CHAPTER 7
Parametric Study and Unified Retrofitting Design
Using Global Energy Balance Approach
The global energy balance approach (GEBA) estimates the critical debonding state as a
function of the structural load and the curtailment location for a FRP-RC beam and it has been
shown able to predict debonding with good accuracy in Chapter 2. The relatively simple
moment-curvature models and the whole-section energy treatment are adequate for the energy
release rate estimate in this approach, which provides a probability for the parametric study
for beams with different designs. It has also been clear from Chapter 3 to 6 that the fracture
process zone in concrete is small and the fracture energy associated with debonding is similar
to the conventional concrete fracture energy independent of the debonding fracture
propagation length. Although there are variations of fracture energy due to the concrete
heterogeneity locally, it is reasonable to use a constant fracture energy for debonding, which
justifies the constant fracture energy assumption in Chapter 2.
From the experimental validations in Chapter 2 and the fracture energy justifications in
Chapter 3 to 6, GEBA is demonstrated to be a reliable and mature debonding analysis method.
Since GEBA depends on many variables in a FRP-RC beam design, a parametric study is
conducted in this chapter for a wide range of design conditions to demonstrate the effects of
various factors. A unified design approach for FRP retrofitting for both flexural strength and
debonding prevention is proposed based on the parametric study results.
7.1 Parametric study uisng GEBA
7.1.1 Determination of key parameters
The GEBA is complex and includes many parameters such as FRP and steel ratios, material
properties and beam depth. It is impossible to give a concise closed-form analytical solution
to include the exact effect of all variables for debonding, so in order to generalise GEBA, a
simplified analytical study is conducted here to determine the key parameters. After
determining the important parameters, a parametric study of their effects on debonding is then
For r
evie
wer u
se o
nly
Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach
187
carried out numerically using GEBA with the whole-section energy treatment and the M-κ
model M1, in Section 2.2 and 2.3. It is worth noting that the M-κ model M3 is a simpler
model giving just slightly less accurate than M1, and it can also be used for the parametric
study of GEBA. It was not used here because the author wanted to keep the highest accuracy
due to the nature of the work that it is a research rather than practical engineering work.
The energy release rate GR is computed from the difference between the external work and the
change of strain energy (Eqs. 2.2 – 2.4) per unit area of debonding fracture. Since strain
energy is determined from the M-κ relationship, GR is related to the beam section design. For
the purpose of deriving the relevant non-dimensional parameters that relate beam design to
debonding, a typical planar section analysis for design use is considered: a fully-bonded FRP-
RC section at the point of first yield, without compression steel. The strains in the various
materials can be determined by considering the section force and moment equilibrium using
the symbols shown in Fig. 2.2 in Section 2.1:
bxfbdEbdf cfffys ' (7.1)
)2/()( fafffysext ttcyxdbdEyxdbdfM (7.2)
where s (= bdAs / ) and
f (= bdA f / ) are the FRP and steel ratios, and is the numerical
coefficient such that 'cf represents the average concrete compressive stress in the
compression zone in an equivalent rectangular stress block. The moment in Eq. 7.2 is taken
about the point of action of the resultant concrete compression force.
Since the section is assumed fully-bonded, from the geometric relationships in Fig. 2.2, there
is:
s
yfa
s
fa
fE
f
xd
ttcxd
xd
ttcxd 2/2/ (7.3)
For r
evie
wer u
se o
nly
Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach
188
In order to deduce to the most important parameters, the concrete cover thickness (c), is
initially ignored; its influence on debonding is discussed in detail later (Section 7.2.5). The
FRP plate and adhesive layer thickness (tf+ta) are negligible (usually < 1% d) compared with
the section dimensions, so they are also ignored. Substituting Eq. 7.3 into Eq. 7.2 gives
d
y
d
x
E
EffbdM
s
f
fysyext 12 (7.4)
Eq.7.4 can be made dimensionless with the conventional flexural design section property
( 2'bdf
c):
d
y
d
x
f
E
E
f
f
f
bdf
M
c
f
s
y
f
c
y
s
c
ext1
'''2
(7.5)
Eq. 7.5 shows that the non-dimensional moment (section capacity) for a FRP-RC beam is
affected by the steel and FRP material ratios ρs and ρf . The strain energy at the designed
section is then:
Bd
y
d
x
E
fEfbd
B
ME
s
y
ffys
ext
strain 212
2
2
2
(7.6)
The effective stiffness B of a FRP-RC section comes from the stiffness of different materials
(steel bars, FRP plate and concrete) but it can be considered as a wholly concrete section with
the effect of other materials included in a single dimensionless parameter .
For r
evie
wer u
se o
nly
Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach
189
32 bdEB c (7.7)
where is the unique stiffness coefficient depending on the section design
Substituting Eq. 7.7 into Eq.7.6 gives
222
11
2
d
y
d
x
E
fEfbd
EB
ME
s
y
ffys
c
ext
strain
(7.8)
Eq. 7.8 gives the strain energy in a unit length of the beam, which is affected by the amount of
steel and FRP material. As with Eq. 7.4, Eq. 7.8 can be made dimensionless by bdf c ' , and
denoted as Ω here:
22
1''
'1
'
d
y
d
x
f
E
E
f
f
f
E
f
bdf
E
c
f
s
y
f
c
y
s
c
c
c
strain
(7.9)
Thus the non-dimensional energy term, Ω, is also dependent on the steel and FRP material
ratios ρs and ρf, in the same way as the non-dimensional moment. However, the energy
release rate GR , which is related to Estrain/b has dimensions, and its value is affected by the
section depth. Estrain represents the energy stored in unit length of the beam, and since the
length of the transfer zone is fixed (= 30tf) and can be determined if the FRP plate thickness is
known, the number of unit lengths in the transfer zone is fixed. The more strain energy that is
stored in one of these short sections, the more there is in the transfer zone, and the more that is
available to be released. Thus GR is reflected in Estrain/b and is largely determined by the term
fc’d Ω. Since fc’d Ω is proportional to d, GR should be proportional to d.
If the material properties of concrete, steel and FRP plate are fixed, it is noted from the
dimensionless part of fc’d Ω (i.e. Ω), that GR for a particular load distribution depends on the
For r
evie
wer u
se o
nly
Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach
190
tension steel ratio (s ), the FRP ratio (
f ) and the Young’s moduli. Thus the tension steel
ratio and FRP ratio are important parameters. In the dimensioned part, the effective beam
depth (d) is the key parameter. Additionally, although compression steel does not affect
Eq. 7.9 due to the simplification in derivation, it is almost invariably present in a beam and
therefore the effect of compression steel ratio (sp ) will also be studied later. Furthermore,
due to the simplified approximation in Eqs. 7.2 – 7.4, there is no term relating to the concrete
cover thickness, but it always exists in RC beams. Hence its influence on GR will also be
examined in the non-dimensional form c/d.
When considering material properties, the concrete compressive strength is usually within the
range from 30 to 60 MPa. For the FRP plate, which never reaches its failure strain and
behaves elastically, the effect of its elastic stiffness should be similar to the effect of its
amount (f ), and therefore the elastic modulus effect is not discussed. The effects of all
these parameters are presented later in Section 7.2.
7.1.2 Construction of parametric space
Since the section moment would change with the load and the location of the section, both the
change of mid-span external moment (which affects the design of the cross-section) and the
curtailment location affect the value of GR. The loading state and the section design vary for
every beam, so it is desirable to determine the appropriate factors so that the behaviour of
many different beams can be covered in the same chart.
In conventional RC beam design and FRP plate retrofitting design, flexural capacity is usually
the primary design target, which is represented by the dimensionless parameter )'/(2
bdfM c,
where the moment here is the maximum moment (or mid-span moment). Conventional RC
beams typically have )'/(2
bdfM c in the range 0.05 – 0.29 in the most heavily loaded section
(Park & Paulay 1975); strengthened beams may be stronger so the range is extended up to 0.4.
This parameter is for the section under maximum loading, which controls the amount of FRP
needed. The principal decision that has to be taken is where to curtail the plate; this is
typically expressed as a fraction of the shear span, (shearcur LL / ), where Lcur is measured from
the support to the plate end.
For r
evie
wer u
se o
nly
Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach
191
It is now possible to produce a three-dimensional plot in this parametric space showing the
variation of GR with the maximum moment and the curtailment location for a typical beam
(Fig. 7.1). The value of Gf , which is a material property shown to be constant in the previous
chapters, appears as a horizontal plane while GR is the curved surface.
Figure 7.1 Fracture energy plane (Gf = 0.15 N/mm) and energy release rate (GR) surface (For
a beam with h = 400 mm, %0.1s and %5.0f )
The intersection line between the curved GR surface and the Gf plane indicates when
debonding would occur and is termed the debonding contour (DBC). This can be plotted on a
2D plot of normalised curtailment (shearcur LL / ) against normalised loading state
( )'/(2
bdfM c) for the mid-span section, which can be used to demonstrate the effects of
changing the various parameters. The beam is likely to fail by debonding if GR > Gf and
should be safe if GR < Gf.
For r
evie
wer u
se o
nly
Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach
192
7.1.3 Design of the standard beam
For the purposes of comparison, a standard beam is considered with the properties given in
Table 7.1.
Table 7.1 Standard beam design parameters
Depth (h) 400 mm Steel Ratio ( )/(bdAss ) 1.0%
Concrete cover (c) 35 mm FRP Ratio ( )/(bdA ff ) 0.5%
Shear span (Lshear) 1500 mm Compression Steel Ratio sp 0
Steel yield strength fy 530 MPa Concrete cylindrical strength f’c 37 MPa
Nominal FRP thickness 2 mm FRP elastic modulus 165 GPa
Nominal adhesive thickness 1.5 mm Adhesive shear modulus 4.8 GPa
Under each load, GR is computed for the curtailment length (Lcur) varying from 200mm to
600mm. The reinforcing steels are input as a ratio rather than as discrete bars. The nominal
FRP and adhesive layer thicknesses are only used to determine the moment arm when
considering the flexural contribution of FRP force in the section. A Gf value of 0.15N/mm is
used according to the test results in Chapter 6, which is similar to the conventional opening
concrete fracture energy. All the other sections considered have one parameter that differs
from the standard beam. The GEBA method using the whole-section M-κ model M1 is used
to investigate the effects on debonding, and the results are presented in the following sections.
7.2 GEBA parametric study results
7.2.1 Effect of tension steel ratio
Fig.7.2 shows the effect on the DBC of varying the tension steel ratio (s ) in the range from
0.6% to 2.0% with a step of 0.2%. The moment in the x-axis is the mid-span (largest)
moment.
For r
evie
wer u
se o
nly
Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach
193
Figure 7.2 Effect of tension steel ratios (0.6 – 2.0%) on DBC with f = 0.5%
The region to the bottom-left of the DBC is the safe zone where GR is smaller than Gf whilst
the region to the top-right of DBC indicates debonding. Curves to the top right of the plot
(such as the line for ρs = 2.0%) show that the section is more resistant to debonding than
curves to the bottom left (such as the line for ρs=0.6%).
It can be concluded from Fig. 7.2 that: (i) Debonding is less likely to occur with an increasing
amount of tension steel, because steel enhances the section stiffness, resulting in lower
curvatures so there is less strain energy available for release. As a result GR is smaller and the
FRP-RC beam is less likely to debond. (ii) The shift of DBC position is relatively even as ρs
changes, because the tension steel is always fully-bonded within the RC section and its
flexural strengthening effect transfers directly into the section without loss.
For r
evie
wer u
se o
nly
Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach
194
7.2.2 Effect of FRP strengthening material ratio
By varying only the FRP ratio (f ) in the standard beam from 0.1% to 1.5% with a step of
0.2%, the changes of the DBC are as shown in Fig. 7.3.
Figure 7.3 Effect of FRP ratios (0.1 – 1.5%) on DBC with s = 1.0 %
Fig. 7.3 is significantly different from Fig. 7.2 in two ways. (i) Debonding occurs more easily
with more FRP material, because a greater f means more energy is stored in the FRP prior
to debonding, all of which can be released to make debonding occur. (ii) The change in the
DBC position is small when f is large (0.7% to 1.5%), while it moves dramatically when
f is small (0.7% to 0.1%). This rapid change is due to the change in the cracking state of
the relevant beam sections; when f = 0.7% the sections may be uncracked, whilst when
f = 0.1% they may well be partially-cracked. Unlike the tension steel, the FRP plate at the
For r
evie
wer u
se o
nly
Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach
195
plate end is not fully-bonded and thus its flexural contribution cannot be completely
integrated into the section. It can be observed from the figures that a beam with FRP ratio
less than 0.3% has little risk of debonding in this case, because the FRP plate attracts a
relatively small load.
It is notable in Figs. 7.2 and 7.3 that changing the amount of steel and FRP material have
opposite effects on the likelihood of debonding. Generally, a beam with a high ratio of
sf / has to transmit a higher proportion of the tensile force through the bonded region,
which thus makes debonding more likely, whereas a lower value makes it less likely. Thus,
changing f and
s has opposite effects. Since both f and
s are also important
parameters in beam strength, a way of optimising the f and
s values should be determined
from both the debonding and beam strength considerations; details will be presented in the
unified design approach in sections 7.4 – 7.6.
7.2.3 Effect of compression steel ratio
By varying only the compression steel ratio (sp ) in the standard beam from 0 to 1.0% with a
step of 0.2%, the changes of the DBC are as shown in Fig. 7.4.
For r
evie
wer u
se o
nly
Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach
196
Figure 7.4 Effect of compression steel ratios (0 – 1.0%) on DBC (s = 1.0 % and f = 0.5 %)
It is observed that: (i) The compression steel ratio has little effect on debonding. Furthermore,
since the amount of compression steel used is small in practice (<< 1.0%), it can be neglected
in debonding consideration, which justifies the exclusion of compression steel in the
derivation of the important dimensionless parameters in Section 7.1.1. (ii) Although its effect
is limited, increasing the amount of compression steel slightly reduces GR and makes the
FRP-RC beam marginally less likely to debond. The reasoning is similar to the effect of
increasing tension steel; more compression steel makes the section stiffer, but because only
the debonding that occurs prior to yielding is taken as premature failure, the strains at the
compression steel level prior to debonding are small and its contribution is negligible
compared with the adjacent concrete.
For r
evie
wer u
se o
nly
Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach
197
7.2.4 Effect of concrete compressive strength
The effect of varying only the concrete compressive strength of the standard beam is shown in
Fig. 7.5. Since the section does not fail in compression, the principal effect is almost certainly
due to the consequent change in the tensile strength of concrete. The change in concrete
tensile strength affects most significantly the first crack moment in the section analysis and
hence the energy state of a section. Varying concrete strength may also influence the concrete
fracture energy Gf, however, this influence should be small provided that the aggregate
properties remain similar (see Eq. 2.14). For normal strength concrete, Gf is commonly
considered dominantly affected by the aggregate type. However, the test results in Chapter 6
suggest that the change of Gf due to concrete properties is less than the random variation of Gf
(Section 6.2). Thus the results from Chapter 6 are used, and Gf is kept constant at 0.15 N/mm
for comparing the effects of varying concrete compressive strength.
Figure 7.5 Effect of concrete compressive strength on DBC (s = 1.0 % and
f = 0.5 %)
For r
evie
wer u
se o
nly
Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach
198
A beam with higher strength concrete is less likely to debond, if the fracture energy is
assumed the same. A higher concrete compressive strength makes the section stiffer so the
curvatures, and hence the strain energy, are lower. It is evident from the figure that when fc’
changes from 30 to 60 MPa, the DBC position changes evenly because the concrete remains
very close to linear-elastic in the transfer zone close to the plate end.
7.2.5 Effect of concrete cover thickness
The concrete cover influences the section analysis via the geometry, which affects the energy
release rate (GR) computation in GEBA. The concrete cover thickness is studied for standard
beams of 400 mm and 800 mm deep. By varying only the cover thickness from 5% to 15% of
the beam depth, the change of DBC is as shown in Fig. 7.6.
For r
evie
wer u
se o
nly
Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach
199
Figure 7.6 Effect of concrete cover thickness on DBC: (a) 400 mm beam; (b) 800 mm beam.
The idealised “zero cover thickness” condition is also presented to show that in the GR
computation the effect of concrete cover thickness is insignificant. Since the overall beam
depth h is fixed, the larger the cover thickness, the smaller the effective beam depth d would
be and the less likely the beam would be to debond (refer to Section 7.1.1). It is evident that
the influence of the cover thickness is small, and its influence is related to the absolute value
of the thickness instead of its ratio to the beam depth. Hence it is reasonable not to consider
the cover thickness separately in practical design, but to take the beam depth (h) to be the
same as the effective beam depth (d).
It is pointed out in Achintha & Burgoyne (2011) and Burgoyne et al. (2012) that a thicker
concrete cover layer may lead to higher uncertainty of the safe plate end location (Lcur) in a
GEBA debonding prediction. The critical flexural-shear crack that initiates debonding
usually develops in the vicinity of the plate end propagating at an angle to the interface, up to
a level between the interface and the tension steel bars in the concrete cover layer, before
going into horizontal debonding direction. This uncertainty in curtailment length due to the
inclined initiation increases with the concrete cover layer thickness. This uncertainty exists in
For r
evie
wer u
se o
nly
Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach
200
all GEBA predictions, and further detailed fracture studies of the concrete cover layer are
needed to quantify it. The parameter effects presented here do not take account of this
uncertainty.
7.2.6 Effect of RC section depth
By varying the beam depth only from 200mm to 1000mm in the standard beam with a step of
100mm (keeping the concrete cover as 35mm), the effect of beam depth on DBC is as shown
in Fig. 7.7: debonding becomes more likely if the beam depth increases. With the constant
rate of beam depth increase, the shifts of the contours become smaller.
Figure 7.7 Effect of section depth on DBC of varying beam depth only
It should be noted that, Lshear is taken here as 1500 mm for beams with all above depths to plot
the contour figures for consistency with previous DBC figures. The deeper beams are thus
For r
evie
wer u
se o
nly
Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach
201
close to being considered as “deep beams” for which different considerations would apply and
which are not covered in this work.
7.3 Combined effect of Gf and beam depth
In order to see if debonding will occur, GR has to be compared with Gf. It was shown in
Section 7.1.1 that GR should be proportional to the effective beam depth d, with a proportional
coefficient with the dimension of N/mm2, if the concrete compressive strength (fc’) is fixed.
This effect has been demonstrated in Section 7.2.6. Since, in practice, the concrete cover
thickness c is small in comparison to the beam depth h and its effect on DBC is negligible as
demonstrated previously, it can also be taken that GR is proportional to the overall beam depth
h.
As a result, the term of h/GR instead of GR should be used to normalise the DBC plot for
beams that only differ in depth, so a plot of the h/GR surfaces, such as Fig. 7.1, should be
provided to consider debonding. Debonding conditions for beams with different
combinations of depth (h) and fracture energy (Gf) are then represented by the same
normalised DBC, if they have the same h/Gf. For instance, if the depth of a reference beam is
defined as href then it would be expected that for beams of other depths h (but with all other
parameters the same), their DBCs obtained for an equivalent fracture energy Gf-q, such that
h/Gf-q = href/Gf, should collapse to a single line. These lines are plotted for a range of beams
with depths that vary from 200 mm to 1000 mm in Fig. 7.8 (taking href as 400 mm, Gf as 0.15
N/mm, and Gf-q as h/(hrefGf)).
For r
evie
wer u
se o
nly
Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach
202
Figure 7.8 GR contours for beams with different depths at Gf-q = (Gf×h)/400
It is clear in Fig. 7.8 that these contours are close to each other as expected. The small
discrepancies in the figure come from keeping the concrete cover the same for all the beams
with different depths. The DBCs in Fig. 7.8 lie at the location of the DBC for the standard
beam in the previous figures. Thus each DBC in the previous studies is effectively the DBC
for all the beams with the same depth-to-fracture-energy ratio (h/Gf), so there is no need to re-
compute the DBC for other beams having the same h/Gf. This provides a concise way to
consider debonding for different beams.
With other properties the same as the standard beam, here the DBCs with various h/Gf values
(1.3 – 13.3×103 MPa
-1) are plotted in Fig. 7.9, roughly corresponding to the combination of
different Gf and h values ranging from 0.07 to 0.2 N/mm and 200 to 1000 mm respectively.
For r
evie
wer u
se o
nly
Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach
203
Figure 7.9 Normalised DBC for beams with different h/Gf values
These DBCs have been obtained by cutting the GR surface of the 400 mm standard beam with
horizontal planes at different Gf values. In this way, instead of computing GR surfaces for
beams with various depths time after time, the GR surface for the standard beam provides
enough information for design use for the beams differing only in depth.
In design of a beam that differ from the standard beam only in depth, after knowing the
particular value of h and Gf the appropriate DBCs contour with the h/Gf value can be
determined in Fig. 7.9. Since M/(fcbd2) can be obtained from conventional section calculation,
the required curtailment can easily be determined. In the design for other beams, charts
similar to Fig. 7.9 are needed to determine the DBCs for each combination with other fc’, s
and f , which means numerous charts are required and impractical to provide. However,
based on these charts, simplified design approach can be worked out using even more concise
charts, which will be explained in the later sections.
For r
evie
wer u
se o
nly