Parametric Study and Unified Retrofitting Design …Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach 187 carried out numerically using

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

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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)

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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 .

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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

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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 %)

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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.

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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

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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

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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)).

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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.

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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.

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Documents

Parametric Study and Unified Retrofitting Design …Chapter 7 Parametric Study and Unified Retrofitting Design using Global Energy Balance Approach 187 carried out numerically using