LUẬN án TIẾN sĩ độ tin cậy chịu uốn của dầm bê tông cốt thép được tăng cường bằng tấm polymer cốt sợi carbon (CFRP)

1
INTRODUCTION
Externally bonded fiber reinforce polyme, FRP, has been appeared for 30 years
and promptly become one of effective methods in retrofitting old RC structures. This
method would bring advantages comparing with traditional ones in term of less
increasing dead load, unchanging general structures, increasing bending capacity,
preventing the appearence of new cracks and widening old cracks in concrete, as well
as easy and faster erection. FRP has high tension strength, light weight, good fatigue
strength, high corrosion capacity and easily sticking on concrete surface, so that
applying FRP in construction has been fast developed over last decades. Carbon fiber
reinforce polyme, CFRP, receives all advantages of fiber reinforce polyme and has
excellent fatigue capacity for retrofitting old bridges, especially RC bridges, more
effectively than conventional methods as placing steeel bars in tension zone,
externally postensioning cables, or externally bonded steel sheets.
Reasons for thesis selection:
CFRP sheets have been used in retrofitting and strengthening bridges in
Vietnam; in contrast, current national code for bridges, 22TCN 272-05, has not been
designed for CFRP material. Some bridges have been designed and erected using
ACI 440.2R-08.
Researches and calculations for retrofitting and strengthening by CFRP have
mostly conducted in semi-reliability method, without considering all statistic
behaviours of design parameters. Furthermore, other researchs on the world have
been using reliability method in several fields. However, there is no reaserch
completely evaluating the reliability of bending beams strengthened with externally
bonded carbon fiber reinforced polymer sheets. Thus, reaserch of applying CFRP
basing on reliability theory is a live problem in Vietnam and all over the world. This
is the main reason for this thesis selection.
Name of thesis: “Flexural reliability of RC Bridge girders strengthened with carbon
fiber reinforced polymer sheets (CFRP)”
Objectives of this study:

to

analyze and promote the factor of CFRP strength
reduction and range of applying externally bonded carbon fiber reinforced polymer
sheets in retrofitting and strengthening RC beams.
Methods of this study:
• Theory method: apply reliability theory with possible distributions and
statistical parameters of random variables for determining reliability index of
bending beams strengthened with externally bonded CFRP sheets.
• Experiment method: carry out room and field experiments for getting possible
distributions and statistical parameters of random variables in flexural resistance
model in ACI 440.2R-08.
Subjects and scopes: RC beams strengthened with externally bonded CFRP
sheets.
Range of study:

2
• Compute and evaluate reliability index of RC beams strengthened with
externally bonded CFRP sheets.
• Carry out room and field experiments for getting possible distributions and
statistical parameters of random variables, icluding section geometry, concrete
compressive strength, steel yield strength, and influence of analysis method to flexual
resistance of RC beams strengthened with externally bonded CFRP sheets according
to ACI 440.2R-08.
Scientific and practical meaning of this study:
•

Theory:
- Deriving a methodology for evaluating level of flexural reliability of RC

Bridge girders strengthened with CFRP sheets, basing on reliability theory and
flexural resistance model in ACI 440.2R-08.
- Suggesting CFRP strength reduction factor and range of applying externally
bonded carbon fiber reinforced polymer sheets in retrofitting and strengthening RC
beams.
• Experiment: define possible distributions and values of statistical parameters
of random variables, icluding section geometry, concrete compressive strength, steel
yield strength, and influence of analysis method to flexual resistance of RC beams
strengthened with externally bonded CFRP sheets according to ACI 440.2R-08
basing on bending sample beams to rupture.
Terms of rasearch include introduction, 4 chapters, and conclusions as following:
Introduction: Introduction of CFRP sheets and name of thesis.
• Chapter 1: General view of reaserchs about structures using FRP.
• Chapter 2: Analyzed reliability index, β, of RC beams strengthened with
externally bonded CFRP sheets according to bending resistance model in ACI
440.2R-08.
• Chapter 3: Studying RC sample beams strengthened with externally bonded
CFRP sheets throught bending to failure.
• Chapter 4: Studying Trần Hưng Đạo bridge beams strengthened with
externally bonded CFRP sheets under bending with calibrated loads
Conclusions and suggestions: presenting results, suggestion, and proposed
topics for future researches.






3

Chapter 1
GENERAL VIEW OF RESEARCH ABOUT STRUCTURES USING FRP
1.1. History of applying FRP in retrofitting and strengthening structures
Reinforce concrete bridges are popular in Vietnam and all over the world.
Nowadays, there are many serious deteriorations, however, there is not enough budget
for replacing with new ones so retrofitting and strengthening bridges seem to be the
best choice. Strengthening methods are varied depending on demand, structure, and e
technical level. According to statistic, bridge strengthening is mainly using
conventional methods as placing steeel bars in tension zone, externally postensioning
cables, or externally bonded steel sheets.
Externally bonded FRP has been appeared for 30 years and promptly become one
of effective methods in retrofitting old RC structures. This method would bring
advantages comparing with conventional ones in term of (1) less increasing dead
load, (2) unchanging general structures, (3) increasing bending capacity, (4)
preventing the appearence of new cracks and widening of old cracks in concrete, and
(5) easy and fast erection. FRP has high tension strength, light weight, good fatigue
strength, high corrosion capacity and easily sticking on concrete surface, so that
applying FRP in construction has been fast develope over last decades.
In 1980s, the first time FRP was applied for retrofitting RC columns in Japans. In
Euro, as early as 1978, German researches had been mentioned the issue of using FRP
in strengthening RC structures. Similarly, Switzer researchers had applied FRP for
strengthening bridge beam in bending in 1987. In The USA, applying FRP was
researched as early as 1930s, but the using of FRP in retrofitting and strengthening has
been started in 1980s.
In Vietnam, FRP has been applied in strengthening bridges as: Sài Gòn-HCM
city; Trần Hưng Đạo - Phan Thiết city, Bình Thuận province; Trần Thị Lý- Đà Nẵng
city, Gián Khẩu- Ninh Bình province, Tô Mậu - Yên Bái province.
1.2. FRP mechanical properties
Unit weight of FRP is from 1.2 g/cm
3

to 2.1 g/cm
3
, about 1/4 to 1/6 that of steel.
Unidirectional FRP has different
thermal coefictions for longitudinal and
transver directions depending on type of
fibers, matrix, and volume of fiber.
FRP in tension has linear
relationship between stress and strain
until rupture and this is truely a brittle
material (Figure 1-1). FRP mechanical
properties decrease under the impact of
environment factors including: high
temperature, humidity, and chemicals.
Figure 1-1. Relationship between typical
FRP stress and strain

4

1.3. FRP application
Main FRP application includes: retrofitting and strengthening structures;
reinforcing for RC; and constructing main frame.
1.4. Existing design manual for FRP
1.4.1 Guide for the Design and Construction of Externally Bonded FRP
Systems for Strengthening Concrete Structures
Design bending RC section strengthening with externally bonded FRP base on
beam theory with reinforce addition of FRP combining for tension. Acorrding to ACI
440.2R-08, flexural expression at limit state is as following:
∅൫M
୬ୱ

+ψ
୤
M
୬୤
൯≥γ
ୈ
M
ୈ
+γ
୐
M
୐

1.4.2 Externally bonded FRP reinforcement for RC structures-FIB Bulletin No.
14
Design bending RC section strengthening with externally bonded FRP acorrding
to FIB is similar to that of ACI:
ܯ
ோௗ
= ܣ
௦ଵ
݂
௬ௗ
ሺ
݀−ߜ
ீ
ݔ
ሻ
+ܣ
௙

ܧ
௙
ߝ
௙
ሺ
ℎ−ߜ
ீ
ݔ
ሻ
+ܣ
௦ଶ
ܧ
௦
ߝ
௦ଶ
ሺ
ߜ
ீ
ݔ−݀
ଶ
ሻ

1.4.3 Strengthening Reinforced Concrete Structures with Externally-Bonded
Fibre Reinforced Polymers-ISIS Design Manual No. 4
ISIS' assumptions is similar to ACI's and added two more assumptions: perfect
bond between concrete and FRP; and FRP are well anchored or extended enough to
ensure the work of FRP to limit state. Flexural expression at limit state is as
following:
ܯ
ோ

=ܶ
௦
ቀ݀−
௔
ଶ
ቁ+ܶ
௙௥௣
ቀℎ−
௔
ଶ
ቁ
1.5. Structure Reliability
1.5.1. Basic of Reliability theory
Probability of Stable, P
s
: P
S
= P{S<R | [0,T]}
and probability of failure: P
f
= P{S>R | [0,T]}
where P{S<R | [0,T]} is the probability for structure not break-down during
operation time, T; P{S>R | [0,T]} is the probability for structure break-down during
operation time, T.
Checking expression: P
S
≥[P
S
] or P
f

≤ [P
f
], where

∫∫
>
=
RS
f
dSdRRfSfP )()(

∫∫
<
=
RS
s
dSdRRfSfP )()(

This is a modern design method; taking into account for unstable and random
characteristics of design variables. However, it is needed to get enough Possible
Distributions and Statistical parameters of all variables.


5
1.5.2. Basic reliability theory for RC bending beams strengthening with externally
bonded CFRP sheets
1.5.2.1 Methodology
RC bending beams strengthening with externally bonded CFRP sheets have been
evaluated in either two directions: Evaluating the reliability of RC structures being
strengthened; or Designing strengthening RC structures with given reliability.

Methodology for Evaluating the reliability of RC structures being strengthened
base on basic reliability theory. Widely accepted model for design is beam theory
with the addition of CFRP as part of tension material.
For given reliability, β, probability of structure fail is defined. With analyzed
model and design beam parameters specified, all demands of FRP are established.
1.5.2.2 Possible distributions and values of statistical parameters of random
variables
1.5.2.3 Elements influencing to reliability of RC bending beams strengthening
with externally bonded CFRP sheets
Reliability of RC bending beams strengthening with externally bonded CFRP
sheets depends on many elements such as: design standards, design level,
construction technology, management and operation, loads, and environment.
Evaluating the reliability of RC structures being strengthened with externally bonded
CFRP sheets is only used to deal with random variables having specified statistical
parameters and describing their influence to structure reliablity through state
functions.
1.5.3. Reliability index,
β
ββ
β

1.5.3.1 Definition of Reliability index,
β
ββ
β


Figure 1- 2. Diagram of state function for Resistance, R,
Load effect, S, and safty reserve, G.
Ratio of ߚ =

ఓ
ಸ
ఙ
ಸ
is called Reliability index of structures.
Reliability index is computed through means and standard deviations of
Resistance, R, and Load effect, S as ߚ =
ఓ
ೃ
ିఓ
ೞ
ට
ఙ
ೃ
మ
ିఙ
ೞ
మ

Probability of structure fail is determined by standard normal distribution
function: P
୤
=Φሺ−βሻ
1.5.3.2 Target Reliability index,
β
ββ
β
T

6
Target Reliability index selected depends on influence level to human-economy-
social when structures are broken. Target Reliability index is different from new and
strengthened structures.
Research Andrzej S Nowak, Maria M Szerszen, and Allen; together with
regulations of ASSHTO LRFD, EC, ACI 318- 05, Target Reliability index is
proposed in this thesis as following:
• β
T
= 3.75: multiple paths beam structures, strengthening age of 10 years.
• β
T
= 3.5: multiple paths beam structures, strengthening age of 5 years.
1.5.3.3 Evaluating Reliability index using Rackwitz-Fiessler method
This method is based on equivalent mean and standard deviation of variables with
unnormal distribution. Susposing random variable, X, has mean of µ
X;
standard
deviation of σ
X;
distribution function of F
X
(x) and density distribution function of
f
X
(x). Equivalent mean of µ
ଡ଼
ୣ
and standard deviation of σ
ଡ଼

ୣ
is defined at point x*
where values of F
X
(x) and f
X
(x) are repectively equal to those of distribution function
and density distribution function of standard normal distribution. Point x* is on the
limiting line or G=0.
F
ଡ଼
ሺ
x
∗
ሻ
= ߔ൬
୶
∗
ିµ
౔
౛
σ
౔
౛
൰ f
ଡ଼
ሺ
x
∗
ሻ

=
ଵ
σ
౔
౛
φ൬
୶
∗
ିµ
౔
౛
σ
౔
౛
൰
Where: Φ and φ are repectively distribution function and density distribution
function of standard normal distribution (µ=0 và σ=1). ). Equivalent mean of µ
ଡ଼
ୣ
and
standard deviation of σ
ଡ଼
ୣ
is defined as following:
µ
ଡ଼
ୣ
=x
∗
−σ

ଡ଼
ୣ
ሾ
ߔ
ିଵ
ሺ
F
ଡ଼
ሺx
∗
ሻ
ሻ
ሿ
σ
ଡ଼
ୣ
=
ଵ
୤
౔
ሺ୶
∗
ሻ
φ
ሾ
ߔ
ିଵ
ሺ
F
ଡ଼

ሺx
∗
ሻ
ሻ
ሿ

Iteration method is used for determining x* and β.
1.5.4. Analyis of statistic parameters
1.5.4.1. Defining minimum sample size
Minimum sample size, ঎
min
, of material, X, having covariance, COV
X
,
approximate constant compared with mean, µ
X
, is defined as following: ঎
୫୧୬
=
൫f
୮
COV
଴
/e൯
ଶ

1.5.4.2. Checking distribution functions of random variables
Checking random variable, X, with a set of ঎ samples for appropriating with
normal distribution includes:
• Graphs (lines or colums for prbability distribution);

• Checking synmetry value of Fisher, g
1
, and kurtosis value of Pearson, g
2
.
• Using standard Shapiro-Wilk for sets of 3 to 20 samples; or extended Shapiro-
Wilk with Royston algorithm for sets of 20 samples or more.
1.6. Analyze, evaluate research about beams strengthening with CFRP sheets
In general, researches from all over the world to Vietnam have been studied
beams strengthening with externally bonded CFRP sheets about many issues,
including:

7
• Type of failure
• Ultimate strength capacity
• Section Stress Distribution
• Advantages of FRP external bonded method for bending and shear
• Model for evaluating peeling of CFRP sheets.
• Finite Element Model for evaluating beams strengthening with externally
bonded CFRP sheets
• Using ductility index to evaluate beams strengthening with externally bonded
CFRP sheets
Terms of research are various but all of them are conducted in semi-reliability
method, not implying all statistic behaviours of design parameters.
1.7. Analyze, evaluate researches about reliability theory
In general, researches from all over the world to Vietnam have been applied
reliability theory in many issues, including:
• Avaluating beam element
• Factors of loads and resistance
• GFRP bar reinforcement for concrete

• Beam with large ratio of reinforcement
• Bridge deck at service limit state
• Box-section RC beams
These researches have been using reliability method in several fields. However,
there is no reaserch completely evaluating the reliability of bending beams
strengthened with externally bonded carbon fiber reinforced polymer sheets.
1.8. Objects of thesis
Evaluating influences of materials, geometry, and analysis model of bending
resistance to reliability of beams strengthening with externally bonded CFRP sheets.
Evaluating and suggesting application range of strengthening with externally
bonded CFRP sheets for single span beams and CFRP strength reduction factor.
1.9. Terms and Methods of this study
Computing and evaluating reliability index of bending beams strengthened with
externally bonded CFRP sheets; suggesting application range of strengthening with
externally bonded CFRP sheets and CFRP strength reduction factor.
Theory method: applying reliability theory with possible distributions and
statistical parameters of random variables for determining reliability index of
bending beams strengthened with externally bonded CFRP sheets.
Experiment method: carrying out room and field experiments for getting possible
distributions and statistical parameters of random variables in flexural resistance
model in ACI 440.2R-08.




8
Chapter 2
EVALUATING FLEXURAL RELIABILITY INDEX OF RC BRIDGE GIRDERS
STRENGTHENED WITH CFRP SHEETS BASING ON FLEXURAL
RESISTANCE MODEL OF ACI 440.2R-08

• Reliability index, β, of bending sections of RC bridge girders strengthened with carbon
fiber reinforced polymer (CFRP) sheets has been computed according to ultimate limit state
(ULS) for flexural design suggested by the ACI 440.2R-08. 2880 strengthened RC beam-
sections have been selected with reasonable sets of random variables (section dimensions,
concrete compressive strength, steel strength, ultimate strength and ultimate relative strain of
CFRP sheets) applying in analyzing model of nominal flexural resistance, M
ୖ
. Monte-Carlo
simulations have been performed to determine the variability in material properties (M) and
fabrication processes (F); whereas experimental data reported in the literature has been used to
quatify the variability related to the analysis method (P). The reliability index, β, caculated using
Rackwitz-Fiessler method with first order function.
2.1. Statistical properties of section geometry and materials
• Geometrical properties: width of section, b, can be measured easily after
construction, so that two extreme nominal values λ were selected 1.01 and 1.00; and
two values COV were selected as 1.78% and 0.60%, respectively (Table 2-1). The
effect of concrete vibrators to the positions of tension steel could hardly to measure
after construction, thus two extreme nominal values λ of d were selected as 0.99 and
1.00, respectively; and two values COV of d were selected as 2.36% and 0.78%,
respectively (Table 2-1). h values are proportionally related to d. Both b and d are
assumed to have Normal distribution.
• Concrete Compressive Strength, ݂
௖
′
, is closely related to construction level and
curing. The bias at sites is usually smaller than in lab and the suggested value is 1.05.
Two nominal values COV of grade 17 and 30 were selected: 15% and 13.5%,
respectively. Compressive strength of concrete is assumed to be normally distributed.
• Tensile Strength of steel bars, ݂
௬

: two types of steel and their extreme nominal values
of ݂
௬
were considered in Table 2-1. Selected bias value is 1.10, near the lower limit;
Selected COV value is 10%, near the upper limit. ݂
௬
is assumed to be normally
distributed. E
ୱ
=200000MPa.
• Tension strength, f
୤୳
∗
, and limit relative strain, ε
୤୳
∗
, of CFRP is selected in
Table 2-1. Selected bias values of them are 1.10, near the lower limit; Selected COV
value of tension strength is 12% ,near the upper limit. Selected COV value of limit
relative strain is 2.5% , near average value. These variables are assumed to agree
with Weibull distribution. FRP modulus of elasticity: E
୤
= 230000MPa.
2.2. Design space

9
Resistance flexural model for Rectangle section RC beam strengthening by CFRP sheet is
suggested by ACI 440.2R-08.
Nominal values of b, d, f’
c

, and f
y
as Table 2-1. Strengthening CFRP layers: ݊
ிோ௉
= 1, 2,
and 3. Steel reinforcement ratio: ߩ
ௌ
= 0.2, 0.3, 0.4, 0.5, 0.6ߩ
௕௟
, where
ρ
ୠ୪
=0.85β
ଵ
୤
ౙ
′
୤
౯
க
ౙ౫
க
ౙ౫
ା୤
౯
୉
౩
⁄

So that design space of section resistance includes 2

4
x 3 x 5 = 240 cases.
Table 2-1. Statistical Properties of Main Variables
Design
Variable
Minimum
norminal Value
(N)
ߤ
&
ߪ

Bias λ
& COV
(%)
Maxnimum
norminal
Value (L)
ߤ
&
ߪ

Bias λ
& COV
(%)
Probability
Distribution

b
(x10

-3
m)
ܾ
ே

=200
ߤ
௕
=
ܾ
ே
+2.34
1.01
ܾ
௅

=500
ߤ
௕
=
ܾ
௅
+2.34
1.00
Normal
ߪ
=
3.60
1.78
ߪ

=3.60
0.60
d
(x10
-3
m)
݀
ே
=
0
.
9
ℎ
ே

=800
ߤ
ௗ
=
݀
ே
−
4
.
41

0.99
݀
௅

=1500
ߤ
ௗ
=
݀
௅
−
4
.
41

1.00
Normal
ߪ
=11.70 2.36
ߪ
=11.70 0.78
݂
௖
′

(MPa)
17
ߤ
௙
೎
′
=
17.9
1.05

30
ߤ
௙
೎
′
=31.5
1.05
Normal
ߪ
=2.7
15.00
ߪ
=4.3
13.50
݂
௬

(MPa)
275
ߤ
௙
೤
=302.50
1.10
420
ߤ
௙
೤
=462.0
1.10

Normal
ߪ
=30.25 10.00
ߪ
=46.20 10.00
݂
௙௨
∗

(MPa)
3000
ߤ
௙
೑ೠ
∗
=3300
1.10
E=230000N/mm
2
Weibull
ߪ
=360 12.00
ߝ
௙௨
∗

(mm/mm)

0.015
ߝ

௙
೑ೠ
∗
=0.0165
1.10

Weibull
ߪ
=0.000375 2.50
2.3. Influenced factors for flexural resistance:

Flexural resistance, M
R
, is a random variable and according to Ellingwood, 2003 [9], the
factors influenced to flexural resistance random characteristic include:
• Material properties (M): strength, modulus of elasticity, relative strain…
• Fabrication (F): dimensions and their effects to geometric properties.
• Analysis method (P): accurate level of selected method in ACI 440.2R-08.
M, F and P are assumed to be independent variables. Effects of M and F are evaluated
together through values of λ
MF
and COV
MF
. These values are computed by randomly generated
variables of material properties and fabrication as Monte Carlo simulation.
Effects of P is evaluated by comparing experimental values of the flexural capacity available
in literature, ܯ
௨,்௘௦௧
with the corresponding analytical value ܯ
஺஼ூ

, derived using the analysis
method proposed by ACI 440.2R-08 (Table 2-2).The following values were chosen: ࣅ
ࡼ
=
૚.૛ and ࡯ࡻࢂ
ࡼ
=૚૙%

Table 2-2. Statistical Properties of P
N
0

Source

Sample No.

ߣ
௉

ܥܱܸ
௉

1

P. Alagusundaramoorthy et al.[116]

12

1.69

13.1%

2

G. Spadea et. al.[57]

8

1.43

3.8%

3

J.G. Dai [65]

6

1.12

19.8%

2.4. Load model

10
Dead load (D) and live load (L) are the two load categories considered in this study.
The dead load considered in the design is the gravity load due to the self weight of the
structure. It is normally treated as a Normal random variables in literature (Cardoso et.al,2007,
Nowak 1993-1995-1999 and Project No. NCHRP 20-7/187 - ASSHTO 2007); because of the
control over construction materials, it is assumed that the accuracy to estimate dead loads is

higher compared to that of live loads. The works in this study induced to adopt a bias ࣅ
ࡰ
=
૚.૙૜ܽ݊݀࡯ࡻࢂ
ࡰ
=૚૙% for dead load.
In this study, live load is HL 93 and its statistical properties are available in Project No.
NCHRP 20-7/187 - ASSHTO 2007. Live load is assumed to agree with Gumbel distribution. In
this study, ૃ
ۺ
=1.20 and COV
L
=18% are selected.
2.5. Evaluating reliability index
2.5.1. State function:
State function to determine flexural reliability index β based on ACI 440.2R-08
concludes 3 random variables: flexural resistace, M
R
, dead load moment, M
D
, and
live load moment, M
L
: G
ሺ
M
ୖ
,M
ୈ
,M

୐
,
ሻ
= M
ୖ
−ሺM
ୈ
+M
୐
ሻ
Assuming ratio of live load moment to dead load moment at specific section is n
୑୐ୈ
=
୑
ై
୑
ీ
.
In this study, investigated values of n
MLD
are 0.25; 0.50; 0.75 and 1.00. In addition, assuming the
section is at limit state or γ
ୈ
M
ୈ
+γ
୐
M
୐
=∅M

୬
, so thatM
ୈ
=
୑
ై
୬
౉ైీ
=
∅୑
౤
γ
ీ
ାγ
ై
୬
౉ైీ

Thus establishing reliability level of function G
ሺ
M
ୖ
,M
ୈ
,M
୐
,
ሻ
= M
ୖ

−ሺM
ୈ
+M
୐
ሻ is
completely defined. This is an linear function of three random variables: M
ୖ
,M
ୈ
, normal
distribution, and M
୐
, Gumbel distribution. Rackwitz-Fiessler method is applied to find reliability
index β.
2.5.2. Computer aided software

Figure 2- 1. Program Figure 2- 2. Block CI

11


Figure 2- 3. Block CIIa Figure 2- 4. Block CIIb

Figure 2- 5. Block CIII Figure 2- 6. Block CIV

12

Programme block diagrams are as Figure 2-1, Figure 2-2, Figure 2-3, Figure 2-4,
Figure 2-5, Figure 2-6. VBA (Visual Basic Application) in Microsoft Excel (Version 2007 or
above) is selected as programmed language.

The name of the mini software is 2TKN with two Menus named "NHAP SO LIEU"
and "TINH TOAN". Details of 2TKN interface are as Figure 2- 7, Figure 2-8,
Figure 2- 9, Figure 2- 10, Figure 2- 11, Figure 2- 12.
2.5.3. Results and evalutions
2.5.3.1. Results
In 2880 basic cases are investigated, 2660 cases inside β investigated region and 220 cases
fail to meet requirement of limits steel stress in service state of (f
ୱ
>0.8f
୷
) as ACI 440.2R-
08. β >3.50 with all cases inside β investigated region.
2.5.3.2. Evaluating group cases fail to meet requirement of limits steel stress in service state
There are 222 cases (7.68%) fail to meet requirement of limits steel stress in service state.
These cases mostly happen low steel ratio ρ
ୱ
= 0.2ρ
ୠ୪
, a few with ρ
ୱ
=0.3ρ
ୠ୪
.
2.5.3.3. Checking the distribution of M
R


Figure 2-7. Probability distribution of M
R

Probability distribution of M
R
is tested with 6 sections randomly selected in the
design space. Extended Shapiro-Wilk Test is applied for every single section with ten
sets of 5000 samples. All 6 sections have p greater than 0.05 and the asumption of
normal distribution of M
R
is accepted (Figure 2-7).
2.5.3.4. Evaluating group cases in investigated region
2.5.3.4.1. Influence of tension steel ratio to
β

Average reliability index of 2660 cases as Figure 2-8. The graphs of ψ
୤
= 0.85
and ψ
୤
= 0.90 are slightly different so that ψ
୤
=0.90 might be used instead of
ψ
୤
= 0.85 as ACI 440.2R-08. Figure 2-8 reveals that β decreases sharply when
increasing live load. Figure 2-9 shows that failure modes depend on tension steel
ratio.
0.00
0.05
0.10
0.15
0

.
20
850
1090
1330
1570
1810
2050
2290
2530
2770
3010
3250
3490
3730
Probability
distribution
M
R
(10
3
Nm)

13


Figure 2-8. Average reliability index of investigated cases
varying with CFRP strength redection factor

Figure 2- 9. Average reliability index of investigated cases

varying with live to dead load ratio



Figure 2- 10. Average reliability index of investigated cases
varying with failure mode

2.5.3.4.2. Comparing
β
with ߰
௙
= 0.85 and ߰
௙
=0.90
All cases inside investigated region have reliability greater than 4.00. These values are
appropriate with strengthening 10 years or more. 876 pairs of cases of ψ
୤
= 0.90 and
ψ
୤
= 0.85 are compared.
Reliabily indexs of 876 pairs are comparatived with variables of live to dead load
ratio and failure mode.
Percentage of strengthening of 876 pairs are comparatived with variable of CFRP
ratio.
2.6 Conclusions
A- General:
1. Despite wide range of design cases for design variables, the carried out research work is
strictly dependent on the specific design cases taken into account.
2. All cases fail to meet requirement of limits steel stress in service state have low

reinforcement ratio of 0.2 to 0.3
4.4
4.6
4.8
5.0
5.2
5.4
5.6
0
.
2
0
.
3
0
.
4
0
.
5
0
.
6
β
β
β
β
0
0
0

0
ρ
s
/ρ
bl
ψ
f=
0
.
80
ψ
f=
0
.
85
ψ
f=
0
.
90
4.0
4.5
5.0
5.5
6.0
6.5
7.0
0
.
2

0
.
3
0
.
4
0
.
5
0
.
6
β
β
β
β
0
0
0
0
ρ
s
/ρ
bl
ML/MD=
0
.
25
ML/MD=
0

.
50
ML/MD=
0
.
75
ML/MD=
1
.
00

14
3. Reliability indexs decrease when CFRP reduction factor increases
4. Reliability indexs slightly increase with ratio of 0.2 to 0.3, and sharply increase with ratio of
0.5 to 0.6
5. Reliability indexs decrease when live to dead load ratio decrease
6. Failure mode depends on reinforcement ratio:
• Reinforcement ratio of 0.2 to 0.3: main failure mode is CFRP rupture, the most
economy mode thanks to large section reduction factor.
• Reinforcement ratio of 0.3 to 0.5: main failure mode is debonding, the largest reliability
mode.
• Reinforcement ratio of 0.5 to 0.6: main failure mode is concrete crushing, the smallest
section reduction factor mode.
B- Comparing Reliabily indexs with ૐ
܎
=૙.ૡ૞ and ૐ
܎
=૙.ૢ૙:
1. β graphs with variable of failure mode in cases of ψ
୤

=0.85 (case 1) and ψ
୤
=0.90
(case 2) are homothetic regarding ρ
ୱ
ρ
ୠ୪
⁄
, M
୐
/M
ୈ
, or A
୊ୖ୔
bd
⁄
.
2. The difference of reliability index β of two cases ψ
୤
=0.85 and ψ
୤
=0.90 is small,
under 0.0015.
3. Reliability indexs increase when reinforcement ratio increase regarding failure
modes.
4. Reliability indexs with reinforcement ratio of 0.6 are large. Reliability indexs with
reinforcement ratio of 0.2 to 0.5 is similar.
C- Comparing Percentage of strengthening with ૐ
܎
= ૙.ૡ૞ and ૐ

܎
=
૙.ૢ૙:
1. Graphs of Percentage of strengthening with variable of CFRP ratio are homothetic
for ψ
୤
= 0.85 and ψ
୤
= 0.90.
2. Percentage of strengthening increase when CFRP ratio increase.
3. Percentage of strengthening with ψ
୤
= 0.90 is larger than that with ψ
୤
=
0.85, and 1.5% is the largest difference.
4. Unit Percentage of strengthening:
• Sharly decreasing when reinforcement ratio of 0.2 to 0.3;
• Averagely decreasing when reinforcement ratio of 0.4; and
Little decreasing when reinforcement ratio of 0.5 to 0.6.

Chapter 3
EXPERIMENTAL RESEARCH OF BENDING BEAMS STRENGTHENED
WITH EXTERNALLY BONDED CFRP SHEETS
3.1. Objectives of experimental research
1. Determining values of statistical parameters of analysis model (P) basing on
testing utimate moment (M
u,test
) and resistance moment deriving from ACI 440.2R -
08 (M

ACI
).
2. Approximating distribution functions of concrete compression strength and
steel yield strength.
3. Investigating behaviour of bending beams with different area of tension steel,
compressive strength of concrete, and layer of CFRP sheets.
4. Investigating beam deflection.
5. Evaluating efficiency of strengthening.

15

3.2. Experimantal places
Civil Engineering LAB -HCM City University of Technology
Address: B6, 268 Lý Thường Kiệt St., District 10, HCM City.
Content of experiment:
• Compressing to failure 12 concrete samples on 09/24/2014
• Tensioning to rupture 9 steel samples on 09/17/2014
• Tensioning to rupture 13 CFRP samples on 11/09/2014
• Bending to rupture 8 sample beams strengthened with externally bonded CFRP
sheets from 09/24/2014 to 09/26/2014.
3.3. Materials
3.3.1. Concrete
a. Minimum size of sample number
Minimum size of sample number, ঎
min
, is defined as equation (1.47), where
COV
0
= (10%+7%+6%+9.1%+9.1%+9.1%)/6=8.38%. Selecting 6 samples of C21
concrete and 6 samples of C25 and getting f

p
=2.923 and p=0.998.
b. Results
Distribution functions of C21 and C25 concrete compressive strength are agrred
with normal distribution through Shapiro-Wilk checking.

3.3.2. Steel reinforcement
a. Minimum size of sample number
Minimum size of sample number, ঎
min
, is defined with p=0.9, f
p
=1.64, e=0.1 và
COV
0
= (5.85%+8.8%+7.7%)=7.45%
Thus,঎
୫୧୬
=
ሺ
1.64x7.45%/0.1
ሻ
ଶ
=1.49 samples
Selecting 3 samples of D12 steel and 3 samples of D16 steel and getting f
p
=2.325
and p=0.975.
b. Results
Distribution functions of steel yield strength are agrred with normal distribution

through Shapiro-Wilk checking.

3.3.3. CFRP sheets
a. Minimum size of sample number
Minimum size of sample number, ঎
min
, is defined with p=0.9, f
p
=1.64, e=0.1 và
COV
0
= (12.1%+23%+12%)/3=15.7%
঎
୫୧୬
=
ሺ
1.64x15.70%/0.1
ሻ
ଶ
= 6.63 mẫu.
Selecting 13 samples of CFRP sheets and getting f
p
=2.29 and p=0.989.

3.4. Geometric propertis of sample beams
Minimum size of sample number, ঎
min
, is defined as equation (1.47) with p=0.90,
f
p

=1.64, e=0.1 và COV
0
= (13.1%+3.8%+19.8%)=12.23%
Thus,঎
୫୧୬
=
ሺ
1.64x12.23%/0.1
ሻ
ଶ
= 4.02 samples.
Selecting 8 sample beams (including 2 control beams) and getting f
p
=2.003 and
p=0.957. Beam parameters as Table 3-1.

16

Figure 3-1. Sample Beam

Table 3-1. Parameters of samle beams
TT Notation

Sample
Type
Strengthening
Type
CFRP

layer

Section
(10
-3
m)
Reinfor-
cement
݂
௙௨
∗
of
CFRP
(MPa)
Concrete
f
c
'
(MPa)
1 RC21 Flexural - - 130x200 2D12 - 18
2 S21-1 Flexural Beam bottom 1 130x200 2D12 3000 18
3 S21-2 Flexural Beam bottom 2 130x200 2D12 3000 18
4 S21-3 Flexural
Beam bottom
+U stirrup
3 130x200 2D12 3000 18
5 RC25 Flexural - - 130x200 2D16 - 22
6 S25-1 Flexural Beam bottom 1 130x200 2D16 3000 22
7 S25-2 Flexural
Beam bottom
+U stirrup

2 130x200 2D16 3000 22
8 S25-3 Flexural Beam bottom 3 130x200 2D16 3000 22
3.5. Carrying out experiment
3.5.1. Preparing concrete surface and CFRP sheets
3.5.2. Adhering CFRP sheets to concrete surface
3.5.3. Arranging of measuring instruments
Sample beams are tested with four-point test (Figure 3-1). 1 sensor, stiking on
CFRP sheets, is used for CFRP strain; 1 sensor, sticking on extremely compression
zone, is used for concrete strain; 1 sensor, stcking on steel reinforcement, is used for
steel strain.
3.5.4. Results
Establishing load and deflection values at: first crack and rupture as Table 3-2.
Figure 3-2 is a typical failure of CFRP sheets debonding. Figure 3-3, 3-4, and 3-5 is
load-deflection and load-strain relationships of sample beams.

Table 3-2. Load-deflection of sample beams
Beam denotation
First crack Rupture Rupture moment
P (KN)
δ
(mm) P (KN)
δ
(mm) KNm
RC21 4 0.38 32 25.31 12.80
S21-1 6 0.69 52 26.7 20.80
S21-2 12 1.88 58 22.16 23.20
S21-3 12 1.90 74 26.27 29.60
RC25 4 1.52 66 68.73 26.40
S25-1 12 1.64 70 40.23 28.00
S25-2 12 1.45 82 >20.41 32.80

S25-3 12 1.44 90 >23.34 36.00

17

Determining values of statistical parameters of analyzed model base on testing
utimate moment (M
u,test
) and resistance moment deriving from ACI 440.2R - 08
(M
ACI
) as Table 3-3. Distribution functions of P are agrred with normal distribution
through Shapiro-Wilk checking.

Figure 3-2.Failure of S21-1 sample
Table 3-3. Determining values of statistical parameters of analysis model
Sample

f'
c

f
y
b d h
ρ
s
/ρ
bl

n
FRP

MPH M
u,Test

M
u,ACI

M
u,Test

/M
u,ACI

μ
୔

ߤ
௉
with
p=0.95

COV
P

S21-1 18

367

130

164

200

0.2 1
Concrete
crushing
20.8 16.14

1.29
1.41

1.14 11.8%

S21-2 18

367

130

164

200

0.2 2 Debonding

23.2 18.62

1.25
S21-3-

ĐAI
18

367

130

164

200

0.2 3 Debonding

29.6 20.32

1.46
S25-1 22

446

130

162

200

0.4 1
Concrete
crushing
28.0 19.19

1.46
S25-2-
ĐAI
22

446

130

162

200

0.4 2 Debonding

32.8 21.92

1.50
S25-3 22

446

130

162

200

0.4 3

Concrete
crushing
36.0 23.89

1.51

a) b)
Figure 3-3: Load-deflection relationships of sample beams
a) Group RC21 b) Group RC25
0
10
20
30
40
50
60
70
80
0.00 10.00 20.00 30.00
Load (10
3
N)
Deflection (10
-3
m)
RC
21
S
21
-

1
S
21
-
2
S
21
-
3
0
20
40
60
80
100
0 20 40 60 80
Load (10
3
N)
Deflection (10
-3
m)
RC
25
S
25
-
1
S
25

-
2
S
25
-
3

18


Figure 3-5.Load-strain relationships of Group RC25.
a) RC25 b) S25-1 c) S25-2 d) S25-3


Efficiency of moment and deflection strengthening (%) of beams strengthening
with externally bonded CFRP sheets are compared with control beam as Table 3-5.

19

Table 3-5. Efficiency of moment and deflection strengthening
Sample beam
Moment Deflection
(10
3
Nm) Efficiency (mm) Efficiency
RC21 12.80 - 8.33 -
S21-1 20.80 63% 6.64 20.3%
S21-2 23.20 81% 6.19 25.7%
S21-3 29.60 131% 6.14 26.3%
RC25 26.40 - 13.33 -

S25-1 28.00 6% 12.18 8.6%
S25-2 32.80 24% 10.87 18.5%
S25-3 36.00 36% 10.73 19.5%

3.6. Conclutions about experiment of beams strengthening with externally bonded
CFRP sheets
1. Values of statistical parameters of analyzed model are ߣ
௉
= 1.14 and
ܥܱܸ
௉
=11.8%. These values are well agreed with the values used in Chapter 2.
2. Distribution of concrete compressive strength and steel yield strength is
normal distribution, agreed with the asumptions in Chapter 2.
3. It is obseved two modes of failure: (1) CFRP sheet debonding at cracks at
beam middle and debonding develope toward supports leading to beam failure; (2)
concrete crushing leading to beam failure.
4. CFRP sheets adhering to beam bottom increase bending capacity from 6% to
63 % for 1 layer of CFRP, from 24% to 81% for 2 layers of CFRP, from 36% to
131% for 3 layers of CFRP.
5. Deflection of beams in Group RC21 decreases 20.3% and 26.3% for load of
28KN. Deflection of beams in Group RC25 decreases 8.6% and 19.5% for load of
60KN.

Chapter 4
EXPERIMENTAL RESEARCH OF TRẦN HƯNG ĐẠO BRIDGE BEAMS
STRENGTHENED WITH EXTERNALLY BONDED CFRP SHEETS
4.1. Objectives of experiment
1. Defining probability distribution of section height and width, concrete
compressive strength, and steel yield strength.

2. Determining and evaluating section reliability index of simple-span RC beams
strengthening with externally bonded CFRP sheets.
4.2. Experimantal places
a. Construction material – Cablirating LAB: LAS-XD 225
Content of experiment:
• Compressing to failure of 12 concrete samples from Trần Hưng Đạo bridge
beams on 30/12/2013
• Tensioning to rupture 3 steel samples from Bông bridge beams on 30/12/2013
b. Trần Hưng Đạo bridge
Address: Phan Thiết City- Bình Thuận province
Content of experiment:
• Measuring geometry dimensions of main beams
• Addresing the percentage of tension reinforcement deterioration

20
• Cablirating bridge after strengthening
Date: June 2012.
4.3. Bridge existing condition before strengthening
Trần Hưng Đạo bridge, across Cà Ty River, had been constructed before 1975,
at Km1704+300 of former 1A high way, which is now Trần Hưng Đạo street, Phan
Thiết City- Bình Thuận province. The bridge is operated with H8, X60 vehicles.
4.4. Strengthening
Strengthening objective is increasing permitted load to 15T-18T vehicle:
• Strenghtening main beams with externally bonded CFRP sheets;
• Retrofitting piers by concrete jacket.
4.5. Cablirating bridge after strengthening
Calibarting results show that after strengthening the bridge could be safely
operated with 18T vehicle.

4.6. Parameters for calculation

4.6.1. Section geometry dimensions
Critial beam is the middle one. Section height and with at middle of spans
namely N1 and N2 are measured 10 times. Probability distribution of section height
and with are agreed with normal distribution through Shapiro-Wilk Test.
Selected statistic parameters of section width as: normal distribution, λ=0.97;
COV= 0.84%.
Selected statistic parameters of section height as: normal distribution, λ=0.98;
COV= 1.57%.
4.6.2. FRP
Using C-Sheet 240, 0.171mm thickness of S&P manufacturer. At the beam
bottom, C-Sheet 240 is adhered with 2 layers of 0.5m width, elonging to suports.
Envioronment factor value, C
E
, is 0.85; CFRP reduction factor for flexural is
0.85. CFRP modulus of elasticty is 230000MPa.
Probability distribution of CFRP tension strength and strain is provided by the
supliers with Weibull distribution.
4.6.3. Concrete
Concrete compressive strength is used the values from 12 cylinder samples from
Trần Hưng Đạo main beams. Concrete compressive strength distribution is agreed
with normal distribution through Shapiro-Wilk Test.
4.6.4. Steel
Tension steel includes 16 bars of φ25. Three cases of tension steel area are
considered: deterioration of 5%, deterioration of 10%, and deterioration of 30%.
Probability parameters of 3 steel samples from Bông bridge and 410 steel samples
from many projects around Vietnam are in Table 4-1.
Table 4-1. Probability parameters of yield steel trength
No. Steel Type Distribution µ (MPa) Σ (MPa) COV(%) Sample No.
1 Miền Nam Normal 420.9 30.2 7.17% 71
2 Pomina Normal 432.6 18.9 7.44% 228

3 Tây Đô Normal 411.0 29.4 7.14% 75
4 Việt Nhật Normal 384.9 32.6 8.48% 36
5
Bông bridge
(Unknow type)
Normal 465.8 28.7 6.47% 3

21


Selected statistic parameters of yield steel strength as: normal distribution,
µ=394.5MPa, COV= 7.44%.
4.6.5. Live load
Maximum moment by live load at middle span is computed from measured
parameters.
Live load probability distribution and its statistical parameters are used those
values in Chapter 2.
4.6.6. Dead load
Maximum moment by dead load at middle span is computed from measured
parameters and beam theory.
Dead load probability distribution and its statistical parameters are used those
values in Chapter 2.
4.7. Determing and evaluating flexural reliability index of middle-span section
There are 18 cases of flexural reliability index of middle-span section (3 load
levels x 3 levels of steel deterioration x 2 cases fore before and after strengthening).
Results computing by 2TKN is showed in Table 4-2 with the asumption of
transversal distribution coefficients being the same for before and after strengthening.
Table 4-2. Flexural reliability index of middle-span section of Trần Hưng Đạo bridge
No.
A

s
(mm
2
)
Load
β
,

before
strengthening
β
, after
strengthening
Failure mode
1 5500 13T 5.3794 6.2328 Debonding
2 5500 18T 3.4078 4.1080 Debonding
3 5500 25T 2.6130 3.0811 Debonding
4 7070 13T 7.0013 7.6683 Debonding

5 7070 18T 4.7738 5.3460 Debonding

6 7070 25T 3.6149 4.1292 Debonding
7 7460 13T 7.3444 7.9817 Debonding
8 7460 18T 5.0691 5.6176 Debonding

9 7460 25T 3.8765 4.3802 Debonding


Figure 4-1. Flexural reliability index of middle-span section
Figure 4-1 tells that β decreases sharly when live load increases. Figure 4-2

reveals that β increases after strengthening. The increase, ∆β, is maximum for 13T
vehicle and minimum for 25T vehicle. This results is agreed with the result in chap
ter 2, β decreasing when live loads increases.
Before strengthening:

22
• For 13T vehicle: minimum β is 5.3794, much larger than target reliabilty
index, β
୘
=3.75; this means that middle-span sections could hardly get flexural
failure.
• For 18T vehicle: minimum β is 3.4078 at tension steel deterioration of 30%,
smaller than target reliabilty index, β
୘
= 3.75; this means that middle-span sections
might get flexural failure for tension steel deterioration of 30% or more.
• For 25T vehicle: minimum β is 2.9130, much smaller than target reliabilty
index, β
୘
=3.75; this means that middle-span sections could easily get flexural
failure.
After strengthening, all reliabily indexs increase:
• For 18T vehicle: minimum β is 4.1080 >β
୘
= 3.75; this means that middle-
span sections could be safely operate with tension steel deterioration of 30% or less.
• For 25T vehicle: minimum β is 3.0811 <β
୘
= 3.75; this means that middle-
span sections might get flexural failure with tension steel deterioration of 30% or

more.

Figure 4-2. Different of reliability index before and after strengthening
4.8. Conclutions
1. Middle-span section geometry measuring at site reveals that reliable
distributions of section height and width are normal.
2. Concrete compressive strength distribution is agreed with normal distribution
through Shapiro-Wilk Test of 12 cylinder samples from Trần Hưng Đạo main beams.
3. Steel yield strength distribution is agreed with normal distribution.
4. Reliability index, β, decreases sharly when live load increases; and increses
after strengthening. This results is well agreed with the result in chapter 2.
5. Before strengthening, reliability index of middle-span section is larger than
target reliabilty index at 13T vehicle; after strengthening, reliability index of middle-
span section is larger than target reliabilty index at 18T. This asumption is similar to
that of bridge calibration before and after strengthening.
CONCLUSIONS AND SUGGESTIONS
Reliability index, β, of sections of bending beams strengthening with externally bonded
CFRP under limit state is determined as ACI 440.2R-08 and reliability theory. 10000 RC
sections in wide design space and random variables have been reasonably selected for defining
moment resistance, M
ୖ
. Monte-Carlo simulation is used to compute statistic factors of M and F;
as well as experimental data of available literature are used for calculating P effect. The reliability

23

indexs, β, are calculated using Rackwitz-Fiessler method with first order function. 2TKN
propramme has been developed for computing β. Lab and field experiments have been carried
out for possible distribution and statistical parameters of random variables.
1. Contributions

a. Deriving a methodology for evaluating level of flexural reliability of RC
Bridge girders strengthened with CFRP sheets, basing on reliability theory and
flexural resistance model in ACI 440.2R-08;
b. Developing 2TKN programme for determining reliability index of rectangular
RC girder sections strengthened with CFRP sheets, basing on VBA language in
Microsoft Excel;
c. Results gaining throught experiments including:
• Defining Analysis Statistical parameters for flexural RC Bridge girders
strengthened with CFRP sheets according to flexural resistance model in
ACI 440.2R-08: µ
P
=1.14 và COV
P
=11.8% through bending to failure 8
rectangular beams;
• Determining possible distributions and statistical parameters of section
width and height of Trần Hưng Đạo bridge;
• Determining possible distributions and statistical parameters of old concrete
throught 12 samples from Trần Hưng Đạo bridge;
• Determining possible distributions and statistical parameters of old steel
yield strength throught 03 samples from Bông bridge;
• Determining possible distributions and statistical parameters of new
concrete throught 12 samples from the 8 rectangular beams;
• Determining possible distributions and statistical parameters of new steel
yield strength throught 06 samples from the 8 rectangular beams and 410
samples from many projects in Việt Nam.
2. Suggestions
a. RC beams Strengthening with externally bonded CFRP sheets should be
applied for tension steel ratio, ρ
ୱ

ρ
ୠ୪
⁄
, of 0.2 to 0.5.
b. CFRP flexural reduction factor, ψ
୤
, should be used value of 0.85 for RC
beam strengthening for live to dead load raito, M
୐
/M
ୈ
, above 1.0.
c. CFRP flexural reduction factor, ψ
୤
, should be used value of 0.90 for RC
beam strengthening for live to dead load raito, M
୐
/M
ୈ
, of 1.0 or less.
3. Recommendations
Further research is deemed necessary to investigate aspects related RC beams
strenghtening with externally bonded CFRP sheets, as:
a. Life service of RC beams Strenghtening with externally bonded CFRP sheets
in Vietnamese conditions.
b. Determining the relationship between reliability index of rectangular section
and other popular sections such as I and T.
c. Determining possible distributions and statistical parameters of dead load and
live load in Vietnam.