70461.00340.00400.000.0190.010KJ-1433.60448.00340.00400.000.0190.006KJ-1536.90434.00340.00400.000.0150.010KJ-1637.20487.00340.00400.000.0150.010


## 4.2. Previous Experimental Investigations

Several experimental studies [15–17] have noted anomalous results for corner joint shear capacities when using the empirical expression proposed by ACI 352-02 [24]. A similar difference [14] is noted in the New Zealand Code of Practice (NZS 3101-2006). This disparity is because of leaving out various key influencing factors from the code proposed expressions [30, 32, 72]. Material, geometry, and reinforcement properties have a significant influence on the joint shear strength [30, 72]. As a result of this precedent, it is necessary to develop an expression that incorporates all these factors as input parameters.


## 4.3. Review of Existing Shear Capacity Models

Extensive variation exists in the proposed empirical expressions of corner joint shear strength. However, many of these expressions produce results that are invalid and impractical. There are many proposed corner joint shear capacity equations; for instance,0.58fc′ [17] and 0.58fc′ [33, 72, 74]. Similarly, other researchers [12, 14] have proposed 0.12fc′ for the joint closing and 0.10fc′ for the opening shear capacity. Some investigators [11] have bounded the principal tensile stress in the corner joint to 0.29fc′ , rather than predicting the joint shear capacity. An identical variable trend in the shear capacity equations is also observed in the expressions of building codes. Some of the empirical models used in the subsequent comparative study are discussed in Table 2Table 2
Previously proposed empirical models.
NameEmperical formulaLimitationsNew Zealand Code of practice (3101-2006) [13]Vjh≤min0.2fc′bjhc;10bjhcExtending the provisions of exterior joint on corner jointChinese Standards 50011:2010) [21]Vjh≤1/γRE0.3ηjβcfcbjhjExtending the provisions of interior joint on corner jointγRE=0.85ηj = 1Architectural Institute of Japan [22]Vju=kϕFjbjDjReduction factor is same for both closing and opening behaviourFj=0.8fc′0.7Vju=0.272fc′0.7bjDjk = 0.4American Concrete Institute [24]Vn=0.083γcfc′AjReduction factor is same for both closing and opening behaviourVn=γcfc′AjEurocode 8 [20]Vjhd=ληfcd1−vd/ηbjhjcExtending the provisions of exterior joint on corner jointη=0.61−fck/250Tariq et al. [25]vjhc=−0.438/−1.23hb/hc2−−1.23−hb/bb+fc′−hb/hc−13.73hb/hc/2.31fc′/fy1/4+ρbfyb/fc′hb2/bjhcMPaDoes not incorporate the effect joint vertical and horizontal joint transverse reinforcementvjho=1/ρbfyfc′−4.9671/2+1.33+hb/hc+hb/bb−0.37bj/hc+1.54fc′/fyb−hb/bbhb/bb1/4+hb/bb4ρbfyb/fc′3MPa


## 5. Fundamentals of Gene Expression Programming

An artificial intelligence-based technique called Gene Expression Programming (GEP) is an evolutionary algorithm that generates mathematical models and processes the input data in a domain-independent manner [75]. GEP differs from the Genetic Algorithms (GAs) and the Genetic Programmings (GPs) in the sequence of chromosome representation. GEP algorithm can produce a linear and nonlinear sequence of chromosomes creating a robust computer-based solution of complex and challenging problems. The algorithm iteratively alters the number of chromosomes and genes, head size, and the linking functions to generate an influential model that can perform a broad range of estimation and prediction tasks. An interesting feature of having more genes and chromosomes is that the function can be more complex, but the results can still be exact. Because of this tradeoff, it is possible to achieve a simplified mathematical model and control the number of genes/chromosomes, as well as achieve the desired level of accuracy [76].As the GEP algorithm progresses, it tries to achieve convergence on the global optimal solution. There may be times when the algorithm is unable to choose the best solution from a pool of possible ones, thereby leading to an endless loop or producing an illogical expression, depending on the state. Changing the number of genes and chromosomes, or changing the linking function, can solve this numerical problem. This numerical problem was effectively resolved herein whilst developing two models for predicting the shear capacity of corner joints, one related to the joint opening and the other to joint closing.


## 6. Development of GEP Based Shear Capacity Model

To develop a GEP model, the population involving 40 data points is randomly extracted from a total of 59 datasets (Table2). The influencing factors of the joint shear strength, including, the column and beam cross-sections, the concrete compressive strength and yielding stress of reinforcing bars, the area of longitudinal tension reinforcement, and the area of longitudinal compression reinforcement, are selected as input parameters to establish two independent GEP models, one for closing action and the other for opening action of joints. These models are validated using the remaining 19 datasets of the experimental database. The following expression is derived for the shear capacity during the joint closing.(1)vjhc=vjhc1+vjhc2+vjhc3MPa,(2)vjhc1=−0.25bjhjdbdcρhjfybfc′fc′−dbdc−ρcbfybfc′−2.16bjhj+ρcbfybfc′,(3)vjhc2=fc′14−bbdb510.49ρcbfybfc′,(4)vjhc3=fc′14−0.942ρvjfybfc′2−bcdb,and the following during the joint opening,(5)vjho=vjho1+vjho2+vjho3MPa,(6)vjho1=0.87fc′bjhj−3ρvjfybfc′ρhjfybfc′bcdc2.44ρtbfybfc′,(7)vjho2=bjhj0.27ρtbfybfc′3bjhj+fc′,(8)vjho3=bbdb25dbdc−bjhj−bcdc−0.1883ρtbfybfc′+0.45,where db = depths of beam member, dc = depth of column members, bj = joint effective width, hj = joint effective depth, ρcb = longitudinal tensile reinforcement under closing, ρtb = longitudinal tensile reinforcement under opening behavior, ρvj = represent the joint vertical shear reinforcement, ρhj represent the joint horizontal shear reinforcement, fc′ = concrete compressive strength, and fyb = reinforcement tensile yield. Moreover, vjhcand vjhorepresent the joint shear capacity in MPa.Figures2 and 3 depict the proposed model for the joint closing and opening capacities, and Table 3 provides some additional details related to the proposed models.Figure 2
Gene expression for the closing model.Figure 3
Gene expression for the opening model.Table 3
Model construction parameter.
Function set+, −,/,×, x1∕2,1∕4,2,3,4,5Chromosomes100Head size5Linking functionAdditionNumber of genes3Mutation rate0.044Inversion rate0.1One-point recombination rate0.3Two-point recombination rate0.3Gene recombination rate0.1Gene transposition rate0.1Constants per gene2Lower/upper bound of constants−20/20


## 7. Statistical Assessment of the Proposed Model

After developing any regression-based model, it is essential to set out criteria for evaluating the model's performance. A variety of statistical indicators are available to assess the predictive ability of the model.The coefficient of variation (CoV) is commonly used to evaluate model performance. CoV can be calculated as follows:(9)CoV%=σμ×100,where σ is the standard deviation and μ is the sample mean. A smaller value of CoV indicates less data spread.The average absolute error (AAE) is another measure of validation. The AAE is defined as follows:(10)AAE%=1n∑VjhExp−VjhEstVjhExp,where nisthenumberoftestspecimens.The performance of the regression model can be challenged by measuring the coefficient of determination (R2), defined as follows:(11)R2=1−∑VjhExp−VjhEst2∑VjhExp−VjhMeanExp2,where the performance of a model is considered very good if R2 is close to 1. The performance of the joint closing and opening models is demonstrated in Figures 4 and 5, respectively. As seen in the joint closing model in Figure 4, the coefficient of determination (R2) is 0.82 for the training, 0.73 for the validation, and 0.79 for the overall data. In the same way, the coefficients for the opening action are 0.72, 0.79, and 0.75, respectively, for the training, the validation, and the overall data. These values clearly show better prediction capability of both the closing and opening models.Figure 4
Estimated shear strength versus experimental shear strength for joint closing. (a) Training. (b) Validation. (c) All data.
(a)(b)(c)Figure 5
Estimated shear strength versus experimental shear strength for joint opening. (a) Training. (b) Validation. (c) All data.
(a)(b)(c)Another measure of the prediction accuracy is the t-statistic [77] for the evaluation and comparison of the corner joints. The null hypothesis is Ho:μD=0, where μD = experimental and predicted shear stress difference (MPa). In comparison, the alternative hypothesis is H1:μD≠0. The t-statistic is defined as follows:(12)t=X¯−μDs/N,where X¯ = Sample mean differenceμD = Population means = Standard deviationN = Sample sizeν=DegreeoffreedomThe significant levelα=0.05 with μ=58 degrees of freedom.Table4 indicates that the tCalculated<t0.025,60, i.e., −0.18<2.00 and −0.32<2.00 for the corner joint under both the closing and the opening, respectively, with a degree of freedom ν=60 and not in the rejection region. Therefore, the null hypothesis is accepted at a 5% significance level.Table 4
t-Statistics results.
DescriptionOpeningClosingvjhExp (MPa)vjhEst (MPa)vjhExp (MPa)vjhEst (MPa)Mean2.162.203.583.60Variance0.500.300.580.33Observations59595959Pearson correlation0.830.90Hypothesized mean difference0.000.00Df58585858t-stat−0.32−0.18t-critical two-tail−2.00/2.00−2.00/2.00In addition to the above-given statistical indicators, the overall performance of both the opening and the closing models can be examined by varying the levels of the primary influencing parameters. Figure6 demonstrates the variation of the joint shear strength ratio (ratio of shear calculated to the experimental value) with ρb, fc′, fyb, ρbfy/fc′,hb/hc, bj/hc, ρvjfyb/fc′, ρhjfyb/fc′, and for opening and closing behavior. This shear strength ratio shows a virtual average of 1.00 within the interval [0.7, 1.5] in Figure 6(a) corresponding to various levels of ρb, which represents the suitable model performance in both closing and opening.Figure 6
Performance of GEP models considering key parameters.
(a)(b)(c)(d)(e)(f)(g)(h)Similarly, in Figure6(b), it is shown that the results of the joint shear strength ratio are comparable in terms of fc′ values. Even though the proposed joint shear model is slightly overestimated for the case of the experimental joint specimens with higher fc′, the model performance is adequate within the interval [0.7, 1.5]. Generally speaking, the closed-joint performed better than their open-joint counterparts.Better results are also obtained forfyb, the index of ρbfy/fc′, bj/hc, hb/hc, horizontal shear reinforcement, and vertical shear reinforcement, as is clear from Figure 6(c) till 6(h). In short, the foregoing shows that the proposed model shows better predictive performance.


## 8. Results and Discussions

The robustness and the predictive capability of the proposed GEP models can be assessed by comparing the results with the existing models. The existing models are either available in literature or adopted by the code of practices. The database comprising the shear strength tests is reported in Tables5 and 6.Table 5
Joint shear capacity under the closing.
Specimen IDfc′ (MPa)vjhEst (MPa)vjhExp (MPa)vjhExp/vjhEstvjhExp/vjhACIvjhExp/vjhEurocodevjhExp/vjhNZS3101vjhExp/vjhAIJvjhExp/vjhGB50010Mogili and Kuang [73]KJ-NO37.603.603.420.980.840.390.450.900.30KJ-N136.803.964.391.021.090.510.601.170.40KJ-F034.503.002.600.850.670.320.380.730.25KJ-F129.203.213.390.820.940.480.581.060.39KJ-S027.802.462.610.740.720.370.450.820.30KJ-S126.802.842.850.760.830.440.530.950.35Zhang [31]KJ-NS38.402.432.600.920.630.290.340.670.23KJ-F36.503.293.161.130.790.370.430.850.29KJ2-H12V1029.303.562.961.010.820.420.510.930.34KJ3-H10V1232.202.992.841.050.750.370.440.830.29KJ-H8V1035.402.923.191.250.810.390.450.880.30KJ-BD50030.904.404.601.361.250.620.741.390.50KJ-CW43030.802.902.751.020.750.370.450.830.30KJ-BD70032.504.024.110.971.080.530.631.200.42KJ-CD50032.302.822.501.110.660.330.390.730.26KJ-CW60033.202.182.150.780.560.270.320.620.22Mazzoni et al. [15]KJ#142.104.134.251.030.990.450.501.030.34KJ#242.104.644.311.041.000.450.511.050.34KJ#332.854.605.571.331.180.720.851.610.57Cote and Wallace [16]KJ#145.703.713.711.060.830.370.410.850.27KJ#249.803.693.881.090.830.360.390.840.26KJ#345.003.403.631.040.820.360.400.840.27KJ#445.603.713.871.100.860.380.420.890.28McConnell and Wallace [17]KJ#531.503.884.101.201.100.550.651.220.43KJ#633.003.523.671.070.960.470.561.060.37KJ#732.903.774.381.281.150.560.671.270.44KJ#836.303.803.701.070.930.440.511.000.34KJ#938.503.423.811.100.920.430.490.990.33KJ#1037.903.703.891.120.950.440.511.020.34KJ#1135.003.823.681.070.940.450.531.020.35KJ#1232.903.533.571.040.940.460.541.030.36KJ#1331.703.793.791.111.010.500.601.120.40Megget [12, 14]KJ-127.803.042.560.810.730.380.460.830.31KJ-227.802.732.640.860.750.390.470.860.32KJ-334.003.573.230.930.830.400.480.910.32KJ-434.003.453.150.910.810.390.460.890.31KJ-533.603.262.410.670.630.300.360.690.24KJ-633.603.543.370.980.880.430.500.960.33KJ-750.004.625.111.161.090.470.511.100.34KJ-840.403.062.540.710.600.280.310.640.21KJ-939.803.283.080.870.740.340.390.780.26KJ-1039.703.283.260.920.780.360.410.830.27KJ-1126.803.383.671.070.970.560.681.220.46Angelakos [33]KJ-145.703.723.140.940.700.310.340.720.23KJ-249.703.703.140.930.670.290.320.680.21KJ-345.003.403.140.950.710.310.350.730.23KJ-445.603.723.140.950.700.310.340.720.23KJ-531.504.844.541.051.220.610.721.350.48KJ-633.004.434.311.001.130.550.651.240.44KJ-732.904.695.271.231.380.680.801.520.53KJ-836.304.134.311.151.080.510.591.160.40KJ-938.503.734.311.151.050.490.561.120.37KJ-1037.904.024.811.281.180.550.631.260.42KJ-1135.004.164.311.151.100.530.621.190.41KJ-1232.903.883.921.051.030.500.601.130.40KJ-1331.704.164.311.161.150.570.681.280.45KJ-1433.604.133.731.020.970.470.561.060.37KJ-1536.903.342.820.880.700.330.380.750.25KJ-1637.203.523.210.960.790.370.430.850.29Average performance factor0.990.900.440.510.980.34Table 6
Joint shear capacity under the opening moment.
Specimen IDfc′ (MPa)vjhEst (MPa)vjhExp (MPa)vjhExp/vjhEstvjhExp/vjhACIvjhExp/vjhEurocodevjhExp/vjhNZS3101vjhExp/vjhAIJvjhExp/vjhGB50010Mogili and Kuang [71]KJ-NO37.602.682.680.920.860.310.360.710.24KJ-N136.802.672.640.900.890.310.360.710.24KJ-F034.503.122.971.050.970.370.430.830.29KJ-F129.202.983.311.001.280.470.571.040.38KJ-S027.802.632.500.831.270.370.450.810.30KJ-S126.802.592.460.801.320.380.460.820.31Zhang [30]KJ-NS38.401.651.711.040.810.190.220.440.15KJ-F36.502.342.501.460.850.290.340.670.23KJ2-H12V1029.302.752.501.080.990.360.430.780.28KJ3-H10V1232.202.212.381.270.880.310.370.700.25KJ-H8V1035.402.092.071.460.770.250.290.570.19KJ-BD50030.902.933.571.681.680.480.581.080.39KJ-CW43030.801.941.790.920.830.240.290.540.19KJ-BD70032.502.582.240.951.450.290.340.650.23KJ-CD50032.302.212.031.320.710.270.310.590.21KJ-CW60033.201.651.70.900.920.220.260.490.17Mazzoni et al. [15]KJ#142.103.042.240.940.730.240.270.540.18KJ#242.103.542.471.040.760.260.290.600.20KJ#332.854.004.771.571.380.610.731.380.48Cote and Wallace [16]KJ#145.701.611.340.950.760.130.150.310.10KJ#249.801.591.761.390.970.160.180.380.12KJ#345.001.501.320.930.790.130.150.310.10KJ#445.601.611.380.980.780.140.150.320.10McConnell and Wallace [17]KJ#531.502.001.860.901.070.250.300.550.20KJ#633.001.761.670.830.910.210.250.480.17KJ#732.901.911.900.940.980.240.290.550.19KJ#836.301.941.30.850.950.190.220.430.15KJ#938.501.731.290.840.830.170.190.390.13KJ#1037.901.861.430.940.910.200.230.450.15KJ#1135.001.951.961.020.910.240.280.540.19KJ#1232.901.762.191.091.040.280.330.630.22KJ#1331.701.922.101.020.960.280.330.620.22Megget [12, 14]KJ-127.802.142.180.841.070.320.390.710.26KJ-227.802.142.200.851.080.330.400.720.26KJ-334.002.632.611.060.940.330.380.740.26KJ-434.001.991.970.940.850.250.290.560.19KJ-533.601.861.320.610.830.170.200.380.13KJ-633.602.032.201.050.940.280.330.630.22KJ-750.003.663.502.050.900.320.350.750.23KJ-840.401.801.540.840.830.170.190.390.13KJ-939.801.902.151.160.870.240.270.540.18KJ-1039.701.902.251.210.910.250.280.570.19KJ-1126.801.942.631.081.230.400.490.880.33Angelakos [32]KJ-145.701.621.200.940.720.130.140.300.09KJ-249.701.601.110.960.650.110.120.260.08KJ-345.001.501.140.930.760.130.140.300.10KJ-445.601.621.110.890.690.120.130.280.09KJ-531.502.582.731.131.550.360.430.810.29KJ-633.002.222.71.121.390.330.390.750.26KJ-732.902.442.771.071.270.320.380.720.25KJ-836.301.941.810.961.060.210.250.490.17KJ-938.501.731.780.990.980.200.230.460.15KJ-1037.901.871.811.000.960.210.240.470.16KJ-1135.002.412.701.061.050.280.330.640.22KJ-1232.902.142.671.051.120.310.360.690.24KJ-1331.702.362.581.041.100.320.380.720.26KJ-1433.602.392.641.041.080.290.340.650.23KJ-1536.902.191.470.940.860.220.260.510.17KJ-1637.202.381.970.930.880.230.260.520.18Average performance factor1.000.990.300.320.600.21


### 8.1. Assessment of Shear Capacity GEP Model during Joint Closing

The overall performance of the proposed joint closing model can be assessed from the comparison results of Table7. The results of the previous models indicate that the closing joint shear strength is not estimated accurately, mainly due to the omission of important influencing factors in the models. For instance, the models proposed by various codes of practices incorporate the influence of only three key parameters, thus resulting in lower accuracy. The table shows that the model proposed by Moiz et al. [25] provides acceptable accuracy, although this model omits the influence of the joint shear reinforcement ratio. The proposed model incorporating all the important influencing parameters possesses the highest value of the coefficient of determination (R2 = 0.79), and the lowest value of both the coefficient of variation (CoV = 12.7) and the average absolute error (AAE = 9.4%), which clearly shows the accuracy and reliability of the model. Apart from the Moiz et al. [25] model, all the other models have the coefficient of determination of less than 50%. Similarly, the performance factor of the formulations proposed by the code of practices is less than 50%. The comparison of these models is also shown in Figures 7 and 8. These results surely confirm robust predictions of the current model compared to all the other models.Table 7
Statistical parameters of shear strength GEP model under the closing moment.
AuthorPF=vjhExp/vjhEstStd. deviationCOV (%)AAE (%)R2MeanMogili et al. [32]1.130.19017.315.00.34Kim et al. [74]1.280.2828.021.00.02ACI 352R [24]0.900.18020.022.00.07Eurocode [20]0.440.10123.0141.80.02Chinese code [21]0.340.08324.5212.60.02Newzealand standard [13]0.510.12524.5108.40.001Architectural institue of Japan [22]0.980.21023.019.70.95Megget [12, 14]0.340.24024.030.40.02McConnell and Wallace [17]1.030.22422.017.80.001Wallace et al. [72]1.200.26022.019.00.02Angelakos [33]1.200.26022.019.00.02Priestly et al. [11]1.260.27422.020.90.02Tariq et al. [25]0.980.13013.010.00.49Proposed1.000.12712.79.40.79Figure 7
Statistical measures of the predicted values against the available models (joint closing).Figure 8
Correlation analysis of corner joint shear strength subject to closing: (a) Proposed equation (b) Mogili et al. (c) Kim et al. (d) American concrete institute (e) Euro code 8 (f) New Zealand standard (g) Chinese seismic standard (h) Architectural Institute of Japan (i) Megget et al. (j) McConnell et al. (k) Anglakos et al. (l) Priestly et al. (m) Moiz et al.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)


### 8.2. Assessment of Shear Capacity GEP Model during Joint Opening

The performance of the joint opening model can be assessed from the comparison results of Table8. Clearly, the previous models do not predict the shear strength accurately because they overlook the influence of key factors in the models. For instance, the models proposed by various codes of practices incorporate only three influencing parameters, thus resulting in lower accuracy. Again, the model proposed by Moiz et al. [25] provides acceptable accuracy during the joint opening, although this model omits the influence of the joint shear reinforcement ratio. The proposed model possesses the highest value of the coefficient of determination (R2 = 0.71), and the lowest value of both the coefficient of variation (CoV = 20) and the average absolute error (AAE = 16%), which clearly shows the accuracy and reliability of the model. Apart from the Moiz et al. [25] model, all the other models have the coefficient of determination of less than 50%. Likewise, the performance factor of the formulations proposed by the code of practices is less than 50%. The comparison of these models is also shown in Figures 9 and 10. These results surely confirm robust predictions of the opening joint model compared to all the other models.Table 8
Statistical parameters of shear strength GEP model under an opening moment.
AuthorPF=vjhExp/vjhEstStd. deviationCOV (%)AAE (%)R2MeanMogili et al. [32]0.980.21022.018.00.45Kim et al. [74]0.790.26031.046.00.13ACI 352R [24]0.550.16029.0101.00.03Eurocode [20]0.270.09736.0323.00.14Chinese code 2011 [21]0.210.07937.0450.30.14Newzealand standard 2006 [13]0.310.11336.0266.80.14Architectural institue of Japan [22]0.610.20033.087.30.14Megget [12, 14]0.620.21434.081.40.13McConnell et al. [17]0.630.19231.073.60.14Wallace et al. [72]0.730.22331.051.40.14Angelakos [33]0.730.22331.051.40.14Priestly et al. [11]0.770.25032.447.10.03Tariq et al. [25]1.000.22022.018.00.50Proposed1.000.20020.016.00.71Figure 9
Statistical measures of the predicted values against the available models (joint opening).Figure 10
Correlation analysis of corner joint shear strength subject to opening (a) Proposed equation (b) Mogili et al. (c) Kim et al. (d) American concrete institute (e) Euro code 8 Proposed equation (b) Mogili et al. (c) Kim et al. (d) American concrete institute (e) Euro code 8 (f) New Zealand standard (g) Chinese seismic standards (h) Architectural Institute of Japan (i) Megget et al. (j) McConnell et al. (k) Anglakos et al. (l) Priestly et al. (m) Moiz et al.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)


### 8.3. Practical Application

Both the opening and closing models can be used for calibration and prediction of corner joint shear strength because the models are based on a wider range of influencing parameters. Therefore, the corner joint shear behavior can be quantitatively assessed for different values of key parameters. Being able to be easily implemented in Excel or MATLAB, the quantitative assessment can offer a useful tool for design and engineering decisions.


## 8.1. Assessment of Shear Capacity GEP Model during Joint Closing

The overall performance of the proposed joint closing model can be assessed from the comparison results of Table7. The results of the previous models indicate that the closing joint shear strength is not estimated accurately, mainly due to the omission of important influencing factors in the models. For instance, the models proposed by various codes of practices incorporate the influence of only three key parameters, thus resulting in lower accuracy. The table shows that the model proposed by Moiz et al. [25] provides acceptable accuracy, although this model omits the influence of the joint shear reinforcement ratio. The proposed model incorporating all the important influencing parameters possesses the highest value of the coefficient of determination (R2 = 0.79), and the lowest value of both the coefficient of variation (CoV = 12.7) and the average absolute error (AAE = 9.4%), which clearly shows the accuracy and reliability of the model. Apart from the Moiz et al. [25] model, all the other models have the coefficient of determination of less than 50%. Similarly, the performance factor of the formulations proposed by the code of practices is less than 50%. The comparison of these models is also shown in Figures 7 and 8. These results surely confirm robust predictions of the current model compared to all the other models.Table 7
Statistical parameters of shear strength GEP model under the closing moment.
AuthorPF=vjhExp/vjhEstStd. deviationCOV (%)AAE (%)R2MeanMogili et al. [32]1.130.19017.315.00.34Kim et al. [74]1.280.2828.021.00.02ACI 352R [24]0.900.18020.022.00.07Eurocode [20]0.440.10123.0141.80.02Chinese code [21]0.340.08324.5212.60.02Newzealand standard [13]0.510.12524.5108.40.001Architectural institue of Japan [22]0.980.21023.019.70.95Megget [12, 14]0.340.24024.030.40.02McConnell and Wallace [17]1.030.22422.017.80.001Wallace et al. [72]1.200.26022.019.00.02Angelakos [33]1.200.26022.019.00.02Priestly et al. [11]1.260.27422.020.90.02Tariq et al. [25]0.980.13013.010.00.49Proposed1.000.12712.79.40.79Figure 7
Statistical measures of the predicted values against the available models (joint closing).Figure 8
Correlation analysis of corner joint shear strength subject to closing: (a) Proposed equation (b) Mogili et al. (c) Kim et al. (d) American concrete institute (e) Euro code 8 (f) New Zealand standard (g) Chinese seismic standard (h) Architectural Institute of Japan (i) Megget et al. (j) McConnell et al. (k) Anglakos et al. (l) Priestly et al. (m) Moiz et al.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)


## 8.2. Assessment of Shear Capacity GEP Model during Joint Opening

The performance of the joint opening model can be assessed from the comparison results of Table8. Clearly, the previous models do not predict the shear strength accurately because they overlook the influence of key factors in the models. For instance, the models proposed by various codes of practices incorporate only three influencing parameters, thus resulting in lower accuracy. Again, the model proposed by Moiz et al. [25] provides acceptable accuracy during the joint opening, although this model omits the influence of the joint shear reinforcement ratio. The proposed model possesses the highest value of the coefficient of determination (R2 = 0.71), and the lowest value of both the coefficient of variation (CoV = 20) and the average absolute error (AAE = 16%), which clearly shows the accuracy and reliability of the model. Apart from the Moiz et al. [25] model, all the other models have the coefficient of determination of less than 50%. Likewise, the performance factor of the formulations proposed by the code of practices is less than 50%. The comparison of these models is also shown in Figures 9 and 10. These results surely confirm robust predictions of the opening joint model compared to all the other models.Table 8
Statistical parameters of shear strength GEP model under an opening moment.
AuthorPF=vjhExp/vjhEstStd. deviationCOV (%)AAE (%)R2MeanMogili et al. [32]0.980.21022.018.00.45Kim et al. [74]0.790.26031.046.00.13ACI 352R [24]0.550.16029.0101.00.03Eurocode [20]0.270.09736.0323.00.14Chinese code 2011 [21]0.210.07937.0450.30.14Newzealand standard 2006 [13]0.310.11336.0266.80.14Architectural institue of Japan [22]0.610.20033.087.30.14Megget [12, 14]0.620.21434.081.40.13McConnell et al. [17]0.630.19231.073.60.14Wallace et al. [72]0.730.22331.051.40.14Angelakos [33]0.730.22331.051.40.14Priestly et al. [11]0.770.25032.447.10.03Tariq et al. [25]1.000.22022.018.00.50Proposed1.000.20020.016.00.71Figure 9
Statistical measures of the predicted values against the available models (joint opening).Figure 10
Correlation analysis of corner joint shear strength subject to opening (a) Proposed equation (b) Mogili et al. (c) Kim et al. (d) American concrete institute (e) Euro code 8 Proposed equation (b) Mogili et al. (c) Kim et al. (d) American concrete institute (e) Euro code 8 (f) New Zealand standard (g) Chinese seismic standards (h) Architectural Institute of Japan (i) Megget et al. (j) McConnell et al. (k) Anglakos et al. (l) Priestly et al. (m) Moiz et al.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)


## 8.3. Practical Application

Both the opening and closing models can be used for calibration and prediction of corner joint shear strength because the models are based on a wider range of influencing parameters. Therefore, the corner joint shear behavior can be quantitatively assessed for different values of key parameters. Being able to be easily implemented in Excel or MATLAB, the quantitative assessment can offer a useful tool for design and engineering decisions.


## 9. Conclusion

Based on the algorithm of the Gene Expression Program (GEP), two shear strength prediction models are established and the results are compared with other models. Specifically, one of the models is capable of determining the shear strength during the corner joint closing, and the other during the joint opening. The conclusions of the current study are outlined as follows:(i)
Sensitivity and parametric study identified the main influencing factors as the geometric properties of the structural elements, the joint aspect ratio, the tensile strength of steel, the concrete compressive strength, and the longitudinal tensile reinforcement ratio.(ii)
Based on the dataset of 59 experiments, the corner joint closing model shows the best performance with the average absolute error (AAE) of 9.4%, Coefficient of variation (CoV) of 12.7%, the coefficient of determination (R2) of 0.79, and the performance factors of 1.01.(iii)
The results of the corner joint opening model show the average absolute error (AAE) of 16.0%, the Coefficient of variation (CoV) of 20% the coefficient of determination (R2) of 0.75, and the performance factors of 1.04. These all prove the efficacy of the model.(iv)
The performance of both models is checked against the expressions proposed by various design building codes. These models show better prediction of corner joint shear strength than codes, such as American Concrete Institute (ACI 352R-02) [24], Architectural Institute of Japan (AIJ) [22], New Zealand Code of practice (NZS 3101-2006) [13] and Chinese Code of practice [21] and Eurocode (EN 1998-1:2004) [20].



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*Source: 7156691-2022-08-04.xml*