structural members and in turn the whole structure when exposed to various ..... square and rectangular columns confined with bonded FRP jackets, and ..... component planes, an example was the close relation of the petal length and.
ARTIFICIAL NEURAL NETWORK MODELING OF CONFINEMENT OF CARBON FIBER REINFORCED POLYMER AS RETROFITTING MATERIALS IN COLUMNS
A Thesis Presented to the faculty of Civil Engineering College of Engineering De La Salle University – Manila
In Partial Fulfillment of the Requirements for the Degree of Bachelor of Science in Civil Engineering Major in Construction Technology and Management
Ken Patrick G. Mendoza Ryan Ray R. Tecson Maila Marie T. Victorino
April 2011
RECOMMENDATION SHEET
“Artificial Neural Network Modeling of Confinement of Carbon Fiber Reinforced Polymer as Retrofitting Materials in Columns”
Prepared and Submitted by:
Mendoza, Ken Patrick G. Tecson, Ryan Ray R. Victorino, Maila Marie T.
In partial fulfillment of the requirements for the degree of Bachelor of Science in Civil Engineering, Major in Construction Technology and Management, has been examined and is recommended for Oral Defense.
Engr. Jason Maximino C. Ongpeng Thesis Adviser
ii
APROVAL SHEET
“Artificial Neural Network Modeling of Confinement of Carbon Fiber Reinforced Polymer as Retrofitting Materials in Columns”
Prepared and Submitted by:
Mendoza, Ken Patrick G. Tecson, Ryan Ray R. Victorino, Maila Marie T.
In partial fulfillment of the requirements for the Degree of Bachelor of Science in Civil Engineering, Major in Hydraulics and Water Resources Engineering, has been examined and is recommended for acceptance. The Oral Defense Panel:
Dr. Andres Winston C. Oreta Chairman of the Panel
Engr. Ronaldo S. Gallardo Panelist
Engr. Alden Paul D. Balili Panelist
Accepted as partial fulfillment of the requirements for the Degree in Bachelor of Science in Civil Engineering, Major in Construction Technology and Management.
Engr. Ronaldo S. Gallardo Chairman, CE Department
Dr. Pag-asa D. Gaspillo Dean, College of Engineering
iii
ACKNOWLEDGEMENTS
This study would not have been possible without the assistance of several individuals who contributed in different ways for the completion of this study and now it is with great pleasure to thank those who made this study possible.
First and foremost, the group is heartily thankful to Engr. Jason Ongpeng, our thesis adviser, whose encouragement, guidance and support from the initial to the final level enabled us to develop an understanding of the study.
Second, the group‘s utmost gratitude to Ms. Kheem Madrid, a fellow student at DLSU-M and our programmer, for sharing her time and extending her assistance on the programming needs of this study.
The group would also like to thank Ms. Phoebee Que for teaching and sharing the group her expertise on the MATLAB program and for sharing her group‘s thesis for reference.
To Mendoza, Tecson, and Victorino family, for the financial support, love, and motivation they have given to us during the course of completing this study.
iv
To the authors of the scientific journals who willingly shared their research studies and data for the completion of this study‘s database.
To the members of the panel namely: Dr. Andres Oreta, Engr. Ronaldo Gallardo, and Engr. Alden Balili for sharing their insights, constructive criticisms and expertise on improving our thesis.
Last but not the least, the one above all of us, the omnipresent God, for answering our prayers and for giving us the strength to plod on despite our constitution wanting to give up, thank you so much Dear Lord.
v
ABSTRACT
The utilization of Carbon Fiber Reinforced Polymer (CFRP) as a retrofitting material has proven its strengthening effects on existing circular, square and rectangular concrete columns. To further develop the use of CFRP in the field of civil engineering, theoretical models are needed to predict the effectiveness of CFRP confinement in columns with regards to its compressive strength.
The Self-Organizing Map (SOM) toolbox was used to classify data according to the parameters that have similarities which is observed by the toolbox. Each grouping classified through SOM that showed evident relationship among its parameters was observed, analyzed and was related to the ultimate confined compressive strength and increase in strength of the confined columns. From the analysis of the best SOM models, parameters which has significant effect on the CFRP-confinement were then chosen to be used for the back-propagation models. These back-propagation models were then compared to existing models by different authors to verify its accuracy.
Three artificial neural networks consisting of circular and non-circular data were developed to predict the ultimate confined compressive strength (f‘cc). The parameters that were considered in the back-propagation model
vi
are volumetric ratio of carbon fiber (ρcfrp), volumetric ratio of steel (ρs), unconfined compressive strength (f‘co),and the column‘s geometrical properties (b, h, and L). The models performed better than some models with regard to their correlation (R).
Ken Patrick G. Mendoza
Ryan Ray R. Tecson
Maila Marie T. Victorino
vii
TABLE OF CONTENTS
CHAPTER
I.
II.
TITLE
PAGE
Problem Setting
1.1 Introduction
1
1.2 Problem Statement
3
1.3 Objectives
4
1.4 Hypothesis
5
1.5 Scope, Limitations and Delimitations
5
1.6 Significance of the Study
6
1.7 Assumptions
8
1.8 Definition of Terms
8
Review of Related Literature
2.1 Carbon Fiber Reinforced Polymer
10
2.2 Experimental Research on CFRP-confined columns
13
viii
2.2.1 Circular Columns
13
2.2.2 Non-Circular Columns
18
2.3 Proposed Models for CFRP-Confined Columns
23
2.4 Other Proposed Models
30
2.4.1 Proposed Models for Circular Columns
30
2.4.2 Proposed Models for Rectangular and Square Columns
34
2.5 Artificial Neural Network
39
2.5.1 Self Organizing Map Toolbox and Backpropagation 40
2.5.2 ANN Application in Concrete Modeling
III.
41
Theoretical and Conceptual Frameworks
3.1 Theoretical Framework
44
3.1.1 CFRP confinement of columns
45
3.1.2 Steel Reinforcement
52
3.1.3 Self Organizing Map
54
3.1.4 Artificial Neural Network
56
ix
3.1.5 Evaluation of the ANN Models
3.2 Conceptual Framework
IV.
V.
59
60
Research Methodology and Research Design
4.1 Data Gathering
62
4.2 Normalization of Data
63
4.3 Self Organizing Map
63
4.4 Training and Testing of the ANN
64
4.5 Selection of Best ANN Model
64
4.6 Comparison with existing models
64
Data Presentation and Analysis
5.1 Data Gathering
67
5.2 Self-Organizing Map
72
5.2.1 Self-Organizing Map Models
73
5.2.1.1 Naming of the SOM Models
73
5.2.1.2 SOM Models
73
x
5.2.2 SOM Results
75
5.2.2.1 Circular Columns
75
5.2.2.2 Non-Circular Columns
88
5.2.2.3 Circular and Non-Circular Columns
5.3 Backpropagation
100
114
5.3.1 Backpropagation Models
116
5.3.2 Backpropagation Results
118
5.3.2.1 Combined circular and non-circular w/o steel
118
5.3.2.1.1 BP-CNC-NS
118
5.3.2.1.2 Comparison with existing models
122
5.3.2.2 Combined circular and non-circular with steel
126
5.3.2.2.1 BP-CNC-S
126
5.3.2.2.2 Comparison with existing models
132
5.3.2.3 Combined circular and non-circular with and without steel
xi
135
5.3.2.3.1 BP-CNC-SNS
5.4 Parametric Studies
VI.
135
139
Conclusions and Recommendations
143
Bibliography
146
Appendix A - Notations
153
Appendix B - Tables of Data Gathered
155
Appendix C - Self-Organizing Map and Backpropagation
167
Appendix D - Java Program
174
xii
LIST OF FIGURES
FIGURE
TITLE
PAGE
3.1
Theoretical Framework
45
3.2
Properties of Different Fibers and Typical Reinforcing Steel
47
3.3
Stress-Strain Diagram of Unconfined Concrete vs CFRPConfined Concrete (Lam & Teng)
48
3.4
Corner Radius of Circular and Non-Circular Columns
51
3.5
Longitudinal Reinforcements and Area of the Concrete Core of Columns
52
3.6
2D Graph of Ungrouped and Grouped
55
3.7
U-matrix, Component Plane, Bar Graph with 2D Graph
56
3.8
Sample Structure of an ANN Model for the Prediction of the Increase in Compressive Strength
58
3.9
Conceptual Framework
61
4.1
Methodology of the Study
66
5.1
SOM 2-3 C-NS (ρcfrp, f‘co: 3 GROUPS)
75
5.2
2D Graph of Ungrouped and Grouped SOM 2-3 C-NS
75
xiii
5.3
Histograms with Bar Graphs (SOM 2-3 C-NS)
76
5.4
U-matrix and Component Planes of SOM 2-3 C-NS
77
5.5
SOM 1-2 C-S (ρs, ρcfrp, f‘co :2 GROUPS)
79
5.6
2D Graph of Ungrouped and Grouped SOM 1-2 C-S
80
5.7
Histograms with Bar Graphs (SOM 1-2 C-S)
81
5.8
U-matrix and Component Planes of SOM 1-2 C-S
81
5.9
SOM 2-2 C-SNS (ρs, ρcfrp: 2 GROUPS)
83
5.10
2D Graph of Ungrouped and Grouped SOM 2-2 C-SNS
84
5.11
Histograms with Label (SOM 2-2 C-SNS)
85
5.12
Histograms with Bar Graphs (SOM 2-2 C-SNS)
85
5.13
U-matrix and Component Planes of SOM 2-2 C-SNS
86
5.14
2D Graph of SOM 2-2 NC-NS
88
5.15
2D Graph of Ungrouped and Grouped SOM 2-2 C-SNS
88
5.16
Histograms with Bar Graphs (SOM 2-2 NC-NS)
89
5.17
U-matrix and Component Planes of SOM 2-2 NC-NS
90
xiv
5.18
2D Graph of Ungrouped and Grouped SOM 4-2 NC-S
92
5.19
Histograms with Bar Graphs (SOM 4-2 NC-S)
93
5.20
U-matrix and Component Planes of SOM 2-2 NC-NS
94
5.21
3D Graph of SOM 3-2 NC-SNS
96
5.22
2D Graph of Ungrouped and Grouped SOM 3-2 NC-SNS
96
5.23
Histograms with Label (SOM 3-2 NC-SNS)
97
5.24
Histograms with Bar Graphs (SOM 3-2 NC-SNS)
97
5.25
U-matrix and Component Planes of SOM 3-2 NC-SNS
98
5.26
3D Graph of SOM 1-2 CNC_NS
100
5.27
2D Graph of Ungrouped and Grouped SOM 1-2 CNC-NS
101
5.28
Histograms with Label (SOM 1-2 CNC-NS)
102
5.29
Histograms with Bar Graphs (SOM 1-2 CNC-NS)
102
5.30
U-matrix and Component Planes of SOM 1-2 CNC-NS
103
5.31
2D Graph of Ungrouped and Grouped SOM 2-2 CNC-S
105
5.32
Histograms with Label (SOM 2-2 CNC-S)
106
xv
5.33
Histograms with Bar Graphs (SOM 2-2 CNC-S)
106
5.34
U-matrix and Component Planes of SOM 2-2 CNC-S
107
5.35
2D Graph of Ungrouped and Grouped SOM 5-3 CNC-SNS 109
5.36
Histograms with Label (SOM 5-3 CNC-SNS)
110
5.37
Histograms with Bar Graphs (SOM 5-3 CNC-SNS)
110
5.38
U-matrix and Component Planes of SOM 5-3 CNC-SNS
112
5.39
Regression Analyses of BP-CNC-NS
119
5.40
BP-CNC-NS Model
121
5.41
Al-Salloum‘s Model
122
5.42
Lam and Teng‘s Model
123
5.43
Ilki et al.‘s Model
124
5.44
Salehian and Kheyroddin‘s Model
125
5.45
Regression Analyses of BP-CNC-S
128
5.46
BP-CNC-S (Circular) Evaluation
130
5.47
BP-CNC-S (Non-Circular) Evaluation
131
xvi
5.48
ACI‘s 440 Model
132
5.49
Hosotani and Kawashima Model
133
5.50
Regression Analyses of BP-CNC-SNS
136
5.51
Evaluation of BP-CNC-SNS
138
5.52
f‘co vs. Predicted f‘cc/f‘co with varying number of CFRP layers
139
5.53
Varying h vs. predicted f‘cc
140
5.54
Varying f‘co and b/h
141
xvii
LIST OF TABLES
TABLE
TITLE
PAGE
2.1
Proposed Models for CFRP-Confined Columns
29
2.2
Efficiency Factor and Shape Factor of the Proposed Shaped Models
29
2.3
Proposed Models for Circular Columns
33
2.4
Proposed Models for Non-Circular Columns
38
3.1
Mechanical Properties of Common Strengthening Materials
47
3.2
Table of Parameters for ANN Modeling
59
4.1
Models Used for the Evaluation of Backpropagation Models
65
5.1
Summary of Database of Circular Columns
68
5.2
Summary of Database of Non-Circular Columns
69
5.3
Statistics of Circular Columns without Steel Reinforcements 70
5.4
Statistics of Circular Columns with Steel Reinforcements
70
5.5
Statistics of Non-Circular Columns without Steel Reinforcements
71
xviii
5.6
Statistics of Non-Circular Columns with Steel Reinforcements
71
5.7
Self-Organizing Map Models
74
5.8
Statistical Parameters for Each Group of SOM 2-3 C-NS
78
5.9
Statistical Parameters for Each Group of SOM 1-2 C-S
82
5.10
Statistical Parameters for Each Group of SOM 2-2 C-SNS
87
5.11
Statistical Parameters for Each Group SOM 2-2 NC-NS
91
5.12
Statistical Parameters for Each Group of SOM 2-2 NC-S
95
5.13
Statistical Parameters for Each Group of SOM 2-2 NC-SNS 99
5.14
Statistical Parameters for Each Group of SOM 1-2 CNC-NS 104
5.15
Statistical Parameters for Each Group of SOM 2-2 CNC-S
108
5.16
Statistical Parameters for Each Group of SOM 5-3 CNC-SNS
113
5.17
Backpropagation Models
116
5.18
Error Analysis of BP-CNC-NS
118
5.19
Weight and Biases obtained from BP-CNC-NS
120
5.20
Other Proposed Models versus BP-CNC-NS
126
xix
5.21
Error Analysis of BP-CNC-S
127
5.22
Weight and Biases obtained from BP-CNC-S
129
5.23
Other Proposed Models versus BP-CNC-S
134
5.24
Error Analysis of BP-CNC-SNS
135
5.25
Weight and Biases obtained from BP-CNC-SNS
137
B.1
Data of Circular Columns
156
B.2
Data of Square and Rectangular Columns
164
xx
CHAPTER I PROBLEM SETTING
1.1 Introduction
The use of fiber reinforced polymers has been esteemed for its strengthening effects in the rehabilitation of existing circular, square and rectangular columns. Previous studies are more focused on circular column designs, as to cope up with this, the study explores on the relationship of the square and rectangular column parameters with that of the circular columns through an artificial neural network model.
Concrete is one of the most essential and commonly used construction material. Civil engineers have been finding ways to improve the performance of concrete structures, and one way is through confinement. To make the behavior of reinforced concrete better in terms of ductility, confinement can be provided by steel in the transverse direction, or with other materials such as Fiber Reinforced Polymer (FRP). FRP‘s are usually applied on the surface of the concrete column by an adhesive material. Confined concrete has more compressive strength and ductility when compared with plain concrete.
The idea of confined concrete has been around since 1906, when it was observed that there was an added benefit in strength and deformation
1
when hydrostatic pressure was applied to concrete. Richart et al determined that the confined axial strength compared to that of unconfined was about 4.1 times the confining pressure value (Lorenzis, 2001).
FRP composites can be classified into three: glass-fiber reinforced polymer (GFRP), aramid-fiber reinforced polymer (AFRP), and carbon-fiber reinforced polymer (CFRP). These three have the same stress-strain behavior which is linear elastic up to final brittle rupture. FRP do not exhibit the ductility that steels have. Nonetheless, when used as confinement for concrete, these can greatly enhance the strength and ductility of columns. Out of the three FRP composites, CFRP demonstrates the best quality. (R. Benzaid et al., 2009)
Carbon Fiber Reinforced Polymer has become a notable material for its applications in retrofitting existing concrete structures and in concrete reinforcement instead of the traditional steel. CFRP have been known to be more cost efficient and more effective than that of steel (Li et al., 2009). This study would analyze the strength performance of CFRP in concrete for further improvements in the material‘s use in construction.
The study uses artificial neural networks for the prediction of the ultimate confined compressive strength of a concrete column confined with CFRP. Artificial neural network was modeled after the human brain thereforeit
2
acts like the mind which is highly complex, nonlinear, and a parallel computer. It has the capability to perform certain computations through its neurons (Haykin, 2009). The consideration of the behavior of concrete is greatly desired, thus the continued study allows those engaged in the behavior to construct models that may be used in the future to accurately predict the behavior of concrete. The goal of accurately predicting the strength response of confined concrete keeps researchers striving to improve the existing models or to make new models that have more correctly predicted results. These many models, studies, and theories reveal that many factors have an effect on the stress-strain diagram of concrete. Some of these items have been identified in the various existing models, but due to the number of items influencing the diagrams, only selected parameters are chosen to be evaluated depending on the scope of the model. To test the validity of the model, Self Organizing Map (SOM) and backpropagation is used for training and investigating the accuracy of the predicted models.
1.2 Problem Statement
Advancements for strengthening concrete structures are being continuously studied for improving the performance of concrete structures. CFRP, being one of the materials most utilized, has been very successful with strengthening and retrofitting concrete structures. Many experimental studies were performed to test its capabilities and these studies have proven its
3
excellent performance. However, more experiments are focused on circular columns where in fact most high rise concrete structures nowadays use rectangular columns for their infrastructures.
Most of the existing models that examine CFRP confined columns use circular cross sections since the confinement of such cross section is easier to investigate than that of non-circular cross sections. There are also existing models that studied non circular columns but the studies need further explorations.
To further develop the use of CFRP in the civil engineering field, a software that would evaluate the strength performance of CFRP on concrete structures is necessitated. This software will be developed through the Artificial Neural Network by predicting the ultimate confined compressive strength of CFRP confined RC columns.
This ANN model can be a
fundamental instrument in the design of the strength of concrete structures.
1.3 Objectives
The study aims to create an Artificial Neural Network model to predict the ultimate confined compressive strength of CFRP-confined circular, square, and rectangular columns.
4
Specific Objectives To develop a neural network model wherein both circular and rectangular column parameters are utilized for a single model. To identify the significant parameters affecting the confinement of CFRP of circular, square, and rectangular columns. To produce a java program which outputs the ultimate confined compressive strength and increase in strength of CFRPconfined circular, square, and rectangular columns.
1.4 Hypothesis
The utilization of ANN can be an effective way in modeling the confinement effect of CFRP confinement of columns with different crosssection: circular, square, and rectangular columns by using the experimental data that the group has gathered.
1.5 Scope, Limitations and Delimitations
The scope of the study is limited to the application of CFRP sheets as a confinement on reinforced concrete columns. It is concentrated mainly in analyzing the confinement effect of CFRP as retrofitting materials in circular, square and rectangular columns. The study is limited on the data from previous researches since the group will not be conducting experimental
5
tests. The data are limited to circular, square and rectangular concrete columns that are axially loaded, non-cyclic and have the significant parameters that can affect the ultimate confined compressive strength of confined concrete columns.
Data gathered from previous researches on carbon fiber reinforced concrete columns are classified into groups using SOM to obtain the best model to be followed by the prediction of the ultimate confined compressive strength through back-propagation method and the evaluation of the ANN model will be made through Pearson product-moment correlation coefficient results analysis and comparison with previous models.
1.6 Significance of the Study
Concrete confinement has become a popular practice for strengthening and retrofitting concrete structures. It improves the compressive strength and ductility of concrete thus extending the structures service life and performance. Steel confinement was the first to be utilized but at present FRP composites such as Carbon-Fiber Reinforced Polymer is now widely used. CFRP have many applications in civil engineering due to their high strengthto-weight ratio that leads to great ease in site handling, reducing labor cost, and interruptions to existing services. It also has high corrosion resistance
6
which
ensures
durable
performance
as
well
as
nonmagnetic
and
nonconductive properties, in comparison to steel plates.
Many studies have been made to study the effectiveness of the confinement of reinforced concrete columns with CFRP and the data results from these studies proved that CFRP confinement improved remarkably the axial load carrying capacity and ductility characteristics of RC columns (Seracino, 2004). Consequently, development of models for the evaluation and prediction of the strength enhancement of CFRP confined RC columns followed. These models can be a fundamental instrument in the design of the strength of concrete structures. Therefore, accurate predictions should be generated by these models.
Previous models have been made before but are limited to some extent possibly due to inadequate data used for training the ANN model, algorithms used and the method in which the ANN model was trained. This study will utilize more data from previous studies on the strengths of circular and rectangular columns to produce an ANN model in which the data from the circular and non-circular columns is utilized.
7
1.7 Assumptions
The researchers assume that the experimental results gathered from previous studies of CFRP-confined reinforced concrete columns with circular, square, and rectangular cross-sections have yielded acceptable outcomes. The data collected will be critically analyzed to have a reliable database.
1.8 Definition of Terms
Carbon Fiber Reinforced Polymer (CFRP) – has high strength to weight ratio, non-corrosive composite material and is much easier to install than steel.
Artificial Neural Network (ANN) – are composed of interconnected artificial neurons that functions like a brain and learns from several given data and algorithms. It is an information processing model that consists of a great number of simple interconnected processing units called neurons.
Normalization of data - makes all input parameters in the same range using a linear equation having a limit that ranges from a minimum value of 0.1 up to a maximum value of 0.9 in the database.
Self-organizing maps (SOM) - are data visualization technique to simplify high dimensional data onto something that humans could recognize easily.
8
Back-propagation Neural Network – in this type of network, it requires a teacher that can calculate the desired output for any given input where the errors propagate backwards from the output nodes to the inner nodes.
Levenberg-Marquardt
Algorithm
-
provides
numerical
solutions
onto
minimizing problems in curve fitting over a space of parameters of the function.
Training set – is the data used to train the artificial neural network
Testing set – is the data use to test the artificial neural network to determine the performance of the network
R (Pearson product-moment correlation coefficient) - is a common measure of the correlation between two variables X and Y that ranges from -1 to +1.
9
CHAPTER II REVIEW OF RELATED LITERATURE
2.1 Carbon Fiber Reinforced Polymer
To make the behavior of reinforced concrete better in terms of ductility, confinement can be provided by steel in the transverse direction, or with other materials such as Fiber Reinforced Polymer (FRP). FRP‘s are usually applied on the surface of the concrete column by an adhesive material. Confined concrete has more compressive strength and ductility when compared with plain concrete.
The idea of confined concrete has been around since 1906, when it was sawn that there was an added benefit in strength and deformation when hydrostatic pressure was applied to concrete. Richart et al determined that the confined axial strength compared to that of unconfined was about 4.1 times the confining pressure value (Lorenzis, 2001).
Over the years, confinement types varied. Traditionally, steel hoops, ties or spirals have been used. However, with the advancement of FRP, the use of this material has increased. The increase is in part due to the inherent properties of FRP, such as extremely high strength-to-weight ratio, high tensile strength and modulus, good corrosion behavior, electromagnetic
10
neutrality, durability, and its ease of use for multiple applications in new construction. FRP can be used to preserve many structures that have sustained damage over time due to environmental conditions, such as damage encountered due to seismic activity.
Application of the lateral pressure was achieved by using hydrostatic pressure in the early experiments and models. The effect of lateral pressure in enhancement of compressive strength and deformation capacity of reinforced concrete, or ductility was documented by some researchers. Confinement is defined as restricting the lateral dilation of concrete. This initially was accomplished by using transverse reinforcement in the form of spirals, circular hoops or rectangular ties, or by encasing the concrete columns into steel tubes that act as permanent formwork (De Lorenzis, 2001). Confinement allows for improvement in concrete strength and deformation. The novelty of FRP allowed for materials other than steel to be used as confinement. FRP is available in many forms from different materials. The FRP allows the concrete to be externally wrapped with sheets, tapes or tubes. The most common materials used for FRP are carbon, glass and aramid. Glass Fiber Reinforced Polymer (GFRP), Carbon Fiber Reinforced Polymer (CFRP) and some other types of reinforcing fibers have been used to retrofit the existing reinforced concrete structural members to enhance their force and displacement capacities, or in new structures to increase their strength, ductility and their resistance against environment. Understanding the behavior of any
11
construction material is the way in design guidelines. The monotonic and cyclic stress-strain response of plain concrete as well as concrete confined by conventional reinforcement or FRP plays a key role in the response of structural members and in turn the whole structure when exposed to various loading conditions. It is important to understand further the properties and behavior of concrete, so that design of concrete structures and structural elements is possible. The consideration of the behavior of concrete is greatly desired, thus the continued study allows those engaged in the behavior to construct models that may be used in the future to accurately predict the behavior of concrete. The goal of accurately predicting the stress-strain response of confined concrete keeps researchers striving to improve the existing models or to make new models that have more correctly predicted results.
These many models, studies, and theories reveal that many factors have an effect on the stress-strain diagram of concrete. Some of these items have been identified in the various existing models, but due to the number of items influencing the diagrams, only selected parameters are chosen to be evaluated depending on the scope of the model. These parameters can include the following: testing conditions (type of machine, loading rate, duration of load, and load history), physical parameters of the specimens (such as size and shape), the size and location of strain gauges, the number of cycles, the age of the concrete, and the material makeup of the concrete
12
including type and quantity of aggregate, as well as concrete strength (Popovics, 1973).
Many models exist to evaluate reinforced and confined concrete. Some of the different parameters incorporated in confinement models include the type of confinement, such as traditional steel type—spirals and hoops, rectilinear ties and concrete filled-steel tubular columns—and FRP. Mander et al (1988) was the first to establish a way to derive a process to model the stress-strain relationship for circular and rectangular reinforced concrete sections confined by conventional reinforcement (Lorenzis, 2001). Loading pattern and loading history, cyclic and monotonic loading patterns, reinforcement type and material, material properties of the concrete and the reinforcement, concrete and steel strength and ductility of each are all variables considered in existing models.
2.2 Experimental Research on CFRP-confined columns
2.2.1 Circular Columns
Park et al. (2008) investigated the effectiveness of applying narrow strips of CFRP laminates with spacing. Spiral wrapping of a narrow strip of CFRP laminates is proposed in this study for concrete reinforcement. Sixty concrete cylinders wrapped with CFRP strips having different spacing and
13
widths were tested under compression load. The effects of several key parameters such as spacing, spliced length, number of layers, and section area of the CFRP laminates are investigated. It was observed that the use of CFRP strips to confine concrete is effective in increasing the strength and ductility of concrete compared to normal concrete unwrapped with CFRP. It was observed that the effect of confinement on the increase of compressive strength is not linear to the number of CFRP layers. The strength increases only 1.54 times, even though the number of layers doubles.
The study made by Issa et al. (2009) used different number of wraps and height of confinement on a total of 30 carbon-wrapped concrete cylinders to verify the finite model. From tests results, the model was compared favorably and concluded that the wider the CFRP wrap, the higher the strength for multi wraps and also the thicker the layers of CFRP, the higher the strength of the fully wrapped concrete cylinder.
Karabinis and Rousakis (2002) presented the behavior of twenty two 200x300 mm cylindrical specimens that were wrapped by CFRP sheets in low volumetric ratios (0.23-0.7%). They found out that CFRP confinement even in low volumetric ratio considerably increases the strength and especially the ductility of concrete. A constitutive model based on plasticity theory was proposed and concluded that the predicted model is accurate.
14
Tamuzs et al. (2008) studied the stability and strength of concrete columns confined by circumferential wrappings strengthened with CFRP wrapped externally. For the test of plain and confined concrete of length 300 and 1500 mm, theoretical predictions show that the addition of CFRP confinement improves the stability of confined columns in the region of moderate slenderness. The prediction for the ultimate strength and stability of the columns agrees with experimental results.
Lam et al. (2006) presented results of an experimental study on the behavior of FRP-confined concrete. Test results from CFRP wrapped concrete cylinders were presented and examined wherein significant conclusions were drawn including the existence of the envelope curve. The results are compared with existing stress-strain model for monotonic loading. The monotonic model had shown accurate predictions of the envelope curve.
In the study of Chastre and Silva (2010), twenty five experimental tests were made on CFRP confined concrete columns that were subjected to axial monotonic compression. The height of the columns was maintained while other parameters were changed like diameter of columns, type of material, steel hoop spacing and number of CFRP layers in order to evaluate the influence of these parameters on the mechanical behavior of columns. A numerical model was proposed to simulate the behavior of CFRP confined concrete columns. Results showed that the model and predictive equations
15
represent very well the axial compression behavior of CFRP confined RC circular columns.
Jiang and Teng (2007) first provided a critical assessment of existing analysis-oriented FRP-confined concrete models with a database of 48 tests that was conducted by the authors. This assessment clarified how each of the key elements forming such a model affected its accuracy and identified a recent model proposed by the author‘s group being the most accurate. The paper then presented a better version of this model which provides accurate predictions of the stress-strain behavior.
Li et al. (2003) modified the L-L model and extended its application to concrete cylinders confined by steel reinforcements only, CFRP only and by both steel and CFRP reinforcements. There were thirty six concrete cylinders tested to the effectiveness of the modified L-L model. They concluded that the modified L-L model provide better prediction than Kawashima models.
Lee et al. (2004) studied the response of RC columns confined with FRP composites and steel spiral ties by testing twenty four RC small-scale columns under pure axial compression. The test results showed that the compressive response of concrete when confined with two materials was different from the compressive response of concrete when confined with only one material.
16
Barros et al. (2009) proposed a uniaxial stress-strain constitutive model and results obtained from the experimental tests were used to calibrate some of the parameters of this model, and to appraise the model performance. Good
agreement
was obtained between
numerical
simulations and
experimental results for monotonic loading tests.
Cu et al. (2006) investigated the axial load, axial stress and the lateral strain of the wrapped RC columns and compared it with the plain RC columns. Circular RC columns with 180 mm diameter and height of 500 mm were wrapped with CFRP and were tested under uniaxial compressive loading until failure. Results showed that the increase in compressive strength for a typical reinforced concrete column was greatest for the columns wrapped with 2 layers of CFRP. The research had shown that the use of CFRP increased the ductility of the column as well as the aesthetics which minimized the amount of maintenance needed.
In the study of Leuterio et al. (2006), RC columns were grouped according to the number of ply of CFRP and spacing of the steel ties. The result of the analysis showed that the 40 mm and 80 mm spacing were able to increase the load carrying capacity of the cylinder unlike the 120 mm spacing. The results also showed that as the volumetric ratio of CFRP increased, the load carrying capacity and ductility of the column increased.
17
Co et al. (2004) analyzed the effectiveness of CFRP as confining material on concrete cylinders. The results of the experiment found out that the effectiveness of the CFRP increases as the number of layers also increases. It was also found out that the effectiveness of CFRP decreases as the unconfined compressive strength of the concrete cylinder increases.
Sarte et al. (2006) investigated the effects of using CFRP strips to confine RC columns when the entire length of the columns was wrapped with CFRP. Thirty-six short compression members were subjected to variables such as presence of steel ties, volumetric ratio, spacing of CFRP strips, and concrete strength. Results had shown that the CFRP strips and presence of steel ties increases the strength of an RC column.
2.2.2 Non-Circular Columns
In the study of Al-Salloum (2006), twenty specimens, with crosssections varied from square to circular, were examined under uniaxial compression. The study aimed to investigate the influence of corner radius confined with CFRP laminate, thus the corner radius of the square columns ranges from 5 to 50 mm which was obtained by inserting foams on the corners of the molds. The specimens‘ 28 th day design strength rangers from 32-35 MPa and after the 28-day curing, CFRP laminates were then applied which was applied unidirectional in the hoop direction with 150 mm overlap for
18
square columns and 100 mm overlap for circular columns. All specimens were stored at room temperature for at least 7 days before testing. Four (4) strain gauges were bonded at the midheight of square columns, two strain gauges to measure lateral strains and the other two for the longitudinal strains. For the circular specimen, two strain gauges were placed 180⁰ apart at the midheight. The testing was performed using the Amsler hydraulic testing machine with a compressive capacity of 10,000 kN. The failure of the circular specimens was characterized by crushing of concrete followed by CFRP rupture at the middle portion of the specimen. For the square columns, the highest increase was attained in the column with 50 mm corner radius but is still lesser as compared to the cylindrical specimens. Still, the effectiveness of CFRP wrapping on square columns were also observed on the increase compressive strength and ultimate strain.
The study of Hassan and Chaallal (2007) summarized six (6) confinement models and three (3) design guidelines for rectangular columns retrofitted with FRP laminates, afterwards, they propose their own confinement model. The predicted values from these models were compared to experimental results of the following studies, for square columns: Demers and Neale, Rochette and Labossierre, Parvin and Wang, Pessiki et al., Suter and Pinzelli, Shehata et al., Lam and Teng, and Chaalal et al., while the database for the rectangular columns are from the following: Rochette and Labossierre, Shehata et al., Lam and Teng, and Chaalal et al.. The total
19
number of specimens was 94 confined externally with FRP laminates: carbon, glass, and aramid and tested under axial compression. The height of the specimens ranges from 300 to 610 mm, the unconfined compressive strength from 21.44 MPa to 55.36 MPa, and the range of the increase is strength is from 0.940 to 4.021. The highest increase in strength was obtained from the specimen with a b/h ratio of 1 and height of 600 mm, a corner radius of 25 mm, without steel reinforcements and the volumetric ratio of CFRP is 1.76%, and an unconfined compressive strength of 24 MPa. This study has concluded that a corner radius of 25.4 mm or 1 in. and higher ensures a suitable confinement of rectangular concrete columns.
The main focus of the research study by Wu et al. (2007) is to examine the effect of corner radius on the CFRP confinement of square concrete columns.
Along with the corner radius, other variables were the CFRP
thickness and concrete grade. The corner radii were varied from 0 to 75 mm while the CFRP jacket thickness is from none to two-ply. The total number of specimens is 108 with design strength from 30 to 50 MPa. After 28 days of curing, the specimens were wrapped with CFRP with tensile strength of 219.0 MPa and 225.7 MPa obtained from the coupon tests. Compression tests were conducted using FORNEY testing machine with a maximum load capacity of 2500 kN while the axial load, vertical displacement, lateral expansion, and strains were recorded by an automatic data acquisition system. Three different failure modes occurred in the specimens: tensile rupture of the CFRP
20
jacket, usually near a corner, delamination of the CFRP jacket, and combination of delamination and tensile rupture of the CFRP jacket. For columns with corner radius 15 mm and 30 mm, the failure mode was CFRP rupture while those with 45 mm, 60 mm, and 75 mm corner radius, failure was due mainly due to delamination or the combination of CFRP rupture and delamination.
The study by Binici et al. (2007) examines the use of carbon fiberreinforced polymer (CFRP) wrapping as a method of retrofitting square reinforced concrete columns with low strength concrete and plain steel bar reinforcements. The specimens had dimension of 350 x 350 x 2000 mm with volumetric ratio of longitudinal steel reinforcements of 1.66% and 0.52% for the transverse steel. The CFRP wrapping were from one to two layers and the corners of the specimens were rounded to 30 mm. Low concrete strength as designed for these specimens, thus having 15 MPa as design strength. The specimens were tested one week after being wrapped with CFRP.
A similar study was conducted by Binici et al. on 2008, now focusing on rectangular columns. The cross-section is 200 x 400 mm with the same height of 2000 mm. The volumetric ratio of longitudinal reinforcement is 2.55% and 0.75% for the transverse reinforcement. The design strength was still low, from 10 to 15 MPa and a corner radius of 30 mm. Seven (7) LVDTs were used to measure the specimen‘s deflection. Three large scale columns were
21
tested with cross-sectional dimension of 400 x 400 mm and a height of 2000 mm under uniaxial loading in the study of Lang et al. (2008). Low concrete compressive strength was designed with the design strength of 20 MPa. The columns were also reinforced with steel bars, longitudinally and also in the transverse direction. The corners were rounded to 25 mm. Nine (9) strain gauges were bonded to the column and then subjected to monotonic uniaxial compression using a 20,000 kN testing machine. The failure mode was CFRP rupture followed by the crushing of concrete and buckling of longitudinal steel reinforcements.
A total of 48 specimens, circular and square columns, were tested under axial compression by Benzaid et al. (2009). The parameters considered are: the shape of cross section shape; the number of wrap layers and the concrete strength. The layers of CFRP are from one to three layers and the unconfined compressive strength is from 25 MPa to 60 MPa. The corners of the square columns were kept sharp having a corner radius equal to zero. The longitudinal steel ratio was equal to 2.25% and the transverse steel ratio was equal to 1.17%. The specimens were cured for 28 days before they were wrapped by CFRP and tested after one week. The instrumentation included one radial linear variable differential transducer (LVDT) placed in the form of a hoop at the mid-height of the specimens. Measurement devices also included three vertical LVDTs to measure the average axial strains and the compressive load was applied at a rate of 0.24 MPa/s. The failure for square
22
columns occurred or initiated near the corner with the sudden rupture of the CFRP wrap. CFRP rupture was also the failure mode for cylindrical columns mainly in the central zone. In conclusion, the CFRP confinement on lowstrength concrete specimens produced higher results in terms of strength and strains than for high-strength concrete similar specimens and efficiency of the CFRP confinement is higher for circular than for square sections.
2.3 Proposed Models for CFRP-Confined Columns
Some proposed models were made by Lam and Teng (2003), Mirmiran et al. (1998), Pantelides and Yan (2007), Al-Salloum (2007), Ilki et al. (2008), and Kheyroddin and Salehian (n.d.). The parameters that were focused on were the shape factor, ks and the corner radius.
The study of Ilki et al. (2008) was composed of experimental testing of columns and modeling the compressive strength. The experimental testing was done on 68 reinforced concrete columns (24 square, 23 rectangular and 21 circular columns) with main test parameters of: thickness of the CFRP jacket, cross-section shape, concrete strength, amount of internal transverse reinforcement, corner radius, existence of predamage, loading type, and the bonding pattern of CFRP sheets. The experimental results confirmed that CFRP confinement is more effective on low strength concrete and circular columns. Though lesser than that of circular columns, square and rectangular
23
columns gained significant increase in its compressive strength when confined with CFRP. As observed from the results, the rectangular columns remarkably exhibited the highest deformability and the reason behind it is the less effective confinement which results in higher axial and transverse plastic deformations. This is the same with the higher deformability of square columns than that of circular columns. However there is an exception with this outcome which is for low strength concrete confined only with one ply of CFRP. The failure mode of the specimens was generally CFRP rupture. Square and rectangular columns CFRP rupture happened around the corners just after the rounded part.
Along with the experimental studies, models were proposed for the prediction of confined strength and corresponding axial strength through utilizing their experimental studies and from other experimental studies as well with a total of 88 specimens. The model proposed should take into account the thickness of CFRP that can provide an ascending branch in the stressstrain diagram. Assumptions made were on the strain corresponding to unconfined concrete strength equal to 0.002 and the tensile rupture strain of FRP jacket was which is equal to 85% of ultimate tensile strain. The contribution of the longitudinal reinforcement steel was disregarded while the contribution of transverse reinforcement steel was taken into account. To evaluate the proposed model, it was compared with proposed models of Lam and Teng, and Samaan et al. by utilizing a compilation of experimental
24
database of 448 specimens. The three models exhibited good performance though the prediction of the proposed model of the compressive strength or rectangular column is unconservative. The model of Samaan et al. produced unconservative results for circular columns and good values for rectangular columns. While Lam and Teng‘s model performed well for both circular and rectangular columns. The proposed model performed well with lesser scatter as compared to the two other models.
Al-Salloum (2006) focused his study on the effect of edge sharpness on square columns and concluded that smoothening edges of square columns delay the rupturing of FRP composites at these corners and that the FRP confinement efficiency is linearly related to the corner radius. Subsequent to testing of columns, the author proposed equations for the prediction of ultimate confined compressive strength of circular and square columns. It introduced the shape factor for the confining pressure of FRP sheets on square columns as a function of the section dimension and corner radius. The maximum value for the shape factor is 1.00 when the ratio of the corner radius to the shape dimension is equal to 0.5. Aside from the shape factor that considers the effect of confined area of the section, a modification factor was introduced. It is the ratio of the length of the side of the square section over the diagonal length of the square section. The maximum value of the modification factor is one which is for circular columns. Thus the proposed
25
equation can be used for circular and square columns. The proposed model agreed well with the experimental results.
The study of Mirmiran et al. (1998) concentrated on the effects of shape, length and bond of the FRP composites. Mirmiran et al. tested circular and square columns. For the square columns, corner radius and the dimensions of the tube affect the confining pressure of the FRP. However, the jacket thickness has minimal effect on the confining pressure of square columns while it greatly affects the confining pressure of circular columns. The MCR, modified confining ratio was introduced defined by the corner radius, dimension of the tube, and the confinement pressure. For MCR less than 15% the jacket will be less effective and no strength enhancement should be expected. The length effect was considered by taking into account the ratio of the height and cross-sectional dimension of the column. The ratio of 2:1 and 5:1 does not have an effect on the strength and ductility of the column. The conclusion for the adhesive bond of FRP does not have a significant effect on the load-carrying capacity of the column.
Pantelides and Yan (2007) proposed a design-oriented model for square and rectangular columns confined with bonded FRP jackets, and shape-modified square and rectangular sections confined with post-tensioned FRP shells. The compressive strength of the FRP-confined concrete was derived using the concrete plasticity theory based on the five parameter
26
Willam and Warnke model while the strain is based on the concepts of secant concrete
modulus
and
strain
dependent
stiffness
established
by
Pantazopoulou and Mills. Added to the efficiency factor and shape factor is the post-tensioning factor which was given as 1.9 for circular; 1.5 for 2:1 elliptical; 1.1 for 3:1 elliptical. The evaluation of the model was done through a comparative study between the calculated and experimental results. Satisfactory performance was exhibited with columns having an aspect ratio of 2:1 however, columns with aspect ratio of 3:1 showed larger deviation. The difference with the calculated and experimental results of the compressive strength is 11% while 6% difference was observed for the strain for the square and rectangular columns. The design model, which is based on a fourparameter concrete stress-strain relation, is relatively easy to use for design purposes; the four parameters are different for columns with hardening or softening behaviour. The design model is general and can describe several cases of FRP-confined concrete including different cross-sectional geometry, bonded FRP jackets, as well as post-tensioned FRP shells.
Kheyroddin and Salehian (n.d.) after evaluating five proposed models and three design guidelines, the authors proposed their own model which accounts for the effect of cross-sectional shape and dimension. The confining pressure of the FRP confinement is affected by the shape factor, which in the proposed model is equal to Lam and Teng‘s proposed model. The efficiency factor of the FRP was obtained through trial and error equal to 0.650. The
27
accuracy of the model was assessed by utilizing experimental data results on FRP confined prismatic columns. In addition, the proposed model was compared with the five proposed models and three design guidelines by applying index error. The proposed model performed better compared to the seven models. It came in second place, with an index error of 1.37, after the model of Ilkin et al. with an index error of 1.36.
As evaluated by the study of Kheyroddin and Salehian (n.d.), the proposed models of Al-Salloum, Lam and Teng , Ilki et al., and the proposed models of the authors exhibited acceptable approximations having index errors of: 1.65, 1.38, 1.36 and 1.37. On the other hand, proposed models of Mirmiran et al. and ACI and CSA underestimated the compressive strength while proposed models of Pantelides and Yan and fib are overestimated for rectangular cross-sections.
28
Table 2.1 Proposed models for CFRP-Confined Columns f‘cc (compressive strength of confined column)
Models
= 1 + 6.0
Mirmiran et al. (1998)
𝑓𝑙 0.7 ′ 2𝑟 𝑓𝑙 𝑓 𝑐0 𝑓𝑜𝑟 ( )( ′ ) 𝑓 ′ 𝑐0 𝑎 𝑓 𝑐0
= (−4.322 + 4.271 1 + 4.193
for
𝑓𝑙 𝑓′ 𝑐0
≥ 0.2
Al-Salloum (2007)
= (1 + 3.14
Lam and Teng (2003)
= (1 + 3.3
Ilki et al. (2008)
= (1 + 2.54
H.R. Salehian, A. Kheyroddin (n.d.) 𝑓′𝑐𝑐0 [0.622 +
𝑓𝑙 𝑓𝑙 − 2 ′ )𝑓′𝑐0 𝑓 ′ 𝑐0 𝑓 𝑐0
𝑓𝑙 )𝑓′ 𝑓 ′ 𝑐0 𝑐0
𝑓𝑙 )𝑓′ 𝑓 ′ 𝑐0 𝑐0 𝑓𝑙 )𝑓′ 𝑓 ′ 𝑐0 𝑐0
𝑓𝑙 + 1.577 𝑓 ′ 𝑐0
𝑓𝑙 + 0.058 ] 𝑓 ′ 𝑐0
Table 2.2 Efficiency factor and shape factor of the proposed shape models Models
Section‘s shape factor, ks
Efficiency factor, kε
Mirmiran et al. (1998)
1.0
Al-Salloum (2007)
1.0
4r/a2
2a
2 13
2
2a-2r
Lam and Teng (2003)
Ilki et al. (2008)
0.57
b a
2
2 a2 +b
0.85
2-1
2
a+b [1ab
H.R. Salehian, A. Kheyroddin 0.65
b a
29
2
2 a2 +b
2
[ 1-
b a
[ 1-
b a
1-2 1- 4-π
r a
2
a b-2r b 2 3 ab- 4-π r a-2r 2 +
a b-2r b 2 3 ab- 4-π r a-2r 2 +
b a
2
r a
]
2
a b-2r b 2 3 ab- 4-π r a-2r 2 +
2
]
2
]
2.4 Other Proposed Models
2.4.1 Proposed Models for Circular Columns
Recent models were proposed by Lam and Teng (2003), Mandal et al. (2005), Matthys et al. (2005), Chastre and Silva (2010) for circular columns which calculates the ultimate confined compressive strength and increase in strength of column specimens.
Many significant variables were utilized by these models. The variables include the columns geometrical characteristic, and concrete properties. Properties of the FRPs and its confinement are also considered important variables in these models. Concrete properties involve the compressive strength of unconfined concrete, lateral strain and modulus of elasticity. The same properties of FRPs, ultimate compressive strength, lateral strain and modulus of elasticity, were utilized by the proposed models. Lastly, the confinement properties include the confinement modulus, confinement pressure and the ratio of the confinement pressure with the column‘s unconfined compressive strength. This ratio was used by most of the models (Lorenzis and Tepfers, 2003).
Lam and Teng (2003) developed a design-oriented stress-strain model using 76 FRP-wrapped plain normal strength concrete circular specimens.
30
The failure modes of all the specimens were rupture of the FRP jacket. Assumptions for the model was a stress-strain curve having a parabolic first portion which is affected to some degree by the confinement and an ascending straight-line second portion. The first and second portions meet smoothly without a change in slope at where they meet. The end of the second portion is equal to both the compressive strength and ultimate axial strain of confined concrete. The confinement coefficient was taken as 3.3. The proposed equation to predict the ultimate strain took into consideration the stiffness and the actual ultimate condition of the jacket.
Another model was proposed by Mandal et al. (2005) which gives emphasis to the unconfined compressive strength of concrete. A range of 2681 MPa concrete strength was used in the study. The stress-strain curve for low to medium strength concrete was a typical bilinear curve while the medium to high strength concrete exhibited strain hardening to flat plateau then to sudden strain softening and decrease in ductility in the second portion. Thus, confining high-strength concrete does not have a significant difference with that of an unconfined concrete. A proposed model was then made based on regression of test data from the study. The study concluded that increase in unconfined compressive strength leads to a decrease of confinement effectiveness. Also, the properties of FRP have little effect in confining highstrength concrete.
31
The study of Matthys et al. (2005), examined the confinement effect of FRP on large-scale columns. The columns used were 400 mm in diameter and 2 meters in height. Existing models were evaluated in their reliability to measure the strength of confined concrete columns. The model proposed by the co-author, Toutanji, was revised to address the effective FRP failure strain.
The latest model proposed by Chastre and Silva (2010) was based on their own experimental tests consisting of twenty five columns with a constant height of 750mm and varying diameter, type of concrete material (plain or reinforced), steel hoop spacing, and layers of CFRP. The authors presented two different approaches: first, equation based on experimental analysis and the second, based stress-strain model for confined concrete in compression and the corresponding axial load vs. strain of the RC column. The confinement coefficient was taken as 5.29. The scale effects on the compressive strength was introduced by α which is in relation with the diameter and height of the column. The stress-strain curve is bilinear where the slope of the first portion is identical to that of unconfined concrete and the slope of the second portion was determined by experimental calibration. The models considered the effect of CFRP and the transversal reinforcement on the lateral confining pressure of the column.
32
Table 2.3 Proposed Models for Circular Columns Proposed Models for Circular Columns Fardis and Khalili Saadatmesh et al.
f'cc f'co
= 2.254 1 + 7.94
ρu '
f co
ρu
-2
'
f co
− 1.254
ρ f'cc = 1 + 1.3485 ' u f'co f co
Miyauchi et al.
f'cc = 1 + 0.0572ρu f'co
Kono et al.
f'cc 𝜌𝑢 0.7 = 1 + 6.0 ' f'co f co
Saamaan et al. Toutanji
f'cc 𝜌𝑢 0.7 = 1 + 3.5 ' f'co f co
0.85
Saafi et al.
f'cc 𝜌𝑢 0.7 = 1 + 2.2 ' f'co f co
0.84
Spoelstra and Monti
f'cc 𝜌𝑢 0.7 = 0.2 + 3 ' f'co f co
Xiao and Wu
0.86
ρ ρ f'cc f'cc =1+4.1 ' u ; =1+3.7 ' u f'co f' co f co f co
0.5
𝑓 ′ 𝑐𝑜 f'cc = 1.1 + [4.1 − 0.75 f'co 𝐸𝑡
2
]
ρu '
f co
𝑓′ 5 𝑓′𝑐𝑢 = 1 + 2.25( ′𝑙𝑢 )4 𝑓′𝑐 𝑓 𝑐
Feng et al. (2007),
f'cc 𝑓𝑙 ,𝑎 = 1 + 3.3 ' f'co f co
Lam and Teng (2003)
Where: 𝑓𝑙,𝑎 = actual maximum confining pressure f'cc 𝑓𝑙 = 1 + 2.3 ' f'co f co
Matthys et al. (2005)
0.85
Where: 𝑓𝑙 = 2𝑡𝑓 𝐸𝑓 𝜀𝑓𝑢 /𝐷
Mandal et al. (2005)
f'cc 𝐸𝑡 𝑓𝑢 𝐸𝑡 𝑓𝑢 = 0.0017( ′ )2 + 0.0232 f'c 𝑅𝑓 𝑐 𝑅𝑓 ′ 𝑐
+1
𝑓𝑐𝑐 = 𝑓𝐷 + 𝑘1 𝑓𝑙𝑢
Chastre and Silva (2010)
Where:𝑓𝐷 (𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑣𝑒 𝑠𝑡𝑟𝑒𝑛𝑔𝑡 𝑜𝑓 𝑐𝑜𝑛𝑐𝑟𝑒𝑡𝑒 = 𝛼𝑓𝑐𝑂 𝛼 𝑠𝑐𝑎𝑙𝑒 𝑒𝑓𝑓𝑒𝑐𝑡𝑠 = (1.5 +
33
𝐷 )/2
2.4.2 Proposed Models for Non-Circular Columns
Models proposed for CFRP confined rectangular columns are very few as compared to that of circular columns. Some proposed models were made by Feng et al. (2007), Harajli et al.(2006), Toutanji et al. (2007), and Cai et al. (2008). The parameters that were focused on were the shape factor, ks and the corner radius.
Proposed models for circular and rectangular concrete columns were prepared by Feng et al. (2007). The parameters utilized are: ultimate strength of FRP-confined concrete and unconfined concrete, ultimate concrete compressive strain, axial stress and strain at the boundary point between the first and second region, the slope of the second region, characteristics and amount of the FRP wrap, and the dimension and cross-section of the concrete specimen. The proposed model for FRP-confined rectangular columns were only compared with two existing models (Hosotani‗s model and Mander‘s model) due to limited available models. The proposed model was known to be more accurate than the other two.
The stress-strain model of the FRP proposed by Harajli et al. (2006) follows a two stage relationship. The first stage can be described using the ascending branch of the stress-strain equations for unconfined or steel confined concrete. It is assumed that this stage follows a second-degree
34
parabola. In the second stage of the response, which includes the intersection of the two stages, the confined concrete compressive strength can be expressed explicitly as a function of the amount of reinforcement and material properties.
According to the study, the rate of increase of the measured average lateral strain with the axial strain tended to decrease as the aspect ratio of the column section and also the area or stiffness of the FRP jacket increase. This observation, similar to Chaallal et al. (2003) and which has been disregarded in the development of earlier stress-strain models, has a substantial implication on the derivation of the confinement effectiveness coefficients, k1 and k2.
Using the experimentally measured axial stress and lateral strains, values of k1 for the various FRP-confined specimens were estimated in the second stage of the response. It was observed that that the magnitude of k1 decreases consistently from a relatively high value in the early stage of the response during which the effective lateral confining pressure is low, to a value close to 2.0 as the confining pressure increases. For k2, the coefficient was not treated as a constant or only a function of the lateral strain but also a function of the volumetric ratio and modulus of elasticity of the FRP jackets, and most importantly, the aspect ratio h/b of the column section.
35
The study showed that improvement on the column axial strength and ductility due to FRP confinement becomes less significant as the aspect ratio (h/b) of the column section increases. For a given aspect ratio of rectangular column section, the rate of increase in lateral strain with axial strain decreases as the stiffness ρf Ef of the FRP jackets increases. However, irrespective of the h/b of the column section or ρf Ef of the FRP jackets, the stress-strain response of FRP confined columns experiences a considerable increase in lateral strain, and consequently, a distinct change in behavior, beyond a confined lateral strain of approximately 0.002.
The proposed model of Toutanji et al. (2007) was applied to both small and three large scale rectangular columns. The axial stress-strain behavior utilized by their study was divided into three different regions. The first region is the linear part in which the axial load is lower than the unconfined compressive strength. Then the second region follows, which is the transition point when the FRP sheets are activated. Lastly, the third region corresponds to a fully activated FRP confinement which can have an ascending or descending trend. In their study, the third region was assumed as an ascending straight line. Their model introduced two coefficients, k 1 and k2 which account for the effect of corner radius and aspect ratio. A third coefficient, ke, effective confinement coefficient was also introduced. The lateral strain of the FRP at point of rupture, εj, was taken as εj = 0.43εfrp. The term εfrp represents the ultimate strain. The value of D, diameter of an
36
equivalent circular column, was taken as D = 2bh/b + h, as specified in the ACI Committee 440. Good correlation of 59 experimental data was observed. The proposed model was also compared with other fourteen existing models.
Cai et al. (2008) proposed a simpler model for the FRP confined core concrete strength. The proposed model utilizes the in-section coefficient of effectiveness, in-height coefficient of effectiveness and the lateral hydrostatic pressure.
The effective confined concrete area accounts for the effectiveness of lateral confinement in confining the concrete in the horizontal plane. The insection coefficient, η1, was derived from the dividing the area of concrete effectively confined by the jacket (Ae) by the gross-cross-sectional area of the rectangular columns (Ag). The column confined with the FRP jacket can be seen as the spiral stirrup column where spacing of the stirrup is zero. Therefore, the in-height coefficient of effectiveness η2 for the FRP-wrapped column is 1 based. The lateral hydrostatic pressure was directly related to the dimensions of the specimen, gross cross-sectional area, and the tensile strength of the FRP.
37
Table 2.4 Proposed Models for Non-Circular Columns Proposed Models for Square and Rectangular Columns Feng et al. (2007),
Harajli et al.(2006)
Toutanji et al. (2007)
Where:
Cai et al. (2008)
fcc = fc + kfr where: fr = η1 η2 σr η1 = in-section coefficient η2 = in-height coefficient σr = lateral hydrostatic pressure k = 2.3
The study had to deal with uncertainties such as the effect of stress concentration close to the corners and the effectiveness of confinement along the longer sectional dimension are added to the inherent uncertainties of the confining mechanism. Based from previously existing works regarding
38
rectangular sections in RC buildings, the study used a constant of 2.3 for the confinement effectiveness coefficient. The proposed design formula showed that it can be used to predict the ultimate load for FRP-wrapped concrete column with high precision.
These existing models were inconsistent with the accuracy of the predicted values. Thus, an ANN model that would efficiently and accurately predict the ultimate confined compressive strength and increase in compressive strength of CFRP confined columns, not just for circular but also for square and rectangular columns is needed.
2.5 Artificial Neural Network
An artificial neural network is a massively parallel distributed processor that has a natural propensity for storing experiential knowledge and making it available for use (Haykin, 1998). It is an information processing model that consists of a great number of simple interconnected processing units called neurons. These neurons essentially work with each other onto solving specific problems. The network is mainly an adaptive model that can be trained through the data obtained and then generalizes these data for the things learned. Neural networks are constructed for specific applications like pattern recognition and data classification, and these are configured through a learning process. An artificial neural network can be compared to adaptations
39
in the biological processes and adjustments between the neuron connections (Stergiou & Siganos, n.d.).
Previous neural network models predicted accurate solutions for the computation of the ultimate compressive strengths of concrete columns wrapped with CFRP. Normalizing the data that would be used for model is important to ensure that the distance measure accords equal weight to each variable. Without normalization, the variable with the largest scale will dominate the measure.
2.5.1 Artificial Neural Network
In a sample study by Vesanto et al. (2000) for the SOM toolbox, they used the well-know Iris data set with four variables which are length and width of sepal and petal leaves. It is consisted of 150 data or iris flowers divided into three groups: 50 Iris-setosa, 50 Iris-versicolor and 50 Iris-virginica. They made use of the visualizations such as U-matrix, component planes, labelling, projection on the graph and bar charts.
From the U-matrix, they have analyzed that three top rows were formed a very clear cluster. They have also correlated the variables using the component planes, an example was the close relation of the petal length and petal width. Also, the component planes were used to know the
40
characteristics of each species. For example, the Iris-sentosa was classified having small petals and short but wide sepals. In the bar chart, the four variables are shown in each hexagon or the map unit. With this, the relative importance of each variable can also be analyzed. The visualizations provided by the SOM toolbox were indeed helpful in data analysis and the likes.
The study made by Crowther and Cox (2005) showed that the suggestion of Haykin (1999) 80/20 split of the train/test data can be potentially unsafe especially for small data sets. For a data set with large number of data points, the exact train/test ratio is less important because there is sufficient data for classification but within the range of 50/50 to 70/30. They demonstrated using empirical tests that when the non-validation data is divided between training and test data using a ratio within the range of 50/50 to 70/30, the resultant accuracy when graphed produces a curve that peaks that represents a good ratio with which to split the data.
2.5.2 ANN Application in Concrete Modeling
Chan et al. (2008) utilized the MATLAB software to apply ANN modeling through Self Organizing Map (SOM) and Back-Propagation (BP) to predict the increase in compressive strength of CFRP. Various parameters were considered in the ANN models such as volumetric ratio of steel, volumetric ratio of CFRP, diameter of column, length of column, unconfined
41
compressive strength and ultimate confined compressive strength. There are two SOM models that were chosen and analyzed based on visible relationship of the groupings. The BP model BP A1-2I-4H-2G was obtained because it has high Rtest, low MSE and it requires lesser parameters to determine the increase in strength. A Visual basic program was designed to calculate the increase in compressive strength with input values of ps and pcfrp.
Ongpeng and Oreta (2005) developed a neural network model considering the confinement effect of steel reinforcements and CFRP on circular RC columns. The effect of various parameters such as ps, pcc, pcfrp, L, d, D, fyh, fcfrp and f‘c are considered in the development of ANN models. The model SCC9-7-1B showed the best predictions in the increase in strength for both steel and CFRP reinforced columns and performed better when compared with existing models.
In the study made by Aquino et al. (2009), three different SOM models were created and these are beams with shear reinforcement, beams without shear reinforcement and the combination of the two using the parameters f‘c, pl, a/d, d and b. The volumetric ratio of the longitudinal steel reinforcements has the great effect on the grouping the data for beams without stirrups. The BP model BPN WS1 with 9 input parameters was chosen because of its high correlation from the target output to the achieved output. The model was then compared with empirical models to verify the model‘s accuracy. It was found
42
out that the models developed by ANN were able to provide better predictions of the shear strength of RC beams. The model was also used to conduct parametric analysis to observe the behavior of the shear stress due to varying parameters. It was found out that there were other parameters that greatly affect the shear strength of RC beams other than the concrete compressive strength (f‘c).
Another study by Ongpeng (2003) investigated the interaction of Carbon fiber reinforced polymer and lateral steel ties in circular concrete columns. The parameters used as input nodes in ANN modeling were the concrete cover, volumetric ratio of lateral steel ties, volumetric ratio of longitudinal steel ties, volumetric ratio of CFRP, fy of steel, fy of CFRP, f‘c, diameter and height of circular columns and f‘cc as output node. After training, testing and validation with existing models, the CC 9-5-1 model was chosen because it was good in prediction of confined ultimate compressive strength.
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CHAPTER III THEORETICAL AND CONCEPTUAL FRAMEWORKS
3.1 Theoretical Framework
The theoretical framework refers to the assumptions grounded in theory that were cogitated by the proponents of the study which is illustrated in Figure 3.1.
The theories on the CFRP-confinement models created by Lam and Teng, Ilki et al., Al-Salloum, Salehian and Kheyroddin, Hosoatani and Kawashima, and ACI 440 model were examined. The models produced showed inconsistencies in the prediction of the confined compressive strength due to the non-linear behavior of CFRP-confined columns. The utilization of the Artificial Neural Network is then considered. Through the Self-Organizing Map, the parameters affecting the CFRP-confinement were analyzed and the results from the SOM models were applied on the Backpropagation Models for the prediction of the confined compressive strength.
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Artificial Neural Network
CFRPconfinement models
Lam & Teng
Self-Organizing Map
Ilki et al. Al-Salloum Salehian & Kheyroddin
BMU Distance Formula Visualizations
Hosotani & Kawashima
Backpropagation
ACI 440 Training Weights & Biases MSE
Figure 3.1 Theoretical Framework
3.1.1 CFRP confinement of columns
FRP materials like CFRP can provide confinement to increase the axial compression strength and ductility of concrete. FRP provide passive confinement to concrete, remaining unstressed until cracking of the wrapped concrete once it is subjected to axial load. For this reason, intimate contact between the CFRP and concrete is important. Confining concrete is done by
45
orienting the fibers transverse to the longitudinal axis of the RC column. In this orientation, the hoop fibers are similar to spiral or tie reinforcing steel. Any contribution of longitudinally aligned fibers to the axial compression strength of concrete should be neglected. Factors such as fiber volume, type of fiber, type of resin, fiber orientation, dimensional effects, and quality control during manufacturing can affect the effectiveness of an FRP material. The FRP jacket can also serve to delay buckling of longitudinal steel reinforcement in compression, and to clamp lap splices of longitudinal steel reinforcement. There are three common types of FRP namely: glass-fiber reinforced polymer (GFRP), aramid-fiber reinforced polymer (AFRP), and carbon-fiber reinforced polymer (CFRP). These three have the same stress-strain behavior which is linear elastic up to final brittle rupture. FRP do not exhibit the ductility that steels have. Nonetheless, when used as confinement for concrete, these can greatly enhance the strength and ductility of columns. Out of the three FRP composites, CFRP demonstrates the best quality as supported by Figure 3.2 and Table 3.1.
46
Figure 3.2 Properties of different fibers and typical reinforcing steel (ACI Committee 440, 2002)
Table 3.1 Mechanical Properties of Common Strengthening Materials Material Modulus of Compressive Tensile Density (kg/m3)
Elasticity
Strength
Strength
(Gpa)
(Mpa)
(Mpa)
20-40
5-60
1-3
2400
Steel
200-210
240-690
240-690
7800
CFRP
200-800
NA
2500-6000
1750-
Concrete
1950
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Figure 3.3 Stress-Strain Diagram of Unconfined Concrete vs CFRP -Confined Concrete (Lam & Teng, 2003)
Lam and Teng (2003) proposed a model that has a parabolic first section with the initial modulus being the same as that of the unconfined concrete‘s initial modulus shown in Figure 3.3. The linear second segment has a modulus that correlates to a reference stress that is equal to that of the unconfined compressive stress. The linear portion meets the parabolic portion at a certain strain, slightly higher than that of the ultimate unconfined compressive strain. This shows that CFRP confined columns perform better in terms of the stress and strain that it can withstand as compared to unconfined columns.
48
FRP jackets are most effective at confining circular members because it provides a circumferentially uniform confining pressure to the radial expansion of the compression member when the fibers are aligned transverse to the longitudinal axis of the member. The volumetric ratio of CFRP wrap is given by this equation (ACI Committee 440, 2002): (Equation 3.1) where: n = number of layers of FRP tf = thickness of FRP D = diameter of circular column
Other test results have shown that confining rectangular members confined with CFRP can provide minor increases in the axial compression strength of the member. Confining square and rectangular sections, while not effective in increasing axial strength, is effective in improving the ductility of compression members. The volumetric ratio of CFRP wrap is given by this equation (ACI Committee 440, 2002): (Equation 3.2) where: n = number of layers of FRP tf = is the thickness of FRP h and b = cross-sectional dimension
49
Relating Equations 1 and 2, the diameter of the circular column which will be addressed in this study as D is given by: (Equation 3.3)
Considering a circular column whose diameter is equal to b or h of a square column, D will then be given as D = b = h. Substituting b = h from equation 3:
D=
or
D=b
(Equation 3.4)
Confinement is less effective with rectangular sections because CFRP is not uniformly confined and the compressive pressure is unevenly distributed unlike in circular sections. The higher stress is usually found at the corners. Rounding the column‘s corners has been a common practice used to reduce the cutting effect on confining sheets where the corner radius is produced. The compressive strength of a confined column increases as the corner radius increases. The confining effect of FRP jackets should be assumed to be negligible for rectangular sections with aspect ratios b/h exceeding 1.5, or face dimensions, b or h, exceeding 36 in. (900 mm), unless testing demonstrates their effectiveness. Higher aspect ratio usually results in a reduction in the confinement pressure. Where fibers wrap around the corners of rectangular cross sections, the corners should be rounded to a
50
minimum 1/2 in. (13 mm) radius to prevent stress concentrations in the FRP system and voids between the FRP system and the concrete.
r r
Figure 3.4 Corner radius of circular and non-circular columns
Sharp corners tend to hinder the effectiveness of the CFRP confinement. High concentrations of stress build up at the corners of square and rectangular columns. The strain values obtained indicate that the confining mechanism has not been fully activated at CFRP rupture. (Wang & Wu 2007) To address this problem, column corners are rounded to certain degrees. Experimental studies showed that the larger the corner radius was, the higher the increase in strength produced by the CFRP confinement. Different values of corner radii were used in different researches. Sharp edged columns (rectangular and square) are taken to have zero corner radii since their corners have not been rounded. Circular columns on the other hand have corner radii of half of their breadth. (Benzaid et al. 2009 and Wang & Wu et al. 2007)
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3.1.2 Steel Reinforcement
The study is comprised of reinforced columns confined with CFRP. The effect of the steel reinforcements will be accounted for by utilizing the parameter volumetric ratio of longitudinal reinforcement ρ l and volumetric ratio of lateral reinforcement ρs. Also, the area confined by steel reinforcement, Acc is utilized.
Figure 3.5 Longitudinal reinforcements and area of the concrete core of columns
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The volumetric ratio of longitudinal steel reinforcement ρ l is given by the equation:
(circular columns)
(square and rectangular columns)
(Equation 3.5)
(Equation 3.6)
where: As = area of longitudinal steel reinforcement multiplied by the number of longitudinal steel
The volumetric ratio for lateral reinforcement ρs, is given by the equation:
(circular columns) (non-circular columns)
(Equation 3.7) (Equation 3.8)
where: d = inner diameter of circular columns s = spacing of lateral reinforcements b‘ and h‘ = cross-sectional dimension minus the concrete cover
53
3.1.3 Self-Organizing Map
The self-organizing map is a popular neural network based on unsupervised learning. This is a good tool for data analysis and correlation. The SOM toolbox is best suited for the data understanding / survey phase and it could also be used for modeling. The developer behind the toolbox aimed the toolbox to be oriented towards powerful visualization functions and can be use to pre-process data, initialize and train SOMs, visualize SOMs in various ways, and analyze the properties of the SOMS and data. Visualizations such as the U-matrix, component planes and the bar graphs will be utilized.
The data are normalized to ensure a better clustering and the best matching unit will be located. The SOM is trained iteratively until the model comes up with the Best-Matching Unit (BMU). The BMU is chosen to be the vector with greatest similarity with the input sample. The similarity is usually defined by means of a distance measure. The best-matching unit, usually noted as mc, is the codebook vector that matches a given input vector x best. It is defined formally as the neuron for which
(Equation 3.9)
The clustered data will be plotted in 2D histograms, shown in Figure 3.6, or 2D graphs. The density of the data on the particular part of the graph is
54
represented by the blue spots for the unclustered graph, and blue, red, and green spots for the clustered graph. Hits
SOM 26-Feb-2011
Figure 3.6 2D Graph of ungrouped (left) and grouped (right)
The Unified distance matrix or the U-matrix is employed on considering the cluster structure of the data sets. From the U-matrix, it can be analyze if the clusters made by the SOM by the distance of each data to neighbouring data. High values of the U-matrix indicate large distance between neighboring map units, uniform areas of low values indicate high degree of similarities or the clusters. The neighborhood functions can also determine how strongly the neurons are connected to each other. The correlation between the parameters will then be analyzed using the component planes. In addition, the component planes can characterize the cluster by knowing the range of values of the parameters for each cluster.
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U-matrix
pcfrp 0.162
0.615
0.0881
0.379
0.0141
0.142 d
f’co
Labels 0.862
Figure 3.7 Left to right: U-matrix, component plane, bar graph with 2D graph 0.511
0.159
To further d SOM 26-Feb-2011
analyze the SOM and evaluate if it produced a good model,
added visualizations will be used. The bar graph, shown in Figure 3.7 (right), is employed to determine the values of the parameters in a particular part of the graph. This is also used to analyze the groupings depending on the values of the parameters that are similar in the group.
3.1.4 Artificial Neural Network
Artificial Neural Networks function like a brain and learns from several given data and algorithms. An ANN is basically an information processing model that consists of a great number of simple interconnected processing units called neurons. These neurons essentially work with each other onto solving specific problems. The model would utilize the data on finding connection patterns for the inputs and outputs.
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An ANN has several ways on finding patterns on the given data. A feed forward system exhibits a forward behavior. Two passes compromises the training of the ANN: the forward pass and backward pass. In the forward pass, the output is being computed from the input data. This is followed by the backward pass in which the error of the output in comparison with the target output is computed. The error will then be transmitted back to the network
and
modifications
are
made
in
the
connection
weights.
Backpropagation training functions were used to train the feed forward neural network. The set of data needed for the model are the inputs and the target output/s. It will then be divided into the training, testing, and validation data set. Before the model undergoes training, weights and biases are first initialized and during the training process, weights and biases are iterated to reduce error, the default error metric for feed forward neural network is mean square error which is measured between the model‘s output and the target output. The testing data set evaluates the performance of the model while the validation data set is used to avoid overfitting. The training is stopped once the error will no longer decrease.
The tan-sigmoid transfer function will be used by multilayer networks. Backpropagation is useful for feed forward networks since they do not have feedbacks and backward propagation computes for errors on the network. A Levenberg-Marquardt algorithm provides numerical solutions onto minimizing
57
problems in curve fitting. The manner of training optimizes problems onto minimizing the errors on the set of training data that was given.
Input
Hidden Layer
x1 x2
w1,1
. . .
a
∑ w1,n
b
f’c c
xn
Figure 3.8 Sample Structure of an ANN Model for the prediction of the increase in compressive strength
Shown in Table 3.2 are the parameters for circular and non-circular columns which that can be chosen as input and the ANN model is the ultimate confined compressive strength of the concrete as shown in table.
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Table 3.2 Table of Parameters for ANN Modeling INDEPENDENT VARIABLES cross-sectional length, b cross-sectional length, h height of the column, L corner radius, r area of concrete core, Acc volumetric ratio of CFRP, ρcfrp, volumetric ratio of longitudinal steel, ρl volumetric ratio of transverse steel, ρs CFRP tensile strength, fcfrp CFRP elastic modulus, Ecfrp CFRP ultimate strain, εcfrp elastic modulus of steel, Es steel tensile strength, fys unconfined compressive strength, f‘co
DEPENDENT VARIABLES
ultimate confined compressive strength, f‘cc
3.1.5 Evaluation of the ANN Models
After the training phase of an ANN model, its performance is tested by different criteria. The Pearson product-moment correlation coefficient (R) will be used for the evaluation of the ANN models and the created models will be compared with other existing models that were previously made. The formula is given by: -
where: = mean of x dataset
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-
(Equation 3.10)
= mean of y dataset = standard deviation of x dataset = standard deviation of y dataset
The Pearson product-moment correlation coefficient (R) is a common measure of the correlation between two variables X and Y. R ranges from -1 to +1. A value of +1 indicates that there is a perfect positive linear relationship between variables while a value of -1 indicates that there is a perfect negative linear relationship between variables and a correlation of 0 indicates that there is no relationship between the two variables.
3.2 Conceptual Framework
Figure 3.9 illustrates the concept and outline of the study. Databases of experiments from previous studies are gathered having different kinds of cross-sectional shapes which are circular, square and rectangular sections. The confinement effect of CFRP in square and rectangular sections of concrete columns is known to be complicated and less effective compared to circular sections. The parameters needed for circular, square and rectangular columns are ρcfrp, ρl, ρs, b, h, L, r, fcfrp, Ecfrp, εcfrp, Es, fys, Eys, Acc and f‘co.
Artificial neural network will be used in this study to test the correlation between the input parameters and the output. The self-organizing map 60
(SOM), backpropagation and Levenberg-Marquardt Algorithm is incorporated in ANN modeling. The Pearson product-moment correlation coefficient (R) is used for the selection of the best model with resulting values nearest to 1. The output will be compared with existing proposed models and is expected to produce constructive results. It will be used for the prediction of the ultimate compressive strengths of circular, square and rectangular columns confined with CFRP.
Output: Ultimate Confined Compressive Strength Parameters: b, h, L, r, Acc, f‘co ρcfrp,, fcfrp, Ecfrp, εcfrp ρl, ρs, Es, fys
Artificial Neural Network
Database of CFRP-confined columns
Figure 3.9 Conceptual Framework
61
CFRPconfinement model
CHAPTER IV RESEARCH METHODOLOGY AND RESEARCH DESIGN
4.1 Data Gathering
An analytical method of research was used in this paper. The group first searched for journals of previous studies which are related to our research about confinement of CFRP in circular, square, and rectangular columns. To conduct this study, a large database is needed which was obtained from journals of previous researches. Since there were already a lot of studies on circular columns, the group also included data that used CFRP confinement in rectangular and square columns. The group selected parameters from circular, square, and rectangular columns that contribute to the prediction of the ultimate confined compressive strength and increase in strength of CFRP confined reinforced concrete columns. Since the data were taken from previous studies related to this one, the group will not be conducting experimental tests. The complete set of data gathered can be found in Appendix B. The parameters considered for this study include ρcfrp, ρl, ρs, b, h, L, r, fcfrp, Ecfrp, εcfrp, Es, fys, Eys, Acc, f‘co and f‘cc since they are the ones available from previous studies.
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4.2 Normalization of Data
The data gathered was normalized to ensure that the database structure is free from undesirable characteristics that would result in a biased network. Normalization of data makes all input parameters in an equal field using a linear equation having a limit that ranges from a minimum value of 0.1 up to a maximum value of 0.9. The group used the equation of a line (y2 - y1) = m (x2 - x1) and y = mx + b to determine the slope and the y-intercept respectively. All data were normalized to have the same range of values before inputting it to the ANN.
4.3 Self Organizing Map
In this study, the parameters were randomly grouped by the SOM according to common properties observed by the program. Every time the parameters were randomly grouped into different combinations, the researchers critically analyzed the classification set by the SOM to identify its grouping. Significant parameters which govern the grouping and that has direct relation with the ultimate confined compressive strength were chosen for the back-propagation model.
63
4.4 Training and Testing of the ANN
Back-propagation was used to train the ANN to analyze the total structure of the program. The normalized data was randomly separated into three input groups, for training set, testing set, and validation set. Each group was trained and tested using separate networks and using different number of nodes to get the most accurate model.
4.5 Selection of Best ANN Model
After the training phase of an ANN models, their performance was tested by different criteria. The evaluation is done through utilizing test data, those that are not been used in the training phase. The Pearson productmoment correlation coefficient (R) will be used for the selection of the best model. To obtain R, the target and the actual data had to be unnormalized. Once all the verification is done, a suitable model can be chosen.
4.6 Comparison with existing models
The ANN model was compared with existing models by different authors: Al-Salloum (2007), Lam and Teng (2003), Illki et al. (2008), Salehian
64
and Kheyroddin (n.d.), Hosotani and Kawashima (1999), and ACI 440 model (2002) to measure the performance of the ANN model. The existing proposed models gathered by the authors were utilized to test the accuracy of the ANN model that the group produced.
Table 4.1 Models used for the evaluation of backpropagation models Models
f‘cc (compressive strength of confined column)
Al-Salloum (2007)
Lam and Teng (2003) Ilki et al. (2008) H.R. Salehian, A. Kheyroddin (n.d.) Hosotani and Kawashima (1999) ACI 440 (2002)
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DATA GATHERING
Parameters include pcfrp, ps, f‘co, f‘cc, diameter (D) and length (L) for circular columns and added parameters like shape dimension (b and h) and corner radius (r) for square and rectangular columns.
NORMALIZATION OF DATA
Using equations (y2 - y10) = m (x2 - x1) and y = mx + b
Using SOM U-matrix Component Planes Neighbourhood Functions Bar Graph
CLASSIFICATION OF DATA
Parameters: Pearson productmoment correlation coefficient (R).
TRAINING AND TESTING THE ANN
COMPARISON WITH EXISTING MODELS
DATA ANALYSIS & CONCLUSION
Figure 4.1 Methodology of the study
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CHAPTER V DATA PRESENTATION AND ANALYSIS
5.1 Data Gathering
The group gathered data of columns with circular and non-circular cross-sections, axially and concentrically loaded, and CFRP-wrapped with and without steel reinforcements. The compiled database was comprised of 422 columns: 310 circular and 112 non-circular (square and rectangular) which is attached in Appendix B. Tables 5.1 and 5.2 shows the summary of data for circular and non-circular columns.
For the artificial neural network model to be acceptable the input parameters for testing data and training data must be the same. Also, the statistical parameters of both sets of data must be of the same range. Statistical parameters such as mean, minimum and maximum values, standard deviation and variance were utilized to show the conformity of the gathered data. Tables 5.3 to 5.6 show the statistical parameters for the compiled database. All the data collected were normalized with 0.1 as the minimum value and 0.9 as the maximum value.
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Table 5.1 Summary of Database of Circular Columns
68
Table 5.2 Summary of Database of Non-Circular Columns
69
Table 5.3 Statistics of circular columns without steel reinforcements 148 Data D L fcfrp ρcfrp f‘co f‘cc f‘cc/f‘co Minimum 100.00 200.00 1908.00 0.15 17.20 23.74 1.00 Maximum 300.00 1500.00 4510.00 3.58 48.80 161.30 4.25 Mean 154.40 382.20 3631.00 0.86 32.14 54.35 1.72 Std. Dev. 41.17 263.80 798.50 0.59 9.32 24.25 0.63 Variance 1695.00 69570.00 6E+05 0.34 86.81 588.00 0.40
Table 5.4 Statistics of circular columns with steel reinforcements L 300.00 600.00
Acc ρs 13273.00 0.00 49088.00 4.16
D 150.00 300.00
Mean Std. Dev. Variance
197.60 529.70 19699.00 1.56 3817.00 0.48 25.19 50.25 38.98 85.52 10894.00 1.31 426.6.00 0.37 7.10 21.10 1520.00 7314.00 1.00E+08 1.72 2.00E+05 0.14 50.43 445.20
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fcfrp 3070.00 4510.00
ρcfrp f‘co f‘cc f‘cc/f‘co 0.12 13.87 27.05 0.99 1.76 36.20 150.80 5.42
162 Data Minimum Maximum
2.08 0.82 0.68
Table 5.5 Statistics of non-circular columns without steel reinforcements 78 Data b h r L fcfrp Minimum 90.00 108.00 0.00 300.00 580.00 Maximum 200.00 230.00 60.00 610.00 4519.00 Mean 150.10 159.90 24.00 417.30 3029.86 Std. Dev. 20.70 23.18 15.00 122.80 1421.02 Variance 428.46 537.30 236.00 15082.00 2019285.00
ρcfrp 0.22 3.93 1.26 1.01 1.02
f‘co f‘cc f‘cc/f‘co 22.60 25.81 1.14 54.10 113.59 2.10 35.68 52.47 1.47 8.85 16.69 0.60 78.33 278.60 0.36
Table 5.6 Statistics of non-circular columns with steel reinforcements 34 Data b h r L Acc ρs fcfrp Minimum 140.00 140.00 0.00 280.00 15006.20 0.09 875.00 Maximum 914.00 1270.00 51.00 2700.00 702244.00 1.17 4300.00 Mean 336.26 401.10 24.00 1286.00 126610.40 0.59 3108.60 Std. Dev. 182.94 237.30 14.00 727.50 161932.20 0.39 916.50 Variance 33466.00 56318.00 201.00 5E+05 2.62E+10 0.15 840061.40
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ρcfrp f‘co f‘cc f‘cc/f‘co 0.18 9.00 9.46 1.05 3.07 63.79 79.59 1.25 0.73 30.42 35.66 1.17 0.68 14.52 18.12 0.14 0.47 210.81 328.15 0.02
5.2 Self-Organizing Map
To further organize the compiled database, the group utilized the selforganizing map. The behavior of circular and non-circular columns, considering the increase in compressive strength was further analyzed by iterating random combination of parameters. Many different models were created, a total of 136 runs and models were made. For the circular columns, models for columns without reinforcement, with reinforcement, and the combination of with and without reinforcements were made. The same modeling was done for non-circular columns and also SOM models for combined circular and non-circular columns.
From the SOM models, weights were generated and were used to group the sets of data using the distance formula.
(Equation 5.1)
where:
d = distance from the reference of a point x1 = 1st parameter coordinate x2 = weight of the first coordinate y1 = 2nd parameter coordinate y2 = weight of the second coordinate
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5.2.1 Self-Organizing Map Models
5.2.1.1 Naming of the SOM Models
The SOM models were named based on the number of groups, the cross-section of the columns used as input data, and the presence of steel reinforcements in the columns. The second number signifies the number of groups. The set of letters after the number could be C for circular, NC for noncircular, and CNC for the combination of circular and non-circular columns as input data in the SOM model. Lastly, the last set of letters signifies the steel reinforcements: NS for without steel reinforcements, S for columns with steel reinforcements, and SNS for the combination of columns with and without steel reinforcements. Given the SOM name, SOM 5-3 CNC-SNS, it can be said that it has three (3) groups and the input data are circular and noncircular columns which is reinforced with steel and unreinforced.
5.2.1.2 SOM Models
The models were chosen from 136 SOM models produced. These models exhibited groupings which is in accordance with most of the proposed models for CFRP-confined columns
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Table 5.7 Self-organizing map models Model
Parameters
No. of Groupings No. of Data
ρcfrp, f‘co
3
148
ρs, ρcfrp, f‘co
2
162
SOM 2-2 C-SNS
ρs, ρcfrp
2
310
SOM 2-2 NC-NS
ρcfrp, f‘co
2
78
r, Acc, ρs, ρcfrp, f‘co
2
34
SOM 3-2 NC-SNS
r, ρcfrp, f‘co
2
112
SOM 1-2 CNC-NS
r, ρcfrp, f‘co
2
226
r, ρs, ρcfrp
2
196
r, L, ρs, ρcfrp, f‘co
3
422
SOM 2-3 C-NS SOM 1-2 C-S
SOM 4-2 NC-S
SOM 2-2 CNC-S SOM 5-3 CNC-SNS
These nine SOM models are subdivided into three: circular columns, non-circular columns, and combined circular and non-circulars. Furthermore, the three divisions can again be divided into three: without steel reinforcements, with steel reinforcements, and the combination of data with and without steel reinforcements. This was done so, so that a comprehensive analysis will be done in each type of column and also with the combination of all the columns.
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5.2.2 SOM Results
5.2.2.1 Circular columns
SOM 2-3 C-NS SOM 2-3 C-NS p = pcfrp, fco 3groups; K-means 0.9 weights group 1 group 2 group 3
0.8 0.7
fco
0.6 0.5 0.4 0.3 0.2 0.1 0.1
0.2
0.3
0.4
0.5 pcfrp
0.6
0.7
0.8
0.9
Figure 5.1: SOM 2-3 C-NS (ρcfrp, f‘co :3 GROUPS) Hits
SOM 26-Feb-2011
Figure 5.2: 2D Graph of ungrouped (left) and grouped (right) SOM 2-3 C-NS
75
The parameters used for SOM 2-3 C-NS are volumetric ratio of CFRP (ρcfrp), and unconfined compressive strength (f‘co). The data set was divided into three (3) groups: Red – 1st group, Green – 2nd group and Blue – 3rd group. The histograms or the 2D graphs on Figure 5.2 made a simple visualization of the ungrouped and grouped data. On Figure 5.2, the figure on the left, blue spots represents the density of data in that part of the 2D graph and these data were clustered on the figure on the right.
Figure 5.3: Histograms with bar graphs (SOM 2-3 C-NS)
To further analyze which parameter has affected the clustering in this SOM model, Figure 5.3 was utilized, wherein each bar represents the input parameters: ρcfrp and f‘co. The blue (3rd) group has consistently high values
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for ρcfrp and low to high values of f‘co while the green (2 nd group) has consistently low values for ρcfrp and very low to moderate values for f‘co. Lastly the red (1st group) has low to moderate values of ρcfrp and f‘co.
U-matrix
pcfrp 0.162
0.615
0.0881
0.379
0.0141
0.142 d
f’co
Labels 0.862
0.511
0.159
SOM 26-Feb-2011
d
Figure 5.4: U-matrix and component planes of SOM 2-3 C-NS
The analysis from the histograms with labels and bar graphs can further be demonstrated in Figure 5.4. On the U-matrix, we can see that the data has dissimilar properties because of moderate to high values. In the component plane of f‘co, it is visibly seen that the highest values are in the third group and the lowest values are within the second group. On the other hand, the values of ρcfrp were scattered within the three (3) groups.
To quantitatively analyze the clustered data, the minimum, maximum, average, standard deviation and variance of each group were obtained. The
77
two (2) parameters were also related with the ultimate confined compressive strength (f‘cc) and the increase in compressive strength (f‘cc/f‘co). The computed statistical values for each group are shown in Table 5.8.
Table 5.8 Statistical parameters for each group of SOM 2-3 C-NS
SOM 2-3-1 C-NS min max var stdev ave SOM 2-3-2 C-NS min max var stdev ave SOM 2-3-3 C-NS min max var stdev ave
pcfrp 0.234 1.467 0.088 0.297 0.623 pcfrp 0.147 1.813 0.233 0.483 0.854 pcfrp 0.289 3.579 0.736 0.858 1.326
f'co 30.530 41.100 13.224 3.636 34.344 f'co 17.200 26.730 7.986 2.826 22.239 f'co 38.000 48.800 15.380 3.922 45.300
f'cc 30.400 104.300 196.138 14.005 51.427 f'cc 23.740 74.920 182.824 13.521 41.216 f'cc 48.100 161.300 941.107 30.677 83.503
f'cc/f'co 0.996 2.881 0.144 0.379 1.498 f'cc/f'co 1.056 3.446 0.366 0.605 1.865 f'cc/f'co 1.088 4.245 0.809 0.900 1.906
In Table 5.8, the input parameters were correlated with its corresponding f‘cc and f‘cc/f‘co. With this, it was analyzed what combination of values will yield the highest ultimate compressive strength and increase in compressive strength. The highest increase in compressive strength, 1.906, was achieved in the 3rd group wherein the amount of CFRP and unconfined compressive strength are both high. However, it was also observed that the next highest increase in strength, 1.865, was obtained from the group with the
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lowest concrete column strength. The difference with the amount of CFRP wrap is also significant. From, the third group, which has the highest increase in compressive strength, has the highest amount of CFRP wrap while the 2 nd group has only a moderate amount of CFRP confinement, still it has attained a significant increase in strength which only has a 0.041 difference from the highest increase in strength.
It can be concluded, that CFRP confinement has significant effects on low-strength column. Even low to moderate amount of CFRP can increase the column‘s compressive strength to a significant level.
SOM 1-2 C-S SOM 1-2 C-S p =ps, pcfrp, fco 2groups; K-means weights group 1 group 2 1 0.8
fco
0.6 0.4 0.2 0 1 1 0.5 pcfrp
0.5 0
0
ps
Figure 5.5 SOM 1-2 C-S (ρs, ρcfrp, f‘co :2 GROUPS)
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Hits
SOM 26-Feb-2011
Figure 5.6 2D Graph of ungrouped (left) and grouped (right) SOM 1-2 C-S
For SOM 1-2 C-S, parameters: volumetric ratio of steel (ρs), volumetric ratio of CFRP (ρcfrp), and unconfined compressive strength (f‘co) were used as input parameters. The SOM divided the data set into two groups: red – 1st group and green – 2nd group for the 2D histograms and blue – 1st group and green – 2nd group for the 3D graph.
From Figure 5.5, the 3D graph, it is very visible that the blue (1st) group has low values of ρs while the green (2nd) group has relatively high amount of steel. The hit histogram showed that more data belonged to the red (1st) group and only few were appointed to the green (2 nd) group, most of which is located at the bottom left of the histogram.
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Figure 5.7 Histograms with bar graphs (SOM 1-2 C-S)
To further analyze which parameter has affected the clustering in this SOM model, Figure 5.7 was utilized, wherein each bar represents the input parameters: ρs, ρcfrp and f‘co. The green (2nd) group has a relatively moderate to high value of ρs and f‘co while the red (1st group) has a relatively low ρs and a higher value of ρcfrp compared to the second group.
ps
U-matrix
pcfrp
0.299
0.854
0.64
0.158
0.48
0.382
0.0176
0.106 d
0.123 d
Labels f'co 0.852 0.486 0.12 d
Figure 5.8 U-matrix and component planes of SOM 1-2 C-S SOM 26-Feb-2011 81
In Figure 5.8, the components planes for the three (3) parameters: ρ s, ρcfrp and f‘co was illustrated. From the component planes, the study had similar findings about each grouping. The 1 st group which is located at the top and bottom left of the histogram had low values of ρs, high values of ρcfrp and low to moderate values of f‘co. It is visibly seen that high values of ρ s are all located in the 2nd group.
Statistical parameters for each grouping were also utilized by this study for quantitative analysis. Table 1.2 shows the statistical parameters for each grouping in SOM 1-2 C-S and the correlation with the ultimate compressive strength (f‘cc) and the increase in compressive strength (f‘cc/f‘co).
Table 5.9 Statistical parameters for each group of SOM 1-2 C-S SOM 1-2-1 C-S min max var stdev ave SOM 1-2-2 C-S min max var stdev ave
ps 0.000 1.933 0.274 0.523 0.919 ps 3.655 4.160 0.021 0.146 3.887
pcfrp 0.120 1.760 0.146 0.382 0.519 pcfrp 0.120 1.470 0.095 0.308 0.346
f'co 13.870 36.200 53.747 7.331 24.432 f'co 20.000 36.200 29.777 5.457 27.921
f'cc f'cc/f'co 27.047 0.985 123.640 5.420 382.637 0.750 19.561 0.866 49.590 2.146 f'cc f'cc/f'co 31.600 1.215 150.800 4.166 682.602 0.358 26.127 0.598 52.647 1.843
In Table 5.9, the highest increase in compressive strength was achieved in the 1st group wherein the amount of CFRP is high and low amount
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of steel reinforcements and unconfined compressive strength. Similar with the previous SOM model, SOM 2-3 C-NS, the amount of CFRP governed the clustering. It gave more increase in compressive strength to columns than that given by steel reinforcements.
SOM 2-2 C-SNS
SOM 2-2 C-SNSp = ps,pcfrp 2 groups; K-means 0.9 0.8 0.7
pcfrp
0.6 0.5 0.4 0.3 0.2 0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ps
Figure 5.9 SOM 2-2 C-SNS (ρs, ρcfrp :2 GROUPS)
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0.9
Hits
SOM 26-Feb-2011
Figure 5.10 2D Graph of ungrouped (left) and grouped (right) SOM 2-2 C-SNS
For SOM 2-2 C-SNS, parameters: volumetric ratio of steel (ρs) and volumetric ratio of CFRP (ρcfrp) were used as input parameters. The SOM divided the data set into two groups: red – 1st group and green – 2nd group for the 2D histograms and plus sign – 1st group and diamond – 2nd group for the 3D graph.
From Figure 5.9, the 3D graph, it is very visible that the 1 st group has low values of ρs while the 2nd group has relatively high amount of steel. The hit histogram showed that more data belonged to the red (1st) group and only few were appointed to the green (2nd) group, most of which is located on the top.
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Figure 5.11 Histograms with label (SOM 2-2 C-SNS)
Figure 5.12 Histograms with bar graphs (SOM 2-2 C-SNS)
Figure 5.11 consists of labels: NS – circular without steel, S – circular with steel. Together with the labels are the color-coded groupings. It was observed that the groupings consist of circular with and without steel reinforcements in the red (1st) group and only circular with steel reinforcements in the green (2nd) group.
The groupings made by the SOM were further analyzed by the use of Figure 5.12. The bar graphs represent the input parameters which are ρ s and ρcfrp. The red (1st) group consists of low ρs and high ρcfrp while for the green (2nd) group has high ρs and low ρcfrp.
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U-matrix
ps 0.243
0.844
0.122
0.422
0.000826
8.35e-007 d Labels
pcfrp 0.683
S S S SS S S S S S S S S S S S SS S S SS S S S S S SS S S SS NS SS NSNS NS NSNSNSNS NS NS NS NS NSNSNS NSNSNS NSNSNS
0.403
0.123
SOM 26-Feb-2011
d
Figure 5.13 U-matrix and component planes of SOM 2-2 C-SNS
The analysis from the histograms with labels and bar graphs can further be demonstrated in Figure 5.13. On the U-matrix, we can see that the data located at the bottom is a good cluster because of the low values on this region. Low values of ρs can be seen in the 1st group while high values of ρ s are located on the 2nd group. In the component plane of ρcfrp, it can be seen that the high values are almost equally divided within the two (2) groups.
To quantitatively analyze the clustered data, the group obtained the minimum, maximum, average, standard deviation and variance of each group. The three (3) parameters were also related with the ultimate compressive strength (f‘cc) and the increase in compressive strength (f‘cc/f‘co). The computed statistical values for each group are shown in Table 5.10.
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Table 5.10 Statistical parameters for each group of SOM 2-2 C-SNS SOM 2-2-1 C-SNS min max var stdev ave SOM 2-2-2 C-SNS min max var stdev ave
ps 0.000 1.289 0.208 0.456 0.328 ps 1.828 4.160 0.912 0.955 3.233
pcfrp 0.120 3.579 0.288 0.537 0.711 pcfrp 0.120 1.467 0.109 0.330 0.410
f'cc f'cc/f'co 23.740 0.985 161.300 5.420 488.533 0.616 22.103 0.785 51.837 1.924 f'cc f'cc/f'co 31.600 1.021 150.800 4.166 659.745 0.363 25.686 0.602 54.048 1.831
In Table 5.10, the highest increase in compressive strength was achieved in the 1st group wherein the amount of CFRP is high and the amount of steel is low, a similar result with that of SOM 1-2 C-S. This is true when the CFRP column is loaded, the steel reinforcement yields easily which then cause the carbon fiber to carry the load which makes the confinement effective.
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5.2.2.2 Non-circular columns
SOM 2-2 NC-NS
Figure 5.14 2D graph of SOM 2-2 NC-NS
Figure 5.15 2D Graph of ungrouped and grouped (right) SOM 2-2 CNC-SNS
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For SOM 2-2 NC-NS, volumetric ratio of CFRP (ρcfrp), and unconfined compressive strength (f‘co) were used as input parameters. The SOM divided the data set into two groups: red – 1st group and green – 2nd group for the 2D histograms and blue – 1st group and green – 2nd group for the 2D graph.
From Figure 5.15, the 2D graph, it is very visible that the blue (1st) group has lower f‘co and high values of f‘co can be found in the green (2 nd ) group. Figures 5.14 and 5.15 also showed that the amount of data was not equal and that one groups is more dominant in quantity than the other.
Figure 5.16 Histograms with bar graphs (SOM 2-2 NC-NS)
The clustering made by the SOM was analyzed by the use of Figure 5.16. The bar graphs represent the input parameters: ρ cfrp, and f‘co. The red
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(1st) group consists of low ρcfrp and f‘co. For the green (2nd) group, the average of the values of ρcfrp was higher than that of the first group as well as the values of f‘co.
Figure 5.17 U-matrix and component planes of SOM 2-2 NC-NS
In Figure 5.17, the component planes for the two (2) parameters: ρ cfrp and f‘co were illustrated. From the component planes, the study had similar findings about each grouping. The 1st group which is located above the histogram has lower values of ρcfrp and f‘co. This also showed that the values of f‘co for the second group were higher than that of the first group and the average values of ρcfrp were higher than that of the other group.
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Table 5.11 Statistical parameters for each group of SOM 2-2 NC-NS SOM 2-2-1 NC-NS min max variance std dev mean SOM 2-2-2 NC-NS min max variance std dev mean
pcfrp 0.220 1.760 0.256 0.506 0.878 pcfrp 0.220 3.930 1.639 1.280 1.883
f'co f'cc f'cc/f'co 22.600 25.809 1.016 36.600 96.504 4.021 15.829 232.772 0.422 3.979 15.257 0.650 29.861 48.756 1.673 f'co f'cc f'cc/f'co 35.800 39.480 0.940 54.100 113.586 2.737 36.490 303.175 0.186 6.041 17.412 0.431 44.997 58.407 1.316
In Table 5.11, the input parameters were correlated with its corresponding f‘cc and f‘cc/f‘co. With this, it can be analyze what combination of values will yield the highest ultimate compressive strength and increase in compressive strength. From the table, it can also be observed that the maximum values of ρcfrp in the second group was a lot higher than that of the first group making the mean higher. The values of f‘cc/f‘co in the first group were higher than that of the second group.
The data that were obtained from the clustering showed that if ρcfrp has a high value as well as the f‘co, the increase in strength would not be very effective comparing it to the cluster with lower values of ρ cfrp \and f‘co which has a higher increase in strength. With this, the application of CFRP for retrofitting is more appropriate in concrete with lower strength since higher
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f‘cc/f‘co was achieved. If retrofitting is to be done to a high strength column, the effect of CFRP is not very much utilized.
SOM 4-2 NC-S
Figure 5.18 2D Graph of ungrouped (left) and grouped (right) SOM 4-2 NC-S
For SOM 4-2 NC-S, corner radius (r), area of the concrete core (Acc), volumetric ratio of transverse steel reinforcements (ρ s), volumetric ratio of CFRP (ρcfrp), and unconfined compressive strength (f‘co) were used as input parameters. The SOM divided the data set into three groups: red – 1st group, green – 2nd and blue - 3rd group for the 2D histograms. The 3D graph for this SOM is not available since the SOM has five (5) input parameters. The histograms or the 2D graphs on Figure 5.18 made a simple visualization of the ungrouped and grouped data. The histograms showed that the red (1 st
92
group) has the most dominant number among the three and the blue (2 nd group) and green (3rd group) has almost the same numbers. The clustering of this particular SOM is not dispersed.
Figure 5.19 Histograms with bar graphs (SOM 4-2 NC-S)
The groupings made by the SOM were further analyzed by the use of Figure 5.19. The bar graphs represent the input parameters: r, Acc, ρ s, ρcfrp, and f‘co. The red (1st) group consists of high r, moderately high f‘co and moderately high cross-sectional area. The green (2nd) group consists of low r and high steel volumetric ratio.
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Figure 5.20 U-matrix and component planes of SOM 2-2 NC-NS
In Figure 5.20, the components planes for the two (5) parameters: r, Acc, ρs, ρcfrp and f‘co were illustrated. From the component planes, the study had similar findings about each grouping. The values that were shown through the histogram and the bar chart is similar with what is shown in this figure. High values of r is on the top portion where red (1 st group) is. High volumetric steel ratio is found on the green (2 nd) group along with high f‘co.
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Table 5.12 Statistical parameters for each group of SOM 2-2 NC-S SOM 4-2-2 NC-S
r
Acc
ps
pcfrp
f'co
f'cc
f'cc/f'co
min
30.000
61504.000
0.092
0.210
30.300
30.704
1.010
max
51.000
702244.000
0.740
3.070
41.200
53.148
1.560
variance
33.763
41603849424.917
0.031
0.782
11.512
49.669
0.024
5.811
203970.217
0.175
0.884
3.393
7.048
0.155
mean
32.813
211167.875
0.275
0.948
32.838
38.997
1.185
SOM 4-2-3 NC-S
r
ps
pcfrp
std dev
Acc
f'co
f'cc
f'cc/f'co
min
0.000
15006.250
1.172
0.371
24.770
27.660
1.030
max
0.000
15006.250
1.172
1.114
63.790
79.590
1.462
variance
0.000
0.000
0.000
0.158
315.120
402.714
0.017
std dev
0.000
0.000
0.000
0.397
17.752
20.068
0.129
mean
0.000
15006.250
1.172
0.743
45.420
54.301
1.210
ps
pcfrp
SOM 4-2-3 NC-S
r
min
25.000
52500.000
0.520
0.180
9.000
9.456
1.036
max
30.000
115600.000
0.748
0.750
20.000
23.000
1.150
variance
4.444
673135470.933
0.012
0.049
18.118
22.744
0.001
std dev
2.108
25944.854
0.109
0.222
4.257
4.769
0.033
29.000
80601.600
0.626
0.355
14.540
15.406
1.056
mean
Acc
f'co
f'cc
f'cc/f'co
In Table 5.12, the input parameters were correlated with its corresponding f‘cc and f‘cc/f‘co. With this, it can be analyze what combination of values will yield the highest ultimate compressive strength and increase in compressive strength. From the table, it can also be observed that the second cluster has the least value of r, next to it is the third cluster and the first cluster has the largest values of r. For the Acc, the second group has the least values while the first group has the greatest values.
As observed, the highest increase in compressive strength is in group two (2) where the values of r are zero which means that they are sharp-edged
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columns. Thus, CFRP confinement is still effective even for sharp-edged columns.
SOM 3-2 NC-SNS
Figure 5.21 3D graph of SOM 3-2 NC-SNS
Figure 5.22 2D Graph of ungrouped and grouped (right) SOM 3-2 CNC-SNS
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For SOM 3-2 NC-SNS, corner radius (r), volumetric ratio of CFRP (ρcfrp), and unconfined compressive strength (f‘co) were used as input parameters. The SOM divided the data set into two groups: red – 1st group and green – 2nd group.
From Figure 5.21, the 3D graph, it is very visible that the blue (1st) group has higher f‘co and low values of f‘co can be found in the green (2nd) group. Figures 5.21 and 5.22 also showed that the amount of data was not equal and that one groups is more dominant in quantity than the other but the data is not dispersed.
Figure 5.23 Histograms with label (SOM 3-2 NC-SNS)
Figure 5.24 Histograms with bar graphs (SOM 3-2 NC-SNS)
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Figure 5.23 consists of labels for non circular sections: NS – without steel and S - with steel. The labels are together with the color-coded groupings.
The groupings made by the SOM were further analyzed by the use of Figure 5.24. The bar graphs represent the input parameters: r, ρcfrp, and f‘co. The values of r from the bar chart are almost the same on the average for the two clusters. The values of ρcfrp, and f‘co for the red (1st) group were both higher than that of the blue (2nd) group.
Figure 5.25 U-matrix and component planes of SOM 3-2 NC-SNS In Figure 5.25, the component planes for the two (3) parameters: r, ρ cfrp and f‘co were illustrated. From the component planes, the study had similar findings about each grouping. The values of r were increasing from left to right
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but the clustering for this particular SOM was divided into top and bottom which explains why the two clusters have similar values of r. For f‘co and ρ cfrp, the values of the second cluster, which was on the top part of the histogram, are lower which explains why the other cluster has bigger values.
Table 5.13 Statistical parameters for each group of SOM 2-2 NC-SNS SOM 3-2-1 NC-SNS r pcfrp f'co f'cc f'cc/f'co min 0.000 0.220 33.500 34.840 0.940 max 60.000 3.930 63.790 113.586 2.737 variance 291.790 1.527 63.553 288.455 0.155 std dev 17.082 1.236 7.972 16.984 0.394 mean 24.649 1.817 46.279 58.722 1.286 SOM 3-2-2 NC-SNS r pcfrp f'co f'cc f'cc/f'co min 0.000 0.180 9.000 9.456 1.010 max 60.000 1.760 36.600 96.504 4.021 variance 192.883 0.234 42.418 289.995 0.339 std dev 13.888 0.483 6.513 17.029 0.582 mean 23.954 0.748 28.067 41.763 1.485
In Table 5.13, the input parameters were correlated with its corresponding f‘cc and f‘cc/f‘co. With this, it can be analyze what combination of values will yield the highest ultimate compressive strength and increase in compressive strength. From the table, it can be also observed that the maximum values of ρcfrp in the first group was a lot higher than that of the second group making the mean higher. The values of f‘cc/f‘co in the second group were higher than that of the first group.
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The data that were obtained from the clustering showed that if ρcfrp has a high value as well as the f‘co, the increase in strength would not be very effective comparing it to the cluster with lower values of ρ cfrp \and f‘co which has a higher increase in strength.
With this, the application of CFRP for retrofitting is more appropriate in concrete with lower strength since higher f‘cc/f‘co was achieved. If retrofitting is to be done to a high strength column, the effect of CFRP is not very much utilized.
5.2.2.3 Combined circular and non-circular columns
SOM 1-2 CNC-NS
Figure 5.26 3D graph of SOM 1-2 CNC-NS
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Figure 5.27 2D Graph of ungrouped and grouped (right) SOM 1-2 CNC-NS
For SOM 1-2 CNC-NS, parameters: corner radius (r), volumetric ratio of CFRP (ρcfrp), and unconfined compressive strength (f‘co) were used as input parameters. The SOM divided the data set into two groups: red – 1st group and green – 2nd group for the 2D histograms and blue – 1st group and green – 2nd group for the 3D graph.
From Figure 5.26, the 3D graph, it is very visible that the blue (1 st) group has higher f‘co and low values of f‘co can be found in the green (2 nd) group. Figures 5.26 and 5.27 also illustrated an almost equal division of data.
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Figure 5.28 Histograms with label (SOM 1-2 CNC-NS)
Figure 5.29 Histograms with bar graphs (SOM 1-2 CNC-NS)
Figure 5.28, has labels C for circular and NC for non-circular. Similar with the previous SOM models for combined circular and non-circular, each group consists of both circular and non-circular. This grouping implies that the cross-section of a column has no significant effect on the clustering of the data.
It can be further analyzed which parameter has affected the grouping of data by examining Figure 5.29 which shows bar graphs and the colorcoded grouping. The bar graph represents the three (3) input parameters: corner radius (r), volumetric ratio of CFRP (ρcfrp), and unconfined compressive strength (f‘co).
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For the red (1st group), the values of r is low to moderate; low to high values of ρcfrp are included in the group and the f‘co has high values. On the other hand, the green (2nd) group has r with low to high values, low to moderate values of ρcfrp and low f‘co. From this, we can conclude that the clustering was based on the values of f‘co.
Figure 5.30 U-matrix and component planes of SOM 1-2 CNC-NS
In the analysis of the component planes in Figure 5.30, the dispersed values of r are illustrated. This explains why the values of r in the groupings made by the SOM are almost similar. For the component plane of ρcfrp, the highest amount of CFRP can be found on the bottom which belongs to the first group. Along with the high values of ρcfrp are also high values of f‘co.
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Whereas the top portion of the component planes of ρcfrp and f‘co, which is the green (2nd) group, has low to moderate values.
In order to relate these parameters with the ultimate compressive strength (f‘cc) and the increase in compressive strength (f‘cc/f‘co), the unnormalized data of the input parameters and the f‘cc and f‘cc/f‘co were correlated along with the respective statistical parameters for the two (2) groups. Table 5.14 shows the computed statistical values for each group.
Table 5.14 Statistical parameters for each group of SOM 1-2 CNC-NS SOM 1-2-1 CNC-NS min max var stdev ave SOM 1-2-2 CNC-NS min max var stdev ave
r
pcfrp
f'co
f'cc
0 100 841.1815 29.00313 62.30208 r 0 150 1099.471 33.15828 56.46965
0.22 3.93 1.049437 1.024421 1.238856 pcfrp 0.146667 1.813333 0.213769 0.462351 0.818173
35.2 54.1 31.66112 5.626822 42.28344 f'co 17.2 36.6 24.37537 4.937142 26.77523
39.48 161.3 613.1859 24.76259 65.70787 f'cc 23.74 96.504 199.6446 14.12956 44.83441
f'cc/f'co 0.94 4.2 0.42432 0.651398 1.574928 f'cc/f'co 1 4.021 0.359776 0.599813 1.71765
To further analyze quantitatively the SOM groupings made, Table 5.14 is used. It is shown, that the values of r for the two (2) groups is almost similar. Thus, the corner radius has no significant effect on the compressive strength of the CFRP confined columns. High values of ρ cfrp and f‘co is on group one whereas the second group has low values for ρcfrp and f‘co.
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However higher increase in strength was obtained in the second group. This observation was also found in SOM 5-3 CNC-SNS. The CFRP confinement really has the capability to strengthen columns however its effectiveness is lesser for high strength columns. The increase in strength is achieved more on low strength columns thus making it suitable for retrofitting. The effect of CFRP wrapping is prominent even for low amount of CFRP.
SOM 2-2 CNC-S
Figure 5.31 2D Graph of ungrouped and grouped (right) SOM 2-2 CNC-S
SOM 2-2 CNC-S has input parameters of corner radius (r), volumetric ratio of transverse steel reinforcements (ρs), and volumetric ratio of CFRP (ρcfrp). The input data were divided into two (2) groups. The red-colored hexagons being the first group while green-colored hexagons as the second
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group. Since the 3D graph is hard to analyze, the group made use of the 2D histograms. From Figure 5.31, the hit histogram showed that more data belonged to the green (2nd) group and only few were appointed to the red (1st) group, most of which is located below the histogram.
Figure 5.32 Histograms with label (SOM 2-2 CNC-S)
Figure 5.33 Histograms with bar graphs (SOM 2-2 CNC-S)
The labels in Figure 5.32 imply the following: C – circular and NC – non-circular. Figure 5.32 exhibited groupings where circular and non-circular belonged in the same group. Hence, it provides evidence that cross-sectional properties of a column have no significant effect on the clustering of the data. With this, r can be assumed having no major importance on the CFRP confinement of columns.
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To further analyze which parameter has affected the clustering in this SOM model, Figure 5.33 was utilized, wherein each bar represents every input parameters: r, ρs, and ρcfrp. The red (1st) group has consistently high values for ρs, low values for r, and low to moderate values of ρ cfrp while the green (2nd group) has wide-ranged values of r and ρcfrp and only low to moderate values for ρs.
Figure 5.34 U-matrix and component planes of SOM 2-2 CNC-S
In Figure 5.34, the components planes for the three (3) parameters: r, ρs, and ρcfrp was illustrated. From the component planes, the study had similar findings about each grouping. The 1st group which is located below the histogram had low values of r, high values of ρs and low to high values of ρcfrp. At the same time, similar result was obtained for the 2 nd group: moderate to high values of r, low to moderate ρcfrp and only low values for ρs.
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Statistical parameters for each grouping were also utilized by this study for quantitative analysis. Table 5.15 shows the statistical parameters for each grouping in SOM 2-2 CNC-S and the correlation with the ultimate compressive strength (f‘cc) and the increase in compressive strength (f‘cc/f‘co).
Table 5.15 Statistical parameters for each group of SOM 2-2 CNC-S SOM 2-2-1 CNC-S r ps pcfrp f'cc f'cc/f'co min 0.000 1.172 0.120 27.660 1.030 max 90.000 4.160 1.467 150.800 4.200 var 1214.145 1.160 0.127 620.126 0.358 stdev 34.845 1.077 0.357 24.902 0.599 ave 71.361 3.382 0.420 52.955 1.734 SOM 2-2-2 CNC-S r ps pcfrp f'cc f'cc/f'co min 25.000 0.000 0.120 9.456 1.000 max 150.000 1.933 3.070 123.640 5.400 var 1070.562 0.273 0.221 400.880 0.773 stdev 32.719 0.522 0.470 20.022 0.879 ave 89.908 0.833 0.553 46.248 1.980 The input parameters: r, ρs, and ρcfrp were linked with the ultimate compressive strength (f‘cc) and the increase in compressive strength (f‘cc/f‘co). From this correlation the researchers can deduce what combination of values for each parameter will yield higher increase in strength and also the parameter which can contribute in increasing the strength of a CFRP-confined column.
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The increase in compressive strength of the first group is also high; however, higher increase in compressive strength (f‘cc/f‘co) was obtained in the second group which has moderate to high values of r, low values of ρ s, and low to high values of ρcfrp. It can then be implied that ρcfrp has the most significant effect in strengthening the columns whereas the presence of transverse steel reinforcements is not that noteworthy with regards to the f‘cc/f‘co. Lastly, the effect of r can be considered negligible since circular and non-circular columns were clustered together. Therefore, the confinement of CFRP for non-circular columns is also effective.
SOM 5-3 CNC-SNS
Figure 5.35 2D Graph of ungrouped and grouped (right) SOM 5-3 CNC-SNS
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The parameters used for SOM 5-3 CNC-SNS are corner radius (r), height of column (L), volumetric ratio of transverse steel reinforcements (ρ s), volumetric ratio of CFRP (ρcfrp), and unconfined compressive strength (f‘co). The data set was divided into three (3) groups: Red – 1st group, Green – 2nd group and Blue – 3rd group. The 3D graph for this SOM is not available since the SOM has five (5) input parameters. The histograms or the 2D graphs on Figure 5.35 made a simple visualization of the ungrouped and grouped data. The histograms showed that the red (1st group) is dispersed while the blue (3 rd group) and green (2nd group) is close to each other. There was also an overlap that happened between the 1st and 2nd group.
Figure 5.36 Histograms with label (SOM 5-3 CNC-SNS)
Figure 5.37 Histograms with bar graphs (SOM 5-3 CNC-SNS)
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Figure 5.36 consists of labels: CNS – circular without steel, CS – circular with steel, NCNS - non-circular without steel, and NCS – non-circular with steel. Together with the labels are the color-coded groupings.
It was observed that the groupings consist of combined circular and non-circular with and without steel reinforcements. Thus, we can assume that the cross-sectional property and the presence of steel reinforcements do not have a significant effect on the grouping of the data.
The groupings made by the SOM were further analyzed by the use of Figure 5.37. The bar graphs represent the input parameters: r, L, ρ s, ρcfrp, and f‘co. The red (1st) group was consists of high ρs and f‘co; the values of r are consistently low while the values of L and ρcfrp are inconsistent, some are high and some are low. For the green (2 nd) group there were low and high values for r. Some of the f‘co values were low but the highest values for the f‘co can be found in this group. Few of the data has high values of ρ cfrp and the values of L and ρs are consistently low. Lastly, the 3rd group is also consisted of low to high values of r and ρcfrp but only low to moderate values of f‘co and ρs. The values of L are consistently high.
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Figure 5.38 U-matrix and component planes of SOM 5-3 CNC-SNS
The analysis from the histograms with labels and bar graphs can further be demonstrated in Figure 5.38. On the U-matrix, we can see that the data on the lower right is a good cluster or similar with each other because of the low values on this region. Moderate values of r are concentrated on the center while the values of L are from high to low from top to bottom. Many of the data has low values for the ρs while the values of ρcfrp are quite scattered. In the component plane of f‘co, it can be seen that the highest values are indeed within the 2nd group and the lowest values are within the 3 rd group. The grouping is quite imperfect since the quantization error is 0.155 which is quite high as compared to others. Thus, the group performed another analysis which is quantitative.
To quantitatively analyze the clustered data, the minimum, maximum, average, standard deviation and variance of each group were obtained. The
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five (5) parameters were also related with the ultimate compressive strength (f‘cc) and the increase in compressive strength (f‘cc/f‘co). The computed statistical values for each group are shown in Table 5.16.
Table 5.16 Statistical parameters for each group of SOM 5-3 CNC-SNS SOM 5-3-1 CNC-SNS min max var stdev ave SOM 5-3-2 CNC-SNS min max var stdev ave SOM 5-3-3 CNC-SNS min
r 0.000 100.000 689.311 26.255 62.147 r 0.000 100.000 744.473 27.285 82.239 r 5.000
L 200.000 1000.000 17401.596 131.915 329.839 L 280.000 1500.000 34603.787 186.021 526.376 L 500.000
ps 0.000 1.220 0.033 0.182 0.032 ps 0.000 4.160 2.039 1.428 1.780 ps 0.000
pcfrp 0.220 3.579 0.295 0.543 0.785 pcfrp 0.120 3.930 0.407 0.638 0.620 pcfrp 0.120
f'co 20.720 54.100 80.488 8.972 33.501 f'co 20.000 63.790 60.681 7.790 31.176 f'co 9.000
f'cc 23.740 161.300 533.318 23.094 54.471 f'cc 27.660 150.800 445.169 21.099 53.729 f'cc 9.456
f'cc/f'co 1.000 4.200 0.383 0.619 1.657 f'cc/f'co 0.940 4.200 0.367 0.605 1.753 f'cc/f'co 0.990
max
150.000
2700.000
1.140
3.790
43.900
113.586
5.400
var
2423.980
262098.75
0.205
0.794
94.528
319.785
0.931
stdev ave
49.234 74.748
511.956 810.609
0.452 0.417
0.891 0.934
9.723 23.090
17.883 42.244
0.965 2.013
In Table 5.16, the input parameters were correlated with its corresponding f‘cc and f‘cc/f‘co. With this, it can be analyze what combination of values will yield the highest ultimate compressive strength and increase in compressive strength. From the table, it can also be observed that the values of r, ρs, and ρcfrp are dispersed among the three (3) groups. The highest increase in compressive strength was achieved in the 3 rd group wherein the
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height of the column (L) is largest and the unconfined compressive strength (f‘co) is lowest.
It was assumed before this study that the increase in compressive strength is directly proportional with the amount of CFRP; however this SOM model refuted the assumption. In clarification, the amount of CFRP affects the strengthening, yet its effectiveness is minimal for high strength columns. For low strength columns, the effect of CFRP wrapping is prominent even for low amount of CFRP.
With this, the application of CFRP for retrofitting is appropriate since higher f‘cc/f‘co was achieved in columns with low f‘co. The corner radius, ρs, and ρcfrp have no significant effect. In addition, it has been determined that the height of the column has no significant effect on the strengthening of the column. Long columns can still be effectively strengthened by CFRP wrapping.
5.3 Backpropagation
Backpropagation is a type of supervised learning neural network. The target output which is the ultimate confined compressive strength is compared with the predicted value of the model. The division of data for training, testing and validation is: 70%, 15% and 15%.
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The selection of the parameters to be included in the Backpropagation models were done through the utilization of the Self-Organizing Map in the analysis of the circular and non-circular columns confined with CFRP.
The SOM models generated outputs that illustrates the parameters effect on the ultimate confined compressive strength, f‘cc and increase in compressive strength, f‘cc/f‘co. The parameters which showed the most significant effect to f‘cc and f‘cc/f‘co are the unconfined compressive strength, f‘co and the volumetric ratio of CFRP, ρcfrp.
The volumetric ratio of CFRP increases the compressive strength of the columns however its effect is also determined by the unconfined compressive strength of the column. Low-strength concrete columns attained higher increase in strength even with low amount of CFRP while for highstrength concrete columns, the CFRP confinement and the amount of CFRP has only minimal effect. Thus, the effectiveness of CFRP is intensely seen on low-strength columns therefore making CFRP a good retrofitting material.
The effect of the corner radius, r was found to be very minimal, since the SOM models produced groupings wherein the circular and non-circular columns are grouped together. Also, an SOM model generated a grouping wherein the highest increase in compressive strength was found in the group with zero corner radius. On the other hand, the amount of transverse steel
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reinforcements was also found to have minimal effect as compared to the amount of CFRP and unconfined compressive strength, however, it is still significant in the differentiation between columns with and without steel reinforcements.
Through the analysis of the SOM models produced, parameters were chosen which will be included in the backpropagation models. These parameters are: volumetric ratio of transverse steel reinforcements, volumetric ratio of CFRP, and unconfined compressive strength. To take into account the geometric properties of the columns, the cross-sectional dimensions and the height of the column were also included.
5.3.1 Backpropagation Models
Table 5.17 Backpropagation Models Model
Parameters
Target Output
BP-CNC-NS
b, h, L, ρcfrp, f‘co
f‘cc
BP-CNC-S
b, h, L, ρs, ρcfrp, f‘co
f‘cc
BP-CNC-SNS
b, h, L, ρs, ρcfrp, f‘co
f‘cc
Six (6) backpropagation models were made in this study. Three (3) of which has the increase in compressive strength (f‘cc/f‘co) as the output while the other three has the ultimate confined compressive strength (f‘cc) as the
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output. It was observed that the models with f‘cc as an output yielded better results. Thus, the three backpropagation with f‘cc as the output was chosen and will be discussed in this section.
The parameters used in the backpropagation models were decided upon the use of the Self-Organizing Map. Parameters ρcfrp and f‘co were the ones that governed the grouping in the SOM models. The parameters: b, h and L were included to correspond to the geometric properties of the column. However, the corner radius (r) was not included since it was found out that it only has minimal effect on the behavior of CFRP confined columns with respect to its confined compressive strength. Lastly, the parameter ρ s was included to characterize the presence of steel reinforcements in the column.
The inputs were first assigned with weights then the sum of the weighted inputs and the bias is entered to the hidden neuron with the tansig or tan-sigmoid transfer function. The output from the hidden neuron is transferred by using the linear transfer function or purelin.
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5.3.2 Backpropagation Results
5.3.2.1 Combined circular and non-circular without steel 5.3.2.1.1 BP-CNC-NS
The input parameters for BP-CNC-NS are: the geometrical properties (b and h), height of the column (L), volumetric ratio of CFRP (ρ cfrp), and unconfined compressive strength (f‘co). Table 5.18 shows the error analysis of BP-CNC-NS.
Table 5.18 Error analysis of BP-CNC-NS Hidden 2
3
4
5
6
7
8
R
0.56985
0.68913
0.57483
0.76956
0.74216
0.62489
0.89661
Rtrain
0.59841
0.71098
0.56089
0.79778
0.74079
0.69341
0.91317
Rtest
0.59386
0.56768
0.64759
0.56979
0.87970
-0.01895
0.74727
Rvalid
0.47262
0.64756
0.57265
0.76472
0.36618
0.82706
0.91171
Nodes
Evaluating the error measurements of BP-CNC-NS in Table 5.18, though, the correlation Rtest of eight (8) hidden nodes is lower as compared to others; the over-all correlation test of 0.89661 is the highest among others. Thus, the regression analysis closest to 1 is chosen which is BP-CNC-NS having eight (8) hidden nodes. Figure 5.39 illustrates the regression analysis
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of BP-CNC-NS. The figure for the regression analysis for the testing showed that the predicted values were not quite near to the actual values. Most of the data points were far from the Y=T line.
Figure 5.39 Regression Analyses of BP-CNC-NS
Weight and biases for each input parameter were obtained after the training of the ANN model. Table 5.19 shows the weight and biases for BPCNC-NS.
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Table 5.19 Weight and biases obtained from BP-CNC-NS Hidden Node
b
h
L
pcfrp
f'co
bias
1
0.091 -3.971
1.774
0.660
2
-3.683 -3.307
0.946
1.423 -0.875
5.084
0.260
3.145 -1.587 -3.805
0.944 -3.294
1.297
1.039
1.841
1.252
-1.618
4.544 -2.849
0.153 -1.555
1.632
0.009 -1.130 -2.610 -0.286
4.506 -1.102
0.722
0.592 -0.023
2.632
2.125 -0.528 -1.879
-0.655
3 4
-1.226
5
-1.277 -0.233
6 7 8
-1.037
1.154 -8.486
1.200 -1.847
2.056 -0.686
6.562
1.606
1.789 -2.234
weights
1.180
-1.621
The last bias is equal to 0.0975. These weights and biases will be used in obtaining the target output ultimate confined compressive strength (f‘cc).
In evaluation of BP-CNC-NS, the database of circular and non-circular columns without steel reinforcements were used to plot the predicted ultimate confined compressive strength of the model versus the actual ultimate confined compressive strength shown in Figure 5.40. A total of 226 data were plotted.
120
180
+10% 160
R=1 -10%
140
Prediicted f'cc
120 100 80 60 40 20 0 0.0
20.0
40.0
60.0
80.0
100.0 120.0 140.0 160.0 180.0
Actualf'cc
Figure 5.40 BP-CNC-NS model
It was observed from Figure 5.40 that the BP-CNC-NS model performed well since most of the plotted data, blue points were near the 45⁰ line. The R value obtained from the model is equal to 0.6505.
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5.3.2.1.2 Comparison with existing models
To analyze if the group has produced a good model, BP-CNC-NS will be compared to other proposed models on CFRP- confined circular and noncircular columns without steel reinforcements. These models are from AlSalloum (2008), Lam and Teng (2003), Ilki et al. (2008), and Kheyroddin and Salehian (n.d.). Al-Salloum (2007) 200 180
+10%
160
R=1 -10%
Predicted f'cc
140 120 100 80 60 40 20 0 0.0
50.0
100.0
150.0
Actual f'cc
Figure 5.41 Al Salloum‘s model (2007)
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200.0
Figure 5.41 shows the graph of the predicted vs. target ultimate confined compressive strength outputs of Al Salloum‘s model which is represented by the blue points. A good model should follow the predicted = target line or the 45⁰ line, however, in the case of Al Salloum‘s models most of the output were far from the predicted = target line. Thus, the regression (R) computed for this comparison was only 0.5167 which indicates that BP-CNCNS performed better than Al Salloum‘s model. Lam and Teng (2003) 180
+10%
160
R=1 -10%
140
Predicted f'cc
120 100 80 60 40 20 0 0.0
20.0
40.0
60.0
80.0
100.0 120.0 140.0 160.0 180.0
Actual f'cc
Figure 5.42 Lam and Teng‘s model (2003)
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On Figure 5.42, the blue points represent the predicted vs. target outputs of Lam and Teng‘s models and was put side by side to the 45⁰ line with R = 1 for evaluation . Just like in Al Salloum‘s model, Lam and Teng‘s model produced a low correlation (R) which is only 0.4935. Again, the graph shows that most of the predicted outputs is not close to the desired target output.
Ilki et al. (2008) 180
+10% 160
R=1 -10%
140
Predicted f'cc
120 100 80 60 40 20 0
0.0
20.0
40.0
60.0
80.0
100.0 120.0 140.0 160.0 180.0
Actual f'cc
Figure 5.43 Ilki et al.‘s model (2008)
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The predicted ultimate confined compressive strength of Ilki et al.‘s model versus the target ultimate confined compressive strength is shown in Figure 5.43 which is represented by the blue points. Again, the graph shows that most of the predicted data points are distant and do not coincide with the red 45⁰ line. The regression for this comparison was 0.4612 which is slightly lower than Lam & Teng‘s model.
Kheyroddin and Salehian (n.d.) 180
+10%
160
R=1 -10%
140
Predicted f'cc
120 100 80 60 40 20 0
0.0
20.0
40.0
60.0
80.0
100.0 120.0 140.0 160.0 180.0
Actual f'cc
Figure 5.44 Kheyroddin and Salehian‘s model (n.d.)
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Figure 5.44 shows the graph comparison of the predicted=target, 45⁰ line, and predicted vs. target ultimate confined compressive strength outputs of Kheyroddin and Salehian‘s model represented by the blue points. Again, the graph shows that most of the predicted vs. target data points are scattered and do not coincide with the 45⁰ line. The regression (R) for Salehian and Kheyroddin‘s model is 0.4009.
Table 5.20 Other proposed models versus BP-CNC-NS
Models
Al-Salloum
Lam and Teng
Illki et al.
R
0.5167
0.4935
0.4612
Salehian and Kheyroddin 0.4009
BP-CNCNS 0.6505
In Table 5.20, the performance of each model was evaluated through the comparison of the correlation R value. Other proposed models have almost similar performances. While the R value of BP-CNC-NS is quite low at 0.6505, it can still be observed that BP-CNC-NS is the best performing model since its R value is the highest as compared to the proposed models.
5.3.2.2 Combined circular and non-circular with steel 5.3.2.2.1 BP-CNC-S
BP-CNC-S has six (6) input parameters which are the following: the geometrical properties (b and h), height of the column (L), volumetric ratio of
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CFRP (ρcfrp), volumetric ratio of transverse steel reinforcement (ρs), and unconfined compressive strength (f‘co). In Table 5.21, the regression analysis of the model‘s training, testing, and validation are shown.
Table 5.21 Error analysis of BP-CNC-S Hidden 2
3
4
5
6
7
8
0.7904
0.95743
Nodes R
0.90296 0.35898 0.88704 0.90154 0.90713
Rtrain
0.91125 0.70877 0.93507 0.90892 0.96457 0.72131 0.95774
Rtest
0.79687 0.25983 0.83585 0.85284 0.86485 0.89739 0.95922
Rvalid
0.96259 0.79472 0.68697 0.91221 0.76279 0.86320 0.96065
Assessing BP-CNC-S using the regression analysis, the model with eight (8) hidden nodes was chosen as the best performing model. Most of the models performed well, but the model with eight (8) hidden nodes was chosen because it consistently performed well in the training, testing and validation having: Rtrain = 0.95774, Rtest = 0.95922, Rvalid = 0.96065 and the over-all performance is 0.95743 which is the highest among the others. The result for BP-CNC-S with eight (8) hidden nodes is illustrated in Figure 5.46. Most of the data points for the training, testing, and validation were near the Y=T line which means that the predicted values of the BP-CNC-S is near the actual values of f‘cc.
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Figure 5.45 Regression Analyses of BP-CNC-S
Weight and biases for each input parameter was obtained after the training of the model. Table 5.22 shows the weight and biases for BP-CNC-S.
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Table 5.22 Weight and biases obtained from BP-CNC-S Hidden Node b
h
L
ps
pcfrp
f'co
bias
weights
1
-3.344 -1.693 -3.004 -0.612 -1.350
1.377
7.901
-1.230
2
-1.119 -1.787
3
0.651 -3.720
1.787
2.577
3.282
1.001
-0.230
1.193 -2.547 -2.886
1.991
1.569
1.103
0.462
4
-2.205
2.266 -2.018
2.706 -0.782
3.334
0.208
5
-2.782 -0.067 -4.732
3.027 -1.633 -4.172
2.207
-0.611
1.957 -0.193
0.579
-0.927
2.507 -0.058 -1.302 -0.282
1.313
6
3.420
7
0.893 -2.435 -1.078
8
-1.738
2.413 -1.999
0.259 -1.443
0.039
0.569
4.410
0.464
7.358 -2.398
-0.150
The last bias was also obtained which is equal to 1.4593. These weights and biases will be used in obtaining the target output ultimate confined compressive strength (f‘cc).
The circular and non-circular columns are separated to evaluate the backpropagation model BP-CNC-S. Figure 5.46 shows the plotted predicted versus actual ultimate confined compressive strength of circular columns.
129
200 180
+10% 160
R=1
Predicted f'cc
140
-10%
120 100 80 60 40 20 0 0
20
40
60
80
100
120
140
160
180
200
Actual f'cc
Figure 5.46 BP-CNC-S (circular) Evaluation
BP-CNC-S is observed as a non-conservative model since most of the data points were above the 45⁰ line. The predicted f‘cc is mostly higher than the actual f‘cc. However, based on the R value obtained, 0.8364, the model can be concluded as a good performing model.
The next evaluation was done on non-circular columns which are shown in Figure 5.47.
130
160
140
120
Predictedl f'cc
100
+10% R=1 -10%
80
60
40
20
0 0
20
40
60
80
100
120
140
160
Actual f'cc
Figure 5.47 BP-CNC-S (non-circular) Evaluation
Unlike the performance of BP-CNC-S on predicting the ultimate confined compressive strength of circular columns with steel reinforcements, the
performance
of
BP-CNC-S for
non-circular columns with
steel
reinforcements yielded a lower R value. As shown in Figure 5.47, most of points were distant to the 45⁰ line. The model only got an R value of 0.7876.
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5.3.2.2.2 Comparison with existing models
Models that predict the ultimate confined compressive strength (f‘cc) for both circular and non-circular columns with steel reinforcements is not available. For that reason, the study made use of the ACI 440 (2002) model which predicts the ultimate confined compressive strength for non-circular columns with steel reinforcements and the Hosotani and Kawashima (1999) model for circular columns with steel reinforcements. ACI 440 Model (2002) 100
+10% 80
R=1
Predicted f'cc
-10% 60
40
20
0 0
20
40
60
80
Actual f'cc
Figure 5.48 ACI‘s 440 model (2002)
132
100
The data of non-circular columns with steel reinforcements were utilized to plot the predicted versus the actual ultimate confined compressive strength in Figure 5.48 with the of ACI 440 model. As observed, the model has quite performed well since most data points were near the 45⁰ line. The regression computed for this comparison was 0.9240 which indicates that ACI‘ model is dependable.
Hosotani and Kawashima (1999) 180
+10% 160
R=1 140
-10%
Predicted f'cc
120 100 80 60 40 20 0
0
20
40
60
80
100
120
140
160
Actual f'cc
Figure 5.49 Hosotani and Kawashima model (1999)
133
180
Hosotani and Kawashima‘s model has performed quite poorly, having an R value of 0.5844. Nonetheless, it can be seen as a conservative model since most of the predicted values were lower than the actual ultimate confined compressive strength.
For better comparison, BP-CNC-S were evaluated along with the proposed models through the comparison of their R values.
Table 5.23 Other proposed models versus BP-CNC-S
Models R
Circular Al-Salloum BP-CNC-S 0.5844 0.8364
Non-circular ACI 440 BP-CNC-S 0.9240 0.7876
From Table 5.23, BP-CNC-S performed better than the model of Hosotani and Kawashima since it obtained a higher R value which is equal to 0.8364. However, its unconservative behavior is observed in Figure 5.46 since most of the plotted points were above the 45⁰ line with an R value of 1.
On the evaluation of BP-CN-S for non-circular columns, it was observed that BP-CNC-S performed poorly, only having an R value of 0.7876 as compared to the R value of ACI 440 which is equal to 0.9240. This poor performance of BP-CNC-S on non-circular columns can be assumed to have been caused by the unbalanced input data used. Only 34 data were noncircular out of the 196 total data plotted.
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5.3.2.3 Combined circular and non-circular with and without steel 5.3.2.3.1 BP-CNC-SNS
BP-CNC-SNS utilized the same input parameters used by BP-CNCSNS, these are the following: geometrical properties (b and h), height of the column (L), volumetric ratio of CFRP (ρcfrp), volumetric ratio of transverse steel reinforcement (ρs), and unconfined compressive strength (f‘co). In Table 5.24, the regression analysis of the model‘s training, testing, and validation is shown. Table 5.24 Error analysis of BP-CNC-SNS Hidden 2
3
4
5
6
7
8
R
0.78112
0.73232
0.86234
0.84234
0.88417
0.65692
0.78736
Rtrain
0.80634
0.74288
0.89375
0.89628
0.89790
0.63948
0.81878
Rtest
0.74119
0.66004
0.78541
0.72982
0.85145
0.65086
0.74054
Rvalid
0.69261
0.74698
0.77291
0.89334
0.85278
0.75707
0.67101
Nodes
For BP-CNC-SNS, the model with six (6) hidden nodes showed good performance by having the highest correlation (R) amongst others for the training, testing and validation. However, the combined ANN of columns with and without transverse steel reinforcements yielded the lowest correlation (R) as compared to BP-CNC-NS and BP-CNC-S. Figure 5.50 illustrates the correlation (R) of BP-CNC-SNS with six (6) hidden nodes.
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Figure 5.50 Regression Analyses of BP-CNC-SNS
Weight and biases for each input parameter were obtained after the training of the ANN model. Table 5.25 shows the weight and biases for BPCNC-SNS.
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Table 5.25 Weight and biases obtained from BP-CNC-SNS Hidden Node
b
h
L
ps
pcfrp
f'co
bias
weights
1
0.257
-0.595
-0.431
1.050
-0.077
0.017
-0.052
0.673
2
-1.037
0.466
3.477
-6.026
-0.755
5.047
-2.788
0.105
3
0.164
1.405
23.472
2.573
2.460
13.279
-12.333
-1.583
4
3.610
-2.331
-8.901
-4.020
-3.589
-3.084
9.958
-0.087
5
-1.786
0.810
-2.055
2.101
-3.870
-1.337
0.683
-0.324
6
2.508
-0.601
23.983
2.789
3.184
13.864
-13.000
1.530
The last bias was also obtained equal to 0.2283. These weights and biases will be used in obtaining the target output ultimate confined compressive strength (f‘cc).
BP-CNC-SNS is evaluated using the database of circular and non-circular columns with and without steel reinforcements with a total of 422 data. Figure 5.51 shows the output graph.
137
200
+10%
180
R=1
160
-10%
Predicted f'cc
140 120
100 80 60 40 20
0 0.0
20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0 200.0 Actual f'cc
Figure 5.51 Evaluation of BP-CNC-SNS
The performance of BP-CNC-SNS on the prediction of the ultimate confined compressive strength was observed as proficient with a computed R value equal to 0.8809. Also, it can be seen in the graph that the linear regression if BP-CNC-SNS, which is the yellow line, coincides with the red lines, which are the 45⁰ line, wherein the predicted is equal to the actual ultimate confined compressive strength, and the -10% line which is the lower limit.
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5.4 Parametric Studies
To identify the effects of each of the parameters in the ultimate confined compressive strength and increase in compressive strength, parametric studies were made with the utilization of BP-CNC-SNS. In Figure 5.52, the effect of the significant parameter unconfined compressive strength
Predicted f'cc/f'co
(f‘co) along with the number of CFRP plies is presented. 8 7 6 5 4 3 2 1 0
1 layer 2 layers 3 layers 0
10
20
30
40
50
f'co, MPa
Figure 5.52 f‘co vs. Predicted f‘cc/f‘co with varying number of CFRP layers (tcfrp = 0.32 mm, b = 150 mm, h = 150 mm, L= 350 mm, ρs = 1.5%) Figure 5.52 shows the trend in the increase in compressive strength with varying values for f‘co while other parameters such as thickness of CFRP, geometric properties: b ,h, and L, and volumetric ratio of transverse steel are held constant. Along with this, the analysis was extended on the variation of CFRP layers from one to three layers. It is observed that the highest increase in strength is achieved in low strength concrete wherein even a one-layer CFRP wrap resulted in an almost 4x increase in strength. On the
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other hand, minimal increase in strength was obtained from high strength concrete column. In addition the effect of the number of layers of CFRP lessened, which was showed when the increase in compressive strength with two layers of CFRP overlapped and was higher than the increase in compressive strength with three layers of CFRP and increase in strength with varying number of layers of CFRP is almost similar. The common mode of failure of the columns is CFRP rupture, however, the sudden increase in compressive strength for columns with f‘co range of 25 to 35 MPa and with two to three layers of CFRP signifies the columns failed due to crushing of concrete since the CFRP wrap is too thick to fail in rupture.
In the evaluation of non-circular columns, the effect of aspect ratio (b/h) in the ultimate confined compressive strength was first analyzed in Figure 5.53. 80
Predicted f'cc, MPa
70 60 50 40
b/h = 1
30
b/h = 0.75
20
b/h = 0.5
10 0 0
100
200
300
400
h, mm
Figure 5.53 Varying h vs. predicted f‘cc (L = 350 mm, ρs = 1.5%, ρcfrp = 0.856%, f‘co = 28 MPa)
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The aspect ratio of the columns was varied from 1 to 0.5. For columns, with aspect ratio of 1 and 0.75, the trend in the ultimate confined compressive strength is increasing, however, for columns with aspect ratio of 0.50, the increase in ultimate confined compressive strength decreases.
Thus, the
confinement effect of CFRP is eminent in columns with low aspect ratio and its effect decreases as the aspect ratio decreases.
To further evaluate the aspect ratio, its effect was analyzed along with unconfined compressive strength.
4.5 4
Predicted f'cc/f'co
3.5 3 2.5 b/h = 1
2
b/h = 0.75
1.5
b/h = 0.5
1 0.5 0 0
10
20
30
40
f'co, MPa
Figure 5.54 Varying f‘co and b/h (L = 350 mm, ρs = 1.5%, ρcfrp = 0.856%)
141
50
Figure 5.54 illustrates the effect in the increase in compressive strength with varying values of f‘co and aspect ratio while the height of the column, volumetric ratio of transverse steel and CFRP were held constant. The same trend in Figure 5.53 was exhibited for columns with aspect ratio of 1 and 0.75, wherein the highest increase in strength was obtained from low strength columns and the lowest increase in strength. For columns with aspect ratio of 0.5, an increase in strength lesser than 1 was observed on columns with unconfined compressive strength of 35 to 45 MPa. Thus, the column failure may be due to CFRP rupture. In addition, a significant increase in stress, almost tripled, was still obtained from columns with an aspect ratio of 0.5.
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CHAPTER VI CONCLUSIONS AND RECOMMENDATIONS
This study was performed to analyze the confinement effect of Carbon Fiber Reinforced Polymer on circular and non-circular columns. The significant parameters for both circular and non-circular columns were examined on how they affect the ultimate confined compressive strength and increase in compressive strength of CFRP-confined columns. The utilization of the Self-Organizing Map helped the researchers choose the significant parameters that will be included in the backpropagation models.
The study conducted 136 SOM runs in total and out of these SOMs, the researchers chose nine (9) SOM models in analyzing the circular and noncircular separately and combined. A new tool, the SOM toolbox was introduced in this study to further analyze the SOM models graphically and quantitatively. Based on the SOM analysis, the volumetric ratio of CFRP (ρcfrp) and unconfined compressive strength (f‘co) were the governing parameters on the SOM grouping. On the other hand, the corner radius (r) was found out to have minimal effect on the CFRP confinement of columns as well as the other parameters.
As a result of the SOM analysis, the study made backpropagation models of combined circular and non-circular columns with the output of
143
ultimate confined compressive strength (f‘cc). The three (3) backpropagation models achieved are: BP-CNC- NS with eight (8) hidden nodes, which is consists of circular and non-circular columns without steel reinforcements, BP-CNC-S with eight (8) hidden nodes having circular and non-circular columns with steel reinforcements data and lastly, BP-CNC-SNS with six (6) hidden nodes which is the combination of with and without steel reinforcement columns. The three backpropagation models had performed well since it obtained higher correlation (R) values as compared to some existing models.
In addition, the researchers prepared a Java program with outputs: ultimate confined compressive strength and increase in compressive strength along with the volumetric ratio of steel and CFRP.
This study has a compiled database comprised of 422 columns: 310 circular and 112 non-circular. The database for circular columns was extensive however non-circular data were of lesser amount. This unbalanced number of data may have been a factor in the results of this study. The group suggests to further expand the database, most especially on non-circular columns. Along with this, a research focus on the modeling of CFRP-confined non-circular columns should be performed to have an enhanced analysis on the effects of CFRP confinement of non-circular columns.
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The backpropagation models made in this study and the other existing models were evaluated using the Pearson Regression Analysis. This analysis is not enough bases to assert the performance of these models. Added error analysis and parametric studies should be performed to broaden the analysis on each model.
The programming tools used in this study were the Self-Organizing Map and the Backpropagation Neural Network which are both under the neural network toolbox in Matlab. A new programming tool, the SOM toolbox, was incorporated in this study to further intensify the analysis of the SOM models made. The toolbox showed superior potential on data analysis with the addition of its visualization features. On that account, the utilization of the SOM toolbox should be expanded to further make use of its components.
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APPENDIX A Notations
Symbols
Description
D
Diameter of the circular cross-section
L
Height of the column specimens
b and h
Side dimension of square & rectangular columns which is also equivalent of diameter of circular columns
r
Corner radius
fl
Confinement pressure
f‘co
Unconfined Compressive Strength
f‘cc
Ultimate Confined Compressive Strength
fcfrp
Tensile strength of CFRP
fys
Yield strength of steel
Acc
Area of concrete core
tcfrp
Thickness of CFRP
εcfrp
Ultimate tensile strain of CFRP
Ecfrp
Elastic modulus of CFRP
Es
Elastic modulus of steel
ke
Efficiency factor
ks
Shape factor
SOM
Self-organizing map
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MSE
Mean squared error
R
Pearson product-moment correlation coefficient
ρcfrp
Volumetric ratio of CFRP
ρl
Volumetric ratio of longitudinal steel
ρs
Volumetric ratio of transverse steel
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APPENDIX B Compilation of Data
155
156
157
158
159
160
161
162
163
164
165
166
Appendix C Self-Organizing Map and Backpropagation
The group started SOM by inputting the codes seen below: P = [‗input parameters‘ (normalized);]; Net = newsom([ ranges of input parameters (normalized);],[no. Of groups]) Wts=net.iw{1,1} net.trainParam.epochs = 1000; net=train(net.p) wts=net.iw{1,1} getting the distance of each vector d =[ w1 input parameter‘ (normalized); w2 input parameter‘ (normalized); wn input parameter‘ (normalized)]; dis = dist(d) To enhance SOM classification, the group used SOM toolbox to better visualize the grouping of data.
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Using SOM in MATLAB
1. First, organize the parameters of the specimens in an excel file. The same parameters should be placed under one column, this is further illustrated in Figure A.1.
Figure C.1 – Sample Data arranged in an Excel file The data should be arranged by rows as shown in Figure A.2 so that it will be read by MATLAB
Figure C.2 – Sample Data arranged horizontally
2. Normalize the data. This provides better results. A sample is shown in Figure C.3.
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Figure C.3 – Normalized Sample Data
Arrange the normalized data by transposing it to the orientation acceptable in MATLAB. This is shown in Figure A.4.
Figure C.4 – Normalized Sample Data for export to MATLAB
3. The format that MATLAB can import is .txt. Just click Save As in the File Menu then click Other Formats to locate Text (Tab Delimited).
169
Figure C.5 – Saving Excel File to Another File Format
4. Import the data using MATLAB. In the File Menu, click Import Data. Import the Text (tab delimited) file saved.
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Figure C.6 – Importing the Data to MATLAB
The tab shown in Figure A.6 will appear. Click Next. Then there will be another prompt shown in Figure A.7, click Finish.
Figure C.7 – Importing the Data to MATLAB
5. Once the SOM finished iterating, the list of data clustered will appear in the command window along with the visualizations.
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Figure C.8 – SOM Model Outputs
Using ANN in MATLAB
1. The input parameters will be normalized in the same manner in normalizing in the SOM.
2.
Input the code.
%CNC-NS % b,h,L,pcfrp,f'co clc; clearvars; som = [input parameters]; % f'cc t = [target output]; %h is for the hidden layers for h = 2:8; net = newff(minmax(som),[h,1],{'tansig','purelin'},'trainlm','learngdm','mse'); init(net); net.divideFcn = 'dividerand'; net.divideParam.trainRatio = 70/100;
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net.divideParam.testRatio = 15/100; net.divideParam.valRatio = 15/100; net.trainParam.show = 100; net.trainParam.epochs = 3000; net.trainParam.goal = 0.00001; % accomplished % NET - New network. % TR - Training record (epoch and perf). [net,tr] = train(net,som,t); % To generate the predicted outputs of the model output = sim(net,som); % view predicted output vs target output % M - Slope of the linear regression. % B - Y intercept of the linear regression. % R - Regression R-value. R=1 means perfect correlation. % [m,b,r] = postreg(a,t) % train trOut = output(tr.trainInd); trTarg = t(tr.trainInd); % test teOut = output(tr.testInd); teTarg = t(tr.testInd); % val valOut = output(tr.valInd); valTarg = t(tr.valInd); plotregression(t, output,['All: ',int2str(h), ' hidden layers'],trTarg,trOut,'Train',teTarg,teOut,'Test',valTarg,valOut,'Validation'); dlmwrite(['CNC_NS_weights_',int2str(h),'.txt'],net.iw{1}, ' '); dlmwrite(['CNC_NS_bias_',int2str(h),'.txt'],net.b{1}, ' '); dlmwrite(['CNC_NS_weights_node_',int2str(h),'.txt'],net.lw{2,1}, ' '); dlmwrite(['CNC_NS_train_',int2str(h),'.txt'],som(:,tr.trainInd), ' '); dlmwrite(['CNC_NS_test_',int2str(h),'.txt'],som(:,tr.testInd), ' '); dlmwrite(['CNC_NS_val_',int2str(h),'.txt'],som(:,tr.valInd), ' ');
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6. Training the Network. Once all the data needed are already imported, training the network begins.
Figure C.9 – Back-Propagation Neural Network Training
7. Backpropagation Output.
Figure C.10 – Back-Propagation Neural Network Output
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APPENDIX D Java Program
Option: o Circular o NonCircular
Option: o w/ steel o w/o steel
Outputs User Inputs
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