Plastic hinge modeling of reinforced concrete beam-column joints using Artificial Neural Networks

Beam-column joints play a crucial role in transferring forces between frame elements and maintaining structural integrity during strong earthquakes. While the behaviors associated with beams and columns are commonly modelled in global frame analyses through plastic hinges, the behaviors of joints are often omitted through the use of rigid end offsets. The objective of this study is to develop an artificial neural network for deriving the plastic hinge curves for modeling the seismic behavior of beam-column joints in global frame analyses using line elements. For this, a feed-forward artificial neural network (FFNN) is developed to predict the shear strengths of beam-column joints. A comprehensive dataset of 598 experimental joint tests is compiled from the literature. The 555 data points, which passed the exploratory data analysis, are used to train, test, and validate the proposed network for applicability to a wide range of input variables and joint configurations. The accuracy and reliability of the proposed FFNN were evaluated using a comprehensive set of evaluation metrics in comparison to three existing networks from the literature. The proposed FFNN is used to derive the shear stress-strain and moment-rotation curves for defining the joint hinges in global frame analyses. The developed spreadsheet executes the network formulations, calculates the joint shear strength, and derives the joint hinge curves for the use of engineers and researchers in seismic analyses.

1. 18
th
World Conference on Earthquake Engineering
July 2024
Plastic Hinge Modeling of Reinforced Concrete
Beam-Column Joints using Artificial Neural
Networks
Serhan Guner, PhD, P.Eng.
Associate Professor
Nirmala Suwal
Structural Engineer
Ohio, USA

2. 2
Plastic Hinge Modeling of Joint using ANN
2
Lumped-plasticity approach
Post-yield behavior of beams and columns
modelled
Behavior of beam-column joints
commonly omitted
– Many different column configurations
– Current formulations are valid for certain
configurations
Global Frame Analysis
Plastic hinge for
beams and columns
Yielding
Maximum
Residual
Force
Deformation
Cracking
Plastic hinge for joint

3. 3
Plastic Hinge Modeling of Joint using ANN
3
Joint Failure Modes Considered
Gombosuren and Maki (2020)
Joint shear failure (S) Bond-slip failure (B)
Sasmal et al. (2011)
Combined joint shear and
beam flexure failure (S-F)
Hwang et al. (2005)

4. 4
Plastic Hinge Modeling of Joint using ANN
4
Existing Joint Models
Five types of joint modeling approaches
Beam-Column
Joint Models
Rotational Spring
Models
Machine Learning
Models
Component Models
Finite Element Models
Rigid Joint Models
Rigid End Offsets Rigid End Offsets
Zero length
rotational Spring
Alath and Kunnath (2005)
Joint core
Spring element
e.g. Lowes and Mitra (2007)
Sagbas et al. (2011)
e.g. Artificial Neural Network

5. 5
Plastic Hinge Modeling of Joint using ANN
5

Develop an artificial neural network (ANN)that predicts shear
strength of beam-column joints for wide range of configurations. 
Using the ANN output,derive joint spring curves(shear stress-
strain and moment-rotation) for global frame analysis. 
Validate joint spring curvesusing never-seen experimental data.
Objectives

6. 6
Plastic Hinge Modeling of Joint using ANN
6
Artificial Neural Network (ANN)
Artificial Neuron
Input Signals
Activation
function
Output Signals
Radial Basis Modular, etc. Convolutional
Recurrent
Feed-Forward (FFNN)
ANN Types
Biological Neuron
Input Neurons
Output Neurons
Information flow

7. 7
Plastic Hinge Modeling of Joint using ANN
7
Proposed Feed
-
Forward Neural Network (FFNN)
1. Training algorithm
Feed-forward training algorithm
with backpropagation techniques
Ser h
1
2
G
u,P
− G
P.uE
)
N
≈ 0
3. Error prediction
Weights (w) and
bias (b)
Backpropagation @ learning rate 0.01
Input
Input value = x
G
P.uE
Output
2. Activation function Net input (u) = wx+b
Activation function = Sigmoid
l h
1
a w 2
os
h G
P.uE
4. Weights and biases updated using
Adaptive moment estimation (Adam)
+
Stochastic momentum gradient descent
Accelerate convergence speed in a
consistent gradient direction
Root mean square propagation
Adjust learning rate and improve
convergence speed

8. 8
Plastic Hinge Modeling of Joint using ANN
8
Proposed Feed
-
forward neural network (FFNN)
8
12
16
20
24
28
32
36
40
0 0.5 1 1.5 2 2.5 3
No. of Neurons
Mean Squared Error
(MSE)
27
0.41
0.78
1.05
0
200
400
600
800
1000
0 1 2 3 4 5
No. of Iterations
Mean Squared Error
(MSE)
Training Testing Validation
100
0.41
0.78
1.05
Efficient configurations = Single hidden layer with 27 neurons
Optimum iterations = 100

9. 9
Plastic Hinge Modeling of Joint using ANN
9
Experimental Database
Total Joint types
273 w Interior
120 w/o
148 w Exterior
57 w/o
598 Total
Total B S-F S Failure Modes
211 2 93 116 w Exterior
110 12 19 79 w/o
132 0 88 44 w Interior
47 0 25 23 w/o
97 Unknown failure modes
598 14 255 262 Total
Total of 598 experimental specimens from 153 studies
Exterior specimen Interior specimen
w/o: without andw: with transverse reinforcement

10. 10
Plastic Hinge Modeling of Joint using ANN
10
Experimental Database
Input variables f’
c= concrete compressive strength (MPa) r
jt= joint transverse reinforcement ratio (%)
f
yjt= joint transverse reinforcement yield
strength (MPa)
r
b= beam longitudinal reinforcement ratio (%)
f
yb= beam longitudinal reinforcement yield
strength (MPa)
b
b, h
b= beam width and depth (mm)
r
c= column longitudinal reinforcement ratio (%)
f
yc = column logitudinal reinforcement strength
(MPa)
b
c, h
c= column width and depth (mm)
ALF = axial load factor (P/f
cb
ch
c)
JT = joint type (exterior or interior)
Output variable t
=
shear strength (MPa)
Exterior specimen Interior specimen

11. 11
Plastic Hinge Modeling of Joint using ANN
11
•To understand the characteristics of database, and
•detect and eliminate outliers before training.
Cook’s Distance:
Threshold limit :
Exploratory Data Analysis (EDA)
2
1
( )
i
red
n
p ipred i
pMSE
D
t
t
       
=
=
-

t
-
=
4
b
0
0.02
0.04
0.06
0.08
0.1
0 100 200 300 400 500 600
Cook's Distance (D)
Total Data Points
I
t= 0.007
43 outliers detected

12. 12
Plastic Hinge Modeling of Joint using ANN
12
Database after EDA
Exploratory Data Analysis (EDA)
Unit Max Min VariableMPa 102.0 15.8
Concrete compressive strength
(f’
c
)
% 2.6 0.0
Beam-column joint transverse
reinforcement ratio, (
r
jt
)
MPa 1374 235
Joint transverse reinforcement
yield strength, (f
yjt
)
% 4.3 0.4
Beam rebar ratio (
r
b
)
MPa 1091 286
Beam rebar yield strength (f
yb
)
mm 750 150
Depth of beam (h
b
)
mm 610 100
Width of beam (b
b
)
% 7.7 0.3
Column rebar ratio ( r
c
)
MPa 1092 274
Column rebar yield strength (f
yc
)
mm 700 140
Depth of column (h
c
)
mm 900 100
Width of column (b
c
)
% 0.7 0.0
Axial load factor (ALF)
1 0
Joint type (JT)
MPa 17.4 1.3
Shear Strength (
t
)
1 1 1
2 2
1 1 1 1
( )( )
[ ( ) ][ ( ) ]
R
n n n
i i i
n n n n
i i i i
n XY x Y
n X Y n X Y
= = =
= = = =
-
- -
=
  
    Correlation analysis between input
and output variables

13. 13
Plastic Hinge Modeling of Joint using ANN
13
•Data split for training, testing, and validation
Training, Testing, and Validation
Total
Exterior Interior
Joint Type
w w/o w w/o
445 204 91 107 43
Training (80%)
55 26 11 13 5
Testing (10%)
55 26 11 13 5
Validation (10%)
555 256 113 133 53
Total
w/o: without andw: with transverse reinforcement
Training of the network uses feed forward and back
propagation for 100 iterations.
At each iteration, calculation either continues to next
iteration or stops if maximum iterations is reached.
Testing and validation are performed by only forward
propagation and computing error.
No back propagation is performed.
Weights, biases not updated, no multiple iterations

14. 14
Plastic Hinge Modeling of Joint using ANN
14
Network Performance Evaluation Metrics
Formula Metrics
R Correlation Coefficient
R
2
Determination Coefficient
MAE Mean Absolute Error
MSE Mean Squared Error
RMSE Root Mean Squared Error
CV Coefficient of Variation
2 2 2 2
[
( )( )
( ) ][ ( ) ]
exp exppred pred
exp exp
pred pred
n
n
n
t t t t
t t t t
-
- -
  
   
2
2
1
( ) ( )
exppred
exp exp
t t t t
-
--
 1
| |
exp
pred
n
t t
-

2
1
( )
exppred
n
t t
-

2
1
( )
s
exppred
n
t t
-

100
1
pred
RMSE
n
t
´


15. 15
Plastic Hinge Modeling of Joint using ANN
15
y = 0.99x
R = 0.98
R² = 0.96
CV = 9.3%
0
3
6
9
12
15
18
0 3 6 9 12 15 18
t
pred
t
exp
Interior (w/o) Interior (w) Exterior (w/o) Exterior (w)
Evaluation of Proposed FFNN
(110 specimens)
Training
(MPa)
w/ois joints without transverse reinforcement,wis joints with transverse reinforcement
y = 0.95x
R = 0.96
R² = 0.91
CV = 14.3%
0 3 6 9 12 15 18
t
exp
(MPa)
Testing
y = 0.95x
R = 0.93
R² = 0.86
CV = 15.1%
0 3 6 9 12 15 18
t
exp
(MPa)
Validation
Meanpredicted-to-experimental shear strength ratio = 0.99
and
CV = 14.7% for 110 specimens from testing and validation
(MPa)

16. 16
Plastic Hinge Modeling of Joint using ANN
16
Comparison with Other Networks
Kotsovou et al.
(2017)
Haido (2022) Alagundi and
Palanisamy (2022)
Proposed FFNN Network
Components
Feed-forward
multilayered
network
FFNN Levenberg-marquardt
ANN
FFNN Training algorithm
Sigmoid Tan-Sigmoid Tan-sigmoid Sigmoid Activation function
MSE MSE MSE MSE Prediction error
Gradient descent Gradient descent Gradient descent Adam Optimization function
Not reported Not reported 0.001 0.01 Learning rate
12-9-01 17-22-01 11-12-01 13-27-01 (Input-
Hidden-Output)
Network configuration
100 1000 100 100 Iterations

17. 17
Plastic Hinge Modeling of Joint using ANN
17
Global performance comparison
Comparison with Other Networks
0
3
6
9
12
15
18
0 3 6 9 12 15 18
t
pred
t
exp
Proposed FFNN Alagundi and Palanisamy (2022) Haido (2022) Kotsovou et al. (2017)
(MPa)
(MPa)
xR=0.98, R
2
=0.96
R=0.97, R
2
=0.95
R=0.98, R
2
=0.96
R=0.95, R
2
=0.93
Training
0 3 6 9 12 15 18
t
exp
(MPa)
xR=0.96, R
2
=0.91
R=0.96, R
2
=0.87
R=0.90, R
2
=0.81
R=0.88, R
2
=0.77
Testing
0 3 6 9 12 15 18
t
exp
x R=0.93, R
2
=0.86
R=0.89, R
2
=0.80
R=0.91, R
2
=0.83
R=0.90, R
2
=0.81
(MPa)
Validation

18. 18
Plastic Hinge Modeling of Joint using ANN
18
Comparison with Other Networks
CV(%) RMSE MSE MAE Network14.7% 0.95 0.92 0.72
Proposed FFNN
17.5% 1.16 1.36 0.84
Alagundi and Palanisamy
(2022)
17.8% 1.21 1.47 0.91
Haido (2022)
19.0% 1.29 1.68 0.94
Kotsovou et al. (2017)
Global performance comparison

19. 19
Plastic Hinge Modeling of Joint using ANN
19
Comparison with Other Networks
0
0.4
0.8
1.2
1.6
0
0.4
0.8
1.2
R R
2
MAE MSE RMSE
TestingValidation
Proposed FFNN
Alagundi and Palanisamy (2022)
Haido (2022)
Kotsovou et al. (2017)
Proposed FFNN
Alagundi and Palanisamy (2022)
Haido (2022)
Kotsovou et al. (2017)
Proposed FFNN Proposed FFNN
Alagundi and Palanisamy (2022) Alagundi and Palanisamy (2022)
Haido (2022) Haido (2022)
Kotsovou et al. (2017) Kotsovou et al. (2017)
R R
2
MAE MSE RMSE
(a) Interior joints without transverse reinforcement
(c) Exterior joints without transverse reinforcement
0
0.3
0.6
0.9
1.2
1.5
0
0.3
0.6
0.9
1.2
R R
2
MAE MSE RMSE R R
2
MAE MSE RMSE
0
0.5
1
1.5
2
2.5
0
0.5
1
1.5
2
2.5
TestingValidation
R R
2
MAE MSE RMSE
R R
2
MAE MSE RMSE
(b) Interior joints with transverse reinforcement 0
0.3
0.6
0.9
1.2
1.5
(
d) Exterior joints with transverse reinforcement
0
0.45
0.9
1.35
1.8
R R
2
MAE MSE RMSE R R
2
MAE MSE RMSE
Higher the better
Lower the better
Testing:
Validation:
Local performance comparison

20. 20
Plastic Hinge Modeling of Joint using ANN
20
Comparison with Other Networks
Exterior Interior
w w/o w w/o
17.1% 14.8% 10.7% 11.9%
Proposed FFNN
20.5% 19.7% 12.5% 13.7%
Alagundi and Palanisamy
(2022)
17.5% 26.9% 14.6% 13.9%
Haido (2022)
21.2% 21.3% 13.2% 17.5%
Kotsovou et al. (2017)
Networks
CV (%)
Local performance comparison

21. 21
Plastic Hinge Modeling of Joint using ANN
21
Predicted-to-experimental shear strength ratio comparisons in terms of
input variables
Comparison with Other Networks
Network
CV (%)
(a)Concrete compressive
strength, f’
c
(MPa)
(c)Joint transverse reinforcement
yield strength, f
yjt
(MPa)
0
0.5
1
1.5
2
2.5
0 200 400 600 800 1000
t
pred/t
exp
0
0.5
1
1.5
2
2.5
0.0% 1.0% 2.0% 3.0%
t
pred/t
exp
(b)Joint transverse
renforcement ratio,
r
jt
0
0.5
1
1.5
2
2.5
0 200 400 600 800 1000
t
pred/t
exp
(e)Beam longitudinal reinforcement
yield strength, f
yb
(MPa)
0
0.5
1
1.5
2
2.5
0.0% 1.5% 3.0%
t
pred/t
exp
(d)Beam longitudinal
reinforcement ratio,
r
b
0
0.5
1
1.5
2
2.5
0 200 400 600 800 1000
t
pred/t
exp
(h)Column longitudinal reinforcement
yield strength, f
yc
(MPa)
0
0.5
1
1.5
2
2.5
0.0% 2.0% 4.0% 6.0%
t
pred/t
exp
(g)Column longitudinal
reinforcement ratio,
r
c
0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8
t
pred/t
exp
(j)Axial load factor (ALF)
(k)Joint type (JT)
0
0.5
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1
1.2
t
pred/t
exp
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5
t
pred/t
exp
(f)Beam aspect ratio (AR
beam
)
AR
beam
= beam depth/beam width
0
1
2
3
4
0 20 40 60 80 100 120
t
pred/t
exp
Proposed FFNN Alagundi (2022) Haido (2022) Kostosvou et al. (2017)
(i)Column aspect ratio (AR
column
)
AR
column
= column depth/width
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5
t
pred/t
exp

22. 22
Plastic Hinge Modeling of Joint using ANN
22
General shear stress-strain curve
Deriving Joint Spring Curves
Strain Shear Damage State Point
g
1
t
crack Cracking 1
g
2 0.95
t
max Yielding 2
g
3
t
max Maximum 3
g
4 0.2
t
max Residual 4
Yielding
Maximum
Residual
Shear stress (
t
)
Cracking
Shear strain (
g
)
where,
t
crack= 0.48√f’
c t
max=
t
pred(Proposed FFNN)
(Anderson et al. 2008)
Rigid End Offsets
Zero length
rotational Spring
Rotational joint springs:
Simple and reasonably accurate

23. 23
Plastic Hinge Modeling of Joint using ANN
23
Shear strain values for different types
of joints
Deriving Joint Spring Curves
Exterior (x10
-3
) Interior (x10
-3
) Damage
State
w w/o w w/o
0.43 Cracking (
g
1)
6 Yielding (
g
2)
20 16 20 19 Maximum( g
3)
185 77 187 117 Residual (
g
4)
Yielding
Maximum
Residual
Shear stress (
t
)
Cracking 0.48√f’
c
0.20
t
pred
0.43
0.95
t
pred
t
pred
6
g
3
g
4
Shear strain (
g
x 10
-3
)
Shear stress-strain curve

24. 24
Plastic Hinge Modeling of Joint using ANN
24
Deriving Joint Spring Curves
Shear stress-strain curve is transformed into equivalent moment
rotation curve
S
v.fvc
=
G
v.fvc

!
1 − "
v
#
$
⁄&⁄ℎ
$
#
v
where,
A
j= joint area (A
j= h
bb
b)
L
b =beam span between points of contraflexure
j =0.875
h
b =beam depth
b
b = beam width
L
c = column height between two points of contraflexure
Celik and Elingwood (2008) S
Rf,
=
G
P.uE

!
1 − "
v
#
$
⁄&⁄ℎ
$
#
v
( = )
Joint Moment (M)
Rotation(
q
x10
-3
)
M
crack
0.2M
max
0.95M
max
M
max
0.436
q
3
=g
3
q
4
=g
4
Yielding
Maximum
Residual
Cracking

25. 25
Plastic Hinge Modeling of Joint using ANN
25
Spreadsheet
:
ANN Joint Hinge Generator

26. 26
Plastic Hinge Modeling of Joint using ANN
26
•Important to demonstrate the load-drift response of joints are captured
accurately using the proposed derived curves.
•Four large-scale specimens were analyzed.
Experimental
Validation
Exterior Interior Variables
w w/o w w/o
1B (Eshani and
Wight, 1985)
Unit 5 (Pantelides
et al., 2002)
BJ3 [54] (Li et al.,
2009)
AL2 (Li et al.,
2009)
Specimen
33.6 31.7 40.0 32.1
f’
c
(MPa)
1.3 0.0 0.5 0.0
r
jt
(%)
437 0 510 0
f
yjt
(MPa)
1.8 3.5 1.3 2.2
r
b
(%)
331 459 510 473
f
yb
(MPa)
480 406 400 400
h
b
(mm)
259 406 300 200
b
b
(mm)
2.5 2.8 6.3 3.1
r
c
(%)
490 470 514 473
f
yc
(MPa)
300 406 350 200
h
c
(mm)
300 406 350 400
b
c
(mm)
0.1 0.3 0.0 0.0
ALF
1 1 0 0
JT
6.8 5.2 9.4 6.3
t
exp
(MPa)
6.3 4.9 8.8 6.0
t
pred
(MPa)
440 346 678 376
M
max
(kNm)

27. 27
Plastic Hinge Modeling of Joint using ANN
27
Validation:
Derived Joint Spring Curves
Interior joint (w/o) Interior joint (w)
Exterior joint (w/o)
Exterior joint (w)

28. 28
Plastic Hinge Modeling of Joint using ANN
28
•SAP2000 software was used.
Validation:
Global Frame Models
Exterior joint Interior joint

29. 29
Plastic Hinge Modeling of Joint using ANN
29
Validation:
Load
-
Drift Responses
Exterior Joint (w/o)
Exterior Joint (w)
Interior Joint (w/o)
Interior Joint (w)
Failure Mode P
pred/P
exp P
pred (kN) P
exp (kN) Specimen Joint type
S 0.93 76 82 AL2
Interior (w/o)
S-F 0.99 192 195 BJ3
Interior (w)
S 0.95 183 194 Unit 5
Exterior (w/o)
S 1.02 156 154 1B
Exterior (w)
0.97
Mean ratio
3.6%
Coefficient of variation (CV)

30. 30
Plastic Hinge Modeling of Joint using ANN
30
SpreadsheetANN Joint Hinge Generator •Open access download link:
www.utoledo.edu/engineering/faculty/serhan-guner/docs/8S-
ANNJointHingeGenerator.xlsx
•YouTube Video:
https://youtu.be/XuIXbS7cPMw
Publications
Open access download link: www.utoledo.edu/engineering/faculty/ser han-guner/publications.html

31. 31
Plastic Hinge Modeling of Joint using ANN
31
•Databaseused in the training and testing of a neural networkplays critical rolein
identifying the optimum parameters and network layout.
•Exploratory data analysisis useful fordetecting and eliminate outliers.
•Proposed FFNN is shown to predict the shear strengths of joints rapidly and
accurately. The predicted-to-experimental shear strength ratios provideda mean of
0.99and acoefficient of variation of 14.7%for 110 specimens.
•The proposed FFNN is shown to provide more accurate and reliable response
simulations than three existing networks:4.3% increasein correlation coefficient:
8.7% increasein determination coefficient; a 18.7% decrease in coeff. ofvariation.
•The approach presented for deriving the shear stress-strain and moment-rotation
curves enables the application of the proposed network in plastic-hinge-based frame
analyses.
•The proposed curves are shown to predict the load-drift responses of four large-
scale joints witha mean of 0.97and acoefficient of variation of 3.6%.
Conclusion

32. Questions?
Plastic Hinge Modeling of Reinforced Concrete Beam-
Column Joints using Artificial Neural Networks
THANK YOU!
Serhan Guner,
Associate Professor
Nirmala Suwal
,
Structural Engineer
Contact:
[email protected]
website:
www.utoledo.edu/engineering/faculty/serhan-guner
University of Toledo
Ohio, USA