Earthquakes occurred in current decades have caused great damages to structures,
particularly structures that were built according to older codes of practice.
Damages and collapses experienced by reinforced concrete structures necessitated
repair and retrofitting of these structures. Reinforced concrete bridges also
suffered damages such as failure in piers, joints and girders. Piers are the
most important structural members in bridges that their failure causes total
system to collapse. Because of inadequate transverse reinforcing or short anchorage
length in piers, particularly in plastic joint regions, these members fail under
low loads. Therefore, retrofitting of bridge piers is inevitable (Solberg
et al., 2009).
In recent decades, several methods were proposed to improve the flexural capacity
and ductility of piers in plastic joint regions. Concrete confinement is a very
useful method for ultimate strain enhancement and increasing the compressive
strength and energy absorption. In design stage, confinement is provided using
closely spaced transverse reinforcements. However, in rehabilitation stage,
FRP jacketing is one of the effective methods to compensate the hoop reinforcement
deficiencies in piers (Monti et al., 2001; Binici,
2007). Therefore, it is desirable to find a relation between the confinement
effects produced by FRP and transverse reinforcements. The present study attempts
to obtain FRP-hoop reinforcement relation in circular bridge piers. Due to multi-parameter
effects, attaining an implicit equation was not possible; thus, Artificial Neural
Networks (ANN) was used to determine the relation existed for small experimental
specimens. Required data for ANN method were produced using Finite Element Program.
A lot of studies are done about confinement provided by transverse reinforcements
and FRP jackets. Early investigators showed that the stress and the corresponding
longitudinal strain at the strength of concrete confined by an active hydrostatic
fluid pressure can be represented by the following simple relationships (Mander
et al., 1988):
and εcc are the maximum concrete stress and the corresponding
strain, respectively under the lateral fluid pressure fl,
and εco are unconfined concrete strength and corresponding strain,
respectively; and k1 and k2 are coefficients that are
functions of concrete mix and the lateral pressure (Mander
et al., 1988).
Richart et al. (1928) found that the average values
of the coefficients for the tests they conducted to be k1 = 4.1 k2
= 5 k1 and also, Balmer (1949) found from
his tests that k1 varied between 4.5 and 7.0 with an average value
of 5.6, the higher values occurring at the lower lateral pressures (Mander
et al., 1988).
Mander et al. (1988) presented an equation for
confinement produced by transverse reinforcements that then became the base
for a lot of models proposed for axial stress-strain curve of concrete confined
by FRP jackets. The model presented by Mander et al.
(1988) included members with both circular and rectangular section under
uniform and cyclic static and dynamic loading and included any types of steel
confinements (Mander et al., 1988). Hoshikuma
et al. (1997) proposed a stress-strain model for concrete including
transverse reinforcements confining effects. This model was based on some compressive
tests on RC samples. Their test results showed that three parameters: peak stress,
strain at peak stress and deterioration rate are important factors in confined
concrete stress-strain curve (Hoshikuma et al., 1997).
First attempt on using composites for confinement is presented by Fardis
and Khalili (1981). They implemented tests on concrete specimens that were
confined using fiberglass and proposed a model for concrete stress-strain curve
based on Richart model. Samaan et al. (1998)
proposed a simple and accurate model based on particular expansion property
of concrete confined with FRP (Samaan et al., 1998).
Li and Sung (2004) studied on shear failure of circular
bridge columns retrofitted by FRP jacket presented an effective confined concrete
constitutive model named Modified L-L model. It is used for determination of
CFRP jacketing effects in retrofitting of bridge piers and to analyze lateral
force-displacement relation in circular columns (Perera,
2006). Perera (2006) presented a simplified damage
model based on continuum damage mechanics for seismic assessment and retrofit
design of columns under flexural-axial combined loading (Hoshikuma
et al., 1997).
The idea of this study is to use the previously proposed models to establish a relation between required transverse reinforcement and thickness of FRP jackets which is addressed as equivalent FRP thickness. This is done so by finding the FRP thickness that produces the same confinement as a specific reinforcement. Using the equivalent FRP thickness has the advantage of omitting somewhat cumbersome design calculation required by FRP design codes. This way the practicing engineer can design and/or retrofit structural members without extensive knowledge of FRP design procedures. Calculation of equivalent FRP thickness requires solving a multi-parameter nonlinear equation. This is done with the help of Artificial Neural Networks and the results are presented as graphs which can easily be used by designers. The data required for ANNs input are produced using calibrated finite element models. Finally a comparison of the results with real data is presented in the study that shows promising agreement and accuracy.
CONFINING EFFECT IN PIERS
In seismic design of reinforced concrete columns of building and bridge substructures,
the potential plastic hinge regions need to be carefully detailed for ductility
in order to ensure that the shaking from large earthquakes will not result in
a collapse. Adequate ductility of members of reinforced concrete frames is also
necessary to ensure that moment redistribution can occur. The most important
design consideration for ductility in plastic hinge regions of reinforced concrete
columns is the provision of sufficient transverse reinforcement in the form
of spirals or circular hoops or of rectangular arrangements of steel, in order
to confine the compressed concrete, to prevent buckling of the longitudinal
bars and to prevent shear failure. Anchorage failure of all reinforcement must
also be prevented (Mander et al., 1988).
Confinement can be achieved by steel jacketing. Steel tubes filled with concrete
have important benefits. Their benefits include high stiffness and strength,
large energy absorption and enhanced ductility and stability. The tube interacts
with the core in three ways: (1) it confines the core, thereby enhancing on
its compressive strength and ductility; (2) it provides additional shear strength
for the core and (3) depending on its bond strength with concrete and its stiffness
in axial direction, it develops some level of composite action, thereby also
enhancing the flexural strength of concrete. The core, in return, prevents buckling
of the tube. Since steel is an isotropic material, its resistance in axial and
transverse direction can be neither uncoupled nor optimized. Also, its high
modulus of elasticity causes a large portion of axial loads to be carried by
the tube, resulting in premature buckling. Furthermore, its Poisson's ratio
is higher than of concrete at early stages of loading (Fardis
and Khalili, 1981). This differential expansion results in partial separation
of two materials, delaying the activation of confinement mechanism. Finally,
outdoor use of steel tubes in corrosive environments may prove costly (Mirmiran
and Shahawy, 1997).
Using Fiber Reinforced Plastic (FRP) materials may eliminate these problems.
Hybrid construction with FRP and concrete combines the mass, stiffness, damping
and low cost of concrete with the speed of construction, lightweight, strength
and durability of FRPs. The orthotropic behavior of FRPs makes them most suitable
for encasing concrete columns. FRP jackets have already been successfully used
in the field of retrofitting of concrete columns (Saadatmanesh
et al., 1994).
FRP-HOOP RELATION IN CIRCULAR COLUMNS
For clarity, it is needed to present a definition of equal confinement. If axial stress-strain curve of concrete confined with hoop reinforcement coincide with stress-strain curve of concrete confined with FRP, it can be said that hoop reinforcements and FRP have an equal confinement effect on concrete. However, it is clear that because of differences in FRP and steel properties, this condition occurrence is not possible. Therefore, it is necessary to consider an equal confinement criterion. Usually in columns under axial compressive load, if concrete strain reaches to ultimate strain, columns failure will be inevitable. Thus, in the present study, equal ultimate strain in concrete was considered as an equal confinement criterion in both columns confined with FRP or hoop reinforcements. On the other hand, if ultimate strain in concrete confined with hoop is equal to ultimate strain in concrete confined with FRP, it can be said that confinement is equal. Attaining to FRP-hoop reinforcement relation some data was produced in finite element program.
Data production in finite element program: In the present study, Finite
Element Program (FEP) was used to produce data needed to find FRP-hoop relation
using Artificial Neural Network. However, it is always necessary to verify a
finite element model. For this, an experimental specimen tested by Hoshikuma
et al. (1997) was used for concrete confined with transverse reinforcements
and sample tested by Samaan et al. (1998) was
used for concrete confined by FRP.
The specimen tested by Hoshikuma et al. (1997)
was 1500 mm in height and 500 mm diameter. Other specimen properties are presented
in Table 1.
Concrete and reinforcements stress-strain curves were assumed as multi-linear and kinematic bilinear, respectively. A 3D finite element model meshed by tetrahedron shaped elements was used for simulation. The pier was fixed at the lower end and the load was applied by displacement control method. The model geometry is shown in Fig. 1.
Specimen was subjected to a uniform compressive load and analyzed statically. Axial stress and strain distribution of modeled sample is shown in Fig. 2. In order to perform buckling control, the model was once analyzed for buckling using the Eigen buckling command.
In Fig. 3, Stress-strain curves obtained from finite element
modeling and relations presented by Mander et al.
(1988) and Hoshikuma et al. (1997) are shown.
|| Modeled column geometry
It can be seen that stress obtained from FEP for an equal strain is greater
than experimental curves. Also, an ultimate strain in FEP is less than that
of experimental models.
Similarly, the result of FEP should be validated for RC columns confined with
FRP jacket. For this, a column tested by Samaan et al.
(1998) was modeled. The sample has a height 305 mm and a diameter 152.5
mm that is confined with GFRP tube and was subjected to axial compressive load.
Other properties of a sample are given in Table 2.
FRP behavior was considered perfectly elastic and Tsai-Wu criterion was used as failure criterion. After analyzing of a sample, axial stress-strain curve was obtained as follows (Fig. 4). It can be seen that similar to concrete confined with hoops, stress obtained from FEP curve for an equal strain is greater than experimental curves. Also, ultimate strain in FEP is less than that of experimental models.
The result of both models, confined with transverse reinforcements and FRP
jacket, are similar in comparison with experimental results. Also, in stress-strain
curves given by Mirmiran et al. (2000) the same
conditions (smaller ultimate strain and higher stress) are seen (Fig.
||Axial stress and strain distribution
||Axial stress-strain curves of RC column confined with hoop
||Axial stress-strain curves of RC column confined with FRP
While, the main objective of this present study is comparison of FRP confined columns with hoop confined columns to obtain the FRP- hoop relation and because of fairly accurate results from FEM, it can be concluded that FEP outputs are correct to extract FRP-hoop relation in small columns.
FEP analyses is based on Finite Element Method. In FEPs, three types of equations are used and solved: equilibrium, consistency and boundary equations. These are fundamentals of any models solution. Therefore, it can be concluded that FEP outputs are valid for columns modeled with any sizes.
After validation of FEP, it was used to create data for ANN. In order to produce each data, two columns with same dimensions and concrete material, one confined with hoop and another confined by FRP jacket, were modeled in FEP. At first, reinforced concrete column was analyzed with a small number of steps and the ultimate strain of concrete is determined at the point of rupture. Then, concrete column confined with FRP was analyzed with a small number of steps. In this stage, tensile strength and FRP modulus of elasticity are given and only its thickness is changed to obtain to the ultimate strain of concrete equal or near to that of reinforced concrete column. Finally, both columns were analyzed with small step intervals to achieve the desired accuracy. In this stage, if the concrete ultimate strain is not equal in columns, the model was changed until equal concrete ultimate strain in both models is reached.
In total, 122 data were created in this way. Table 3 shows several data that were made using FEP.
Regression analysis: The data produced by FEP are plotted in Fig.
6. In Fig. 6 steel reinforcement and FRP volumetric ratio
are plotted against each other and each data represents the two FEP models that
have reached a similar ultimate strain and thus showing the FRP amount that
has an equal effect as steel reinforcement.
||Stress-strain curves given from FEP in comparison with experimental
curves (Mirmiran et al., 2000)
It is clear that no explicit trend or relationship exists that describes variations of FRP thickness with respect to transverse reinforcements.
Proposed equation based on experimental models: In this section in order to find a relation between FRP thickness and transverse reinforcement, models proposed based on experiments are used to define and parameterize the problem.
A concrete axial stress-strain model considered by Hoshikuma
et al. (1997) included two branches (ascending and falling branches).
Equation for ascending branch was given as:
where fcc, εcc and are peak stress, strain at peak
stress and initial stiffness, respectively. εcc and fcc
for circular columns were presented as:
|| FRP-hoop equation from regression analysis
|| Several data made with FEP
In above equations ρs, fyh, fco are hoop volumetric ratio, hoop yield stress and plane concrete compressive strength, respectively. Falling branch equation was presented as follows:
where, Edes is the deterioration rate which is developed from regression
analysis of test data in the range εc to εcu.
The definition of ultimate strain (εcu) is important. In the
tests, crushing of core content and buckling of longitudinal reinforcement were
observed when the compressive stress dropped to less than 0.5 fcc.
Because such damage is excessive and not repairable, the strain corresponding
to 50% of the peak stress fcc is assumed as the ultimate strainand
obtained as (Hoshikuma et al., 1997):
In Hoshikuma et al. (1997) model, substituting
Eq. 5, 6 and 8 in Eq.
9, it will be as follows:
Samaan et al. (1998) presented a bilinear model
for concrete confined by FRP. Richard and Abbott (1975)
presented it as a four-parameter equation and calibrated it as:
where, fc and εc are axial stress and strain in
confined concrete, respectively; E1 and E2 are initial
and secondary slopes, respectively; fo is the reference plastic stress
at the interception of the second slope with the stress axis; and n a curve-shape
parameter that mainly controls the curvature in the transition zone. It can
be shown that the model is not very sensitive to the curve-shape parameter n
and a constant value of 1.5 was used. The first slope (E1) depends
solely on concrete and to evaluate it, the following formula for the secant
modulus as proposed by Ahmad and Shah (1982) was adopted
(Samaan et al., 1998):
The second slope (E2) is a function of the stiffness of the confining tube and to a lesser extent, the unconfined strength of concrete core, as follows:
In above equations
is unconfined concrete strength (MPa), Ej effective modulus of elasticity
of the tube in the hoop direction and the interception stress fo
is a function of the strength of unconfined concrete and the confining pressure
provided by the tube and was estimated as:
fr is the confinement pressure as given by:
where, fj = hoop strength of the tube; and D = core diameter:
Samaan et al. (1998) estimated peak stress and
strain of concrete confined with FRP tube as follows:
Substituting Eq. 13, 14 and 16
in Eq. 16, it will be as:
Now, if equal ultimate strain in concrete is considered as an equal confinement
criterion, equating Eq. 10 and 18:
It should be noted that the above equation is valid only for test specimens with small sizes. But two points are derived from above equation:
||It is found that there are 8 parameters affecting FRR-hoop
relation that are: column height (H), column diameter (D), hoop diameter
(d), hoop spacing (s), FRP yield strength (fj), FRP modulus of
elasticity (Ej), compressive strength of concrete ()
and hoop yield strength (fy). FRP thickness (tj) is
considered as a parameter that its relation with hoop needs be obtained
||Equation 19 can be modified to use for
tall columns. Thus a correction coefficient K was added to the right hand
side of the equilibrium to account for the difference. Therefore, Eq.
19 takes the form below:
Once the coefficient K is determined, the equivalent FRP thickness (tj) can be calculated by Eq. 20. ANN was used to derive a relationship between K and the rest of the parameters, which are discussed in the following.
DETERMINATION OF COEFFICIENT K
Determination of K coefficient was implemented by using ANN and the produced data in FEP were used to train the network. Usually, ANN methods consist of two steps: 1-traininig 2- testing. In training step, 100 produced data consisting of 8 inputs and 1 output was applied to ANN. 22 remaining data were also applied in the testing step. If ANN outputs correspond to real outputs, it can be said that ANN can predict FRP-hoop relation correctively.
Figure 7 shows ANN outputs in comparison with available data for training and testing steps, respectively. It is clear that ANN outputs for small amount of FRP volume are not predicted very well. Therefore, in spite of good correlation coefficient, ANN outputs are not valid.
There are two methods to solve this problem: (1) normalization of data (2) increase of data. If ANN outputs are not improved with data normalizing, increase of data is inevitable.
Figure 8 shows ANN normalized outputs in comparison with FEP data that were normalized in range of 0.1 to 1 and applied to ANN. It can be seen that data normalization is an effective method and ANN to FEP output ratios are better than that of un-normalized data. Therefore, application of normalized data to ANN, results in better prediction of outputs. On the other hand, ANN is only valid for normalized data in range of 0.1 to 1.
Now, ANN can be used to determine K coefficient. For this, each of 8 effective
parameters was considered as a constant and the other 7 parameters were changed
to obtain a relation between K and the constant parameters. However, it did
not lead to best results. Then, two parameters of 8 effective parameters were
considered as constants and other 6 parameters were changed. It was found that
if concrete compressive strength ()
and steel yield strength (fy) was considered as constant parameters,
curves as follow can be obtain for
Similar graphs can be plotted for different values of
and fy. Figure 9 only shows results for four different
For example, considering a reinforced concrete column with the parameters:
height 5000 mm, diameter 1000 mm, concrete strength 50 MPa, reinforcement ratio
0.004, steel yield stress 325 MPa; using Eq. 10 the ultimate
strain is estimated as follows:
Using graphs for
= 30 MPa, fy = 325 MPa (Fig. 9):
Now using a GFRP of tensile strength 800 MPa and modulus of elasticity 30000
MPa the equivalent FRP thickness is calculated from Eq. 20:
||ANN outputs in comparison with available data for training
and testing steps
||ANN normalized outputs in comparison with available normalized
data for training and testing steps
Solution of the above equation results in an equivalent FRP thickness of 0.77 mm. This is the thickness that if used, results in the same ultimate strain as when transverse reinforcement is present. If according to a code of practice (perhaps a newer code) this column would require more reinforcement to ensure required performance, then the equivalent FRP thickness can be calculated in the same way. The difference between the two thicknesses is the thickness of FRP material needed to retrofit the pier.
In recent decades, several methods were proposed to improve the flexural capacity
and ductility of piers in plastic joint regions. Concrete confinement is a very
useful method for fracture strain enhancing and increasing in compressive strength
and energy absorption. In design stage, confinement is provided using closely
spaced transverse reinforcements. In retrofitting of concrete structures, particularly
bridges, one of effective methods to compensate for hoop reinforcements in piers
is FRP jacketing. Using a relation between FRP and transverse reinforcements,
FRP amount to compensate for hoop reinforcements deficiency can be obtained.
In the present study, FRP-hoop reinforcement equation in circular bridge piers
was derived from existing experimental models. Because of multi-parameter effects,
an explicit relation was not possible, therefore Artificial Neural Network (ANN)
was used to popularize the equation existed for experimental models. Required
data for ANN were produced using FEP. It was found that if concrete compressive
and steel yield strength (fy) were considered as constants, a relation
is possible. Comparison of FEP created data with results obtained from proposed
equation and plotted graphs, good agreements between experimental results and
the proposed were observed. In this way equivalent FRP thickness corresponding
to hoop reinforcements can be calculated and used for design and retrofitting