Introduction

Strengthening or retrofitting of the damaged concrete structures due to upgradation of loads and severe environmental condition has been significantly addressed in the last few decades. Reinforced concrete beams which form an integral part of structural system are more prone to the damage during their service lifetime. To address these issues, various strengthening techniques and new repair materials are developed. Most of the repair materials supplemented with suitable strengthening techniques tend to satisfy the criteria for strength restoration or upgradation. But they fail to satisfy the environmental durability criterion resulting in the loss of efficiency of the retrofit system. In addition to predict the debonding behaviour of retrofitted beams under monotonic loading, it is also essential to predict the debonding behaviour under fatigue loading. This becomes significant in situations such as bridges subjected to traffic loads and buildings subjected to wind-induced fatigue.

In a typical strengthening system, the performance of strip–beam interface in providing the means for effective stress transfer is of utmost importance. A number of failure modes in strip-strengthened beams are directly caused by the interfacial debonding between the composite strip of materials and the concrete beam [1,2,3]. One of the failure modes, referred to as intermediate crack-induced debonding (IC debonding), involves debonding of the composite plate or strip which initiates at a major crack such as flexural or flexure-shear crack and propagates along the strip-to-concrete interface. Debonding is likely to happen at the ends of the strip due to the combination of high normal and shear stresses. Predominantly, failure occurs due to the latter which qualifies for the mode of failure to be referred as Mode II [4,5,6].

The intermediate crack(IC)-induced debonding may be classified into two types. The first type involves initiation of debonding from a crack in a strip under tension and propagation towards the free end of the strip [7]. The studies reported in the literature reveal that the main failure mode of this type of strengthening system is the concrete failure under shear occurring near the concrete–adhesive interface. The ultimate load that the interface can carry depends predominantly on the concrete strength beyond which a particular value of bond length cannot increase which is termed as the effective bond length.

In normal RC beams bonded with an FRP soffit plate, series of cracks are generally distributed along the length of the beam. As a result, between the major flexural or flexural-shear crack where debonding initiates and the stress-free ends of the plate, causes initiation of other cracks. The mechanics of the debonding process in such beams can be significantly different from that of a simple pull-out test specimen. These failures may be termed as the second type of IC debonding failures to distinguish them from the first type of IC debonding failures.

When it comes to the case of fatigue loading, it was reported in the literature that introduction of composite material in the tension zone of the reinforced concrete beams resulted in significant reduction in longitudinal steel stress and increase in the fatigue life of the beams [8, 9]. But it was also observed that the premature detachment of the composite strips from the concrete substrate again increased the stress range in the reinforcing steel. This can be attributed to the debonding of the strip which redistributes the stress back to the internal steel reinforcement.

The cases of debonding have seldom been reported in the literature regarding ultra-high-performance concrete (UHPC)-strengthened structures. This is due to the fact that the tensile strength and the stiffness of UHPC are more similar to those of the concrete substrate in comparison with the other repair materials such as FRP and steel. There is a need to determine the influence of shear stresses developed at the edge of the strips and between the cracks in a view to determine the optimum bond length and thickness of the strips. Also accurate predictions of interfacial stresses are important for designing guidelines for composite strip-strengthened RC beams against the debonding failures. Some articles reported in the literature have modelled the thick laminated composite plates based on higher-order shear deformation theory [10,11,12]. However, finite element and analytical studies on the ultimate load failure prediction and modes of failure of the laminated plates have been carried out by the researchers in the recent past [13, 14]. In the present study, the popular models such as Tounsi model (2006), Teng model (2006) and Diab model (2009) are employed to predict the debonding behaviour of retrofitted RC beams under monotonic and fatigue loading [15,16,17].

Models for Debonding Studies

This section discusses the various analytical models employed in the present study for debonding behaviour of damaged RC beams strengthened with UHPFRC precast strips.

Tounsi Model (2006)

The intermediate crack-induced debonding (first type) was addressed by suggesting a closed-form solution which is an improved method of Teng model [4, 15]. The model reasonably assumes that the shear stress distribution is continuous across the adhesive–adherend interface and the shear stress be zero at the free surface to satisfy the equilibrium. Shear deformation in the adherents is ignored as suggested in the literature [18, 19].

To start with a differential element of size dx is considered from the strengthened beam (Figs. 1, 2). The beam is comprised of three layers of materials namely reinforced concrete, adhesive and UHPFRC.

Fig. 1
figure 1

Simply supported beam strengthened with a composite strip

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Fig. 2
figure 2

Forces in an infinitesimal element of a strengthened beam

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The basic assumptions made in the formulation are:

  • All materials behave in a linear elastic manner.

  • The adhesive is responsible only for transferring stresses from one adherend to other.

  • The stresses in the adhesive layer remain constant throughout its thickness.

Following the above assumptions, the strains at the top and bottom faces of the adhesive can be expressed as follows:

$$ \varepsilon_{1} \left( x \right) = \frac{{{\text{d}}u_{1} \left( x \right)}}{{{\text{d}}x}} = \varepsilon_{1}^{M} \left( x \right) + \varepsilon_{1}^{N} \left( x \right) $$
(1)
$$ \varepsilon_{2} \left( x \right) = \frac{{{\text{d}}u_{2} \left( x \right)}}{{{\text{d}}x}} = \varepsilon_{2}^{M} \left( x \right) + \varepsilon_{2}^{N} \left( x \right) $$
(2)

where \( u_{1} \left( x \right) \) and \( u_{2} \left( x \right) \) are the longitudinal displacements at the base of concrete and top of UHPFRC layer, respectively. \( \varepsilon_{1}^{M} \left( x \right)\;{\text{and}}\;\varepsilon_{2}^{M} \left( x \right)\;{\text{and}}\; \varepsilon_{1}^{N} \left( x \right)\;{\text{and}}\;\varepsilon_{2}^{N} \left( x \right) \) are the respective strains induced due to bending moments and the longitudinal forces at the concrete and UHPFRC layer, respectively, which are given as follows:

$$ \varepsilon_{1}^{M} \left( x \right) = \frac{{M_{1} \left( x \right)y_{1} }}{{E_{1} I_{1} }}; \;\varepsilon_{2}^{M} \left( x \right) = \frac{{ - M_{2} \left( x \right)y_{2} }}{{E_{2} I_{2} }} $$
(3)
$$ \varepsilon_{1}^{N} \left( x \right) = \frac{{{\text{d}}u_{1}^{N} \left( x \right)}}{{{\text{d}}x}}; \;\varepsilon_{2}^{N} \left( x \right) = \frac{{{\text{d}}u_{2}^{N} \left( x \right)}}{{{\text{d}}x}} $$
(4)

where \( y_{1} \;{\text{and}}\;y_{2} \) are distances from the bottom of the concrete layer and the top of UHPFRC layer to their respective centroids and \( u_{1}^{N} \;{\text{and}}\;u_{2}^{N} \) are the longitudinal force-induced displacements of the respective adherends.

Complete details are available in Tounsi model [15].

Teng Model (2006)

The problem of the second type of IC debonding discussed earlier is addressed by making use of a closed-form formulation [16]. This formulation is originally developed to understand the behaviour of FRP-to-concrete interface between two adjacent cracks, but can be extended to problems involving other composite materials. It presents the behaviour of the interface where the composite strip is subjected to different tensile forces at the two ends while the concrete beam or prism is subjected to two different axial forces which may be either compressive or tensile at the two ends as shown in Fig. 3. A bilinear bond–slip model is assumed for the prediction of the debonding process in the model.

Fig. 3
figure 3

Idealized model of UHPFRC concrete bonded joint between two adjacent cracks

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The model is constructed with the notion of the strip as a plate and beam as a prism. The plate is subjected to two tensile forces \( P_{1} \;{\text{and}}\;P_{2} \) at the left and right end, respectively, whereas the concrete prism is subjected to two forces \( P_{3} \;{\text{and}}\;P_{4} \) which can be either compressive or tensile.

The basic assumptions considered in this formulation are as follows:

  1. (1)

    All the forces remain proportional to each other throughout the loading process.

  2. (2)

    The cross-sectional dimensions of the plate and prism remain constant along the length.

  3. (3)

    The adhesive layer is of constant thickness, and the whole model is in a plane stress state.

The adhesive layer is thus subjected to shear deformations while the adherents are subjected to axial deformations in the light of the above specified assumptions. So the bending deformation of the adherents is neglected, and the shear stress across the adhesive layer is assumed to be constant.

The equilibrium considerations will result in the following equations based on the above assumptions [20]:

$$ \frac{{{\text{d}}\sigma_{p} }}{{{\text{d}}x}} - \frac{\tau }{{\tau_{p} }} = 0 $$
(5)
$$ \sigma_{p} t_{p} \sigma_{p} + \sigma_{c} t_{c} \sigma_{c} = P = P_{1} - P_{3} = P_{2} - P_{4} $$
(6)

where \( \sigma_{p} \) is the axial stress in the plate, \( \sigma_{c} \) is the axial stress in the concrete prism, \( \tau \) is the shear stress in the adhesive layer, \( t_{p} \) is the thickness of the plate and \( t_{c} \) is the thickness of the concrete prism.

The interfacial shear stress can be expressed as a function of interfacial slip (constitutive equation for the adhesive) which is given as follows:

$$ \tau = f\left( \delta \right) $$
(7)

Similarly, the constitutive equations for both the adherents are given as follows:

$$ \sigma_{p} = E_{p} \frac{{{\text{d}}u_{p} }}{{{\text{d}}x}} $$
(8)
$$ \sigma_{c} = E_{c} \frac{{{\text{d}}u_{c} }}{{{\text{d}}x}} $$
(9)

The interfacial slip \( \delta \) in Eq. (7) is the relative displacement between the two adherends

$$ \delta = u_{p} - u_{c} $$
(10)

After substituting equations Eqs. (6)–(10) in Eq. (5), we get

$$ \frac{{{\text{d}}^{2} \delta }}{{{\text{d}}x^{2} }} - f\left( \delta \right)\left( {\frac{ 1}{{E_{p} t_{p} }} + \frac{{b_{p} }}{{b_{c} E_{c} t_{c} }}} \right) = 0 $$
(11)

and

$$ \sigma_{p} = \frac{{\left( {\frac{{{\text{d}}\delta }}{{{\text{d}}x}} + \frac{P}{{b_{c} E_{c} t_{c} }}} \right)}}{{t_{p} \left( {\frac{ 1}{{E_{p} t_{p} }} + \frac{{b_{p} }}{{b_{c} E_{c} t_{c} }}} \right)}} $$
(12)

where \( b_{p} \) and \( b_{c} \) are the widths of the plate and the concrete prism, respectively.

Introducing the fracture parameters of local bond strength \( \tau_{f} \) and interfacial fracture energy \( G_{f} \) in Eqs. (11) and (12), we get

$$ \frac{{{\text{d}}^{2} \delta }}{{{\text{d}}x^{2} }} - \frac{{2G_{f} }}{{\tau_{f}^{2} }}\lambda^{2} f\left( \delta \right) = 0 $$
(13)
$$ \sigma_{p} = \frac{{\tau_{f}^{2} }}{{2G_{f} t_{p} \lambda^{2} }}\left( {\frac{ 1}{{E_{p} t_{p} }} + \frac{{b_{p} }}{{b_{c} E_{c} t_{c} }}} \right) $$
(14)

where

$$ \lambda^{2} = \frac{{\tau_{f}^{2} }}{{2G_{f} }}\left( {\frac{ 1}{{E_{p} t_{p} }} + \frac{{b_{p} }}{{b_{c} E_{c} t_{c} }}} \right) $$
(15)

Details related to complete formulation can be found in Teng model [13].

Diab Model (2009)

The evolution of the debonding of FRP strips bonded to the reinforced concrete beams under fatigue loading was studied [17]. The fatigue performance of the FRP–concrete interface is investigated through an analytical procedure by considering the effect of cycles in the bond–slip behaviour. This procedure can also be extended to other composite materials.

The formulation used in this work is an extension of the above-discussed debonding model which can be used for the case of fatigue loading. Similar to the previously discussed models, the adhesive layer in this model is assumed to be subjected to pure shear and the adherents are subjected to pure axial deformation. The thickness and width of the layer are \( t_{1} \;{\text{and}}\; b_{1} \) for UHPFRC strip and \( t_{2} \;{\text{and}}\; b_{2} \) for the concrete beam, respectively. The bonded length of the strip is taken as L.

The governing differential equations for the slip are governed by the following equations which are same as those of the previous formulation:

$$ \frac{{{\text{d}}^{2} \delta }}{{{\text{d}}x^{2} }} - \frac{{2G_{c} }}{{\tau_{f}^{2} }}\lambda^{2} f\left( \delta \right) = 0 $$
(16)
$$ \sigma_{1} = \frac{{\tau_{f}^{2} }}{{2G_{c} t_{1} \lambda^{2} }}\frac{{{\text{d}}\delta }}{{{\text{d}}x}} $$
(17)
$$ \lambda^{2} = \frac{{\tau_{f}^{2} }}{{2G_{c} }}\left( {\frac{ 1}{{E_{1} t_{1} }} + \frac{{b_{1} }}{{b_{2} E_{2} t_{2} }}} \right) $$
(18)

where \( \sigma_{1} \) is the normal stress at the strip, E1 and E2 are the Young’s moduli of the strip and beam, respectively, and \( G_{c} \) is the interfacial fracture energy.

As discussed earlier, a bilinear model as shown in Fig. 4 is used for the representation of bond–slip behaviour which has been modified to accommodate the degradation of stiffness and strength of the interface while the fracture energy is kept constant.

Fig. 4
figure 4

Cycle-dependent bilinear slip model

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The cycle dependence is inducted into the model by introducing creep-fatigue coefficient \( \phi \) obtained from the experiments to represent the degradation of the interfacial stiffness [17]. The cycle-dependent bilinear model is given below:

$$ f\left( x \right) = \left\{ {\begin{array}{*{20}c} {\frac{{\tau_{ft} }}{{\delta_{1} }}\delta ,} & {0 \le \delta \le \delta_{1} } \\ {\frac{{\tau_{ft} }}{{(\delta_{ft} - \delta_{1} )}}\left( {\delta_{ft} - \delta } \right),} & {\delta_{1} \le \delta \le \delta_{ft} } \\ \end{array} } \right\} $$
(19)

where

$$ \tau_{ft} = \frac{{\tau_{f0} }}{1 + \phi .\alpha } $$
(20)
$$ \phi = \left\{ {\begin{array}{*{20}c} {0.046t^{0.4} , } & { t \le 2000} \\ {1.2\left( {1 - \exp \left( { - t/1200} \right)} \right),} & {t > 2000} \\ \end{array} } \right\} $$
(21)

where \( \tau_{f0} ,\tau_{ft} \) are cycle-dependent bond strengths at time zero and time \( t \) in min, \( \delta_{ft} = \tau_{f0} \delta_{t0} /\tau_{ft} \) is the cycle-dependent slip and \( \alpha \) is the frequency-dependent factor.

Substituting the above bilinear model in Eq. (16), the following equations are obtained.

$$ \frac{{{\text{d}}^{2} \delta }}{{{\text{d}}x^{2} }} - \lambda_{1}^{2} \delta = 0\quad {\text{for}}\;0 \le \delta \le \delta_{1} $$
(22)
$$ \frac{{{\text{d}}^{2} \delta }}{{{\text{d}}x^{2} }} + \lambda_{2}^{2} \delta = \lambda_{2}^{2} \delta_{ft} \quad {\text{for}}\;\delta_{1} \le \delta \le \delta_{ft} $$
(23)

where

$$ \lambda_{1}^{2} = \frac{{\tau_{ft} }}{{2G_{f} }}\left( {\frac{ 1}{{E_{p} t_{p} }} + \frac{{b_{p} }}{{b_{c} E_{c} t_{c} }}} \right) $$
(24)
$$ \lambda_{2}^{2} = \frac{{\tau_{ft} }}{{\delta_{ft} - \delta_{1} }}\left( {\frac{ 1}{{E_{p} t_{p} }} + \frac{{b_{p} }}{{b_{c} E_{c} t_{c} }}} \right) $$
(25)

Using the appropriate boundary conditions [17], the solution of the stress state can be expressed as follows:

For \( 0 \le \delta \le \delta_{1} \)

$$ \delta = \delta_{1} \frac{{\cosh \left( {\lambda_{1} x} \right)}}{{\cosh \left[ {\lambda_{1} \left( {L - z} \right)} \right]}} $$
(26)
$$ \tau = \tau_{ft} \frac{{\cosh \left( {\lambda_{1} x} \right)}}{{\cosh \left[ {\lambda_{1} \left( {L - z} \right)} \right]}} $$
(27)
$$ \sigma_{1} = \frac{{\tau_{ft} }}{{t_{1} \lambda_{1} }}\frac{{\sinh \left( {\lambda_{1} x} \right)}}{{\cosh \left[ {\lambda_{1} \left( {L - z} \right)} \right]}} $$
(28)

For \( \delta_{1} \le \delta \le \delta_{ft} \)

$$ \delta = \left( {\delta_{ft} - \delta_{1} } \right)\left\{ {\frac{{\lambda_{2} }}{{\lambda_{1} }}\tanh \left[ {\lambda_{1} \left( {L - z} \right)} \right] \times \sin \left[ {\lambda_{2} \left( {x - L + z} \right)} \right] - \cos \left[ {\lambda_{2} \left( {x - L + z} \right)} \right] + \frac{{\delta_{ft} }}{{\left( {\delta_{ft} - \delta_{1} } \right)}}} \right\} $$
(29)
$$ \tau = - \tau_{ft} \left\{ {\frac{{\lambda_{2} }}{{\lambda_{1} }}\tanh \left[ {\lambda_{1} \left( {L - z} \right)} \right] \times \sin \left[ {\lambda_{2} \left( {x - L + z} \right)} \right] - \cos \left[ {\lambda_{2} \left( {x - L + z} \right)} \right]} \right\} $$
(30)
$$ \sigma_{1} = \frac{{\tau_{ft} }}{{\lambda_{2} t_{1} }}\left\{ {\frac{{\lambda_{2} }}{{\lambda_{1} }}\tanh \left[ {\lambda_{1} \left( {L - z} \right)} \right] \times \cos \left[ {\lambda_{2} \left( {x - L + z} \right)} \right] + \sin \left[ {\lambda_{2} \left( {x - L + z} \right)} \right]} \right\} $$
(31)

where \( x \) is the distance from the free end of the strip as shown in Fig. 5 which shows the shear stress distribution due to fatigue crack and \( z \) is the length of the softening zone which increases with the increase in load P before the initiation of the macro-debonding.

Fig. 5
figure 5

Shear stress distribution due to fatigue crack

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The softening length \( z \) and the load \( P \) can be related by the following equation:

$$ P = \frac{{\tau_{ft} b_{1} }}{{\lambda_{2} }}\left\{ {\frac{{\lambda_{2} }}{{\lambda_{1} }}\tanh \left[ {\lambda_{1} \left( {L - z} \right)} \right] \times \cos \left( {\lambda_{2} z} \right) + \sin \left( {\lambda_{2} z} \right)} \right\} $$
(32)

The maximum value of P can be obtained from the following equation:

$$ P_{{\hbox{max} ,\;{\text{capacity}}}} = \frac{{\tau_{ft} b_{1} }}{{\lambda_{2} }}\left\{ {\frac{{\delta_{1} }}{{(\delta_{ft} - \delta_{1} )}}\tan^{2} h\left[ {\lambda_{1} \left( {L - z_{m} } \right)} \right] + 1} \right\}\sin \left( {\lambda_{2} z_{m} } \right) $$
(33)

where \( z_{m} \) is the maximum length of the softening zone for a given bond–slip curve and can be determined from the following equation through iterative process:

$$ \tanh \left[ {\lambda_{1} \left( {L - z_{m} } \right)} \right] = \frac{{\lambda_{2} }}{{\lambda_{1} }}\tanh \left[ {\lambda_{2} z_{m} } \right] $$
(34)

The fracture mechanics-based approach for remaining life prediction assumes that the cycle-fatigue life of the bonded length is dominated by debonding propagation and debonding growth, rather than the initiation of such cracks. Furthermore, ultimate failure occurs when the remaining bonded length of the FRP–concrete interface can no longer sustain the maximum applied load. It is conventional in fracture mechanics to relate rate of debonding per cycle \( {\text{d}}a/{\text{d}}N \) to the maximum strain energy release rate \( G_{\hbox{max} } \) which can be calculated using the following formula:

$$ G_{\hbox{max} } = \frac{{P_{\hbox{max} }^{2} }}{{2b_{1}^{2} E_{1} t_{1} }} $$
(35)

Since the maximum strain energy release rate depends upon the parameters such as \( E_{1} \;{\text{and}}\;t_{1} \) of the strip, the debonding rate can be expressed in relation to the ratio \( G_{\hbox{max} } /G_{c} \) where \( G_{c} \) is the quasi-static fracture energy.

$$ \frac{{{\text{d}}a}}{{{\text{d}}N}} = m\left( {\frac{{G_{\hbox{max} } }}{{G_{c} }}} \right)^{n} $$
(36)

where \( m \) and \( n \) are material constants.

Experimental results in the literature have shown that the debonding propagation rate decreases significantly with increasing debonding length, a [17]. To consider the effect of the bonding length to the bonding growth, the energy ratio is expressed in terms of the applied load ratio which is given below.

$$ \frac{{G_{\hbox{max} } }}{{G_{c} }} = \frac{{P_{\hbox{max} }^{2} }}{{P_{{\hbox{max} ,\;{\text{capacity}}}}^{2} }} = \frac{{P_{\hbox{max} }^{2} }}{{\left[ {\left[ {\frac{{\tau_{ft} b_{1} }}{{\lambda_{2} }}\left\{ {\frac{{\delta_{1} }}{{(\delta_{ft} - \delta_{1} )}}\tan^{2} h\left[ {\lambda_{1} \left( {L - z_{m} } \right)} \right] + 1} \right\}\sin \left( {\lambda_{2} z_{m} } \right) } \right]} \right]^{2} }} $$
(37)

It is to be noted that the above equation is not valid for larger bond lengths. Also this model can be construed as a simulation of a concrete prism bonded with strip of smaller bond length subjected to pull-out tests. This model is employed in the present study to predict the behaviour of damaged RC beam strengthened with UHPFRC strip under fatigue loading.

In addition to expressing the energy ratio in terms of applied load ratio, an additional constant called the fatigue debonding growth coefficient \( \beta \) is

$$ \frac{{{\text{d}}a}}{{{\text{d}}N}} = m_{1} \left( {\frac{{G_{\hbox{max} } }}{{G_{c} }}} \right)^{{n_{1} }} . \beta $$
(38)

From the above equation, the debonded length can be obtained with respect to the number of cycles until the occurrence of total debonding or failure of the specimen.

Results and Discussion

Experimental investigations have already been carried out by the authors for the beams made of normal strength concrete (NSC) strengthened with repair concrete (UHPFRC). Details regarding the mix details and mechanical properties of both NSC and UHPFRC could be referred from the previous work as shown in Table 1 [21, 22]. Tests were conducted for the cases of both static and fatigue loadings. The minimum shear and normal strengths of the UHPFRC mix determined from splitting prism and slant shear tests, respectively, are reported as 11.05 MPa and 3.21 MPa [23]. No cases of debonding were observed during the experimental investigations on strengthened RC beams after failure as shown in Fig. 6. These experimental investigations have been taken as a datum to validate the analytical models studied in the present work. Parametric studies have been carried out for the load values of 60 kN and 80 kN representing the states of the beam post-yielding and prior to failure, respectively. The loads values 60kN and 80kN were based on the earlier experimental studies conducted by the authors [24]. This section discusses the effect of length and thickness of UHPFRC strip on its interfacial shear and normal stresses calculated using Tounsi model [15]. Studies have also been carried out comparing the shear and normal strengths of UHPFRC with those of the conventional retrofit materials such as GFRP and CFRP. This section also discusses the possibility of debonding in UHPFRC-strengthened beams due to the presence of intermediate cracks based on Teng model [16]. The same is extended to the case of fatigue loading as well based on the Diab model [17].

Table 1 Materials, mix proportion and mechanical properties
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Fig. 6
figure 6

Typical bonding of UHPFRC on NSC beams after failure

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Figures 7 and 8 present the variation of interfacial shear and normal stresses with the distance from the strip end for the retrofit materials UHPFRC, GFRP and CFRP. In this analysis, the Young’s moduli of CFRP and GFRP are taken as 140 GPa and 52 GPa, respectively, and shear modulus as 5000 MPa for both the materials [4, 15]. It can be seen from Figs. 7 and 8 that UHPFRC is found to develop lower values of shear (0.55 MPa) and normal (0.21 MPa) stresses at the interface for a given length and thickness of the strip. It is apparent from these figures that the maximum stresses occur at the free edge of the strip making it a vulnerable location for debonding. But the values of stresses developed in UHPFRC are well within their permissible limits, and hence, the chances of debonding are less as compared to other retrofit materials.

Fig. 7
figure 7

Effect of strip material on interfacial shear stress at aP = 60 kN. bP = 80 kN

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Fig. 8
figure 8

Effect of strip material on interfacial normal stresses at aP = 60 kN. bP = 80 kN

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The effect of adhesive thickness (ta = 2, 3 and 5 mm, respectively) on the interfacial shear stresses is shown in Figs. 9 and 10. Both the interfacial stress distributions are hardly affected by the thickness of the adhesive layer except for the change in their concentration levels. The effect of the strip thickness (ts = 5, 8 and 10 mm, respectively) on the interfacial shear stresses is shown in Figs. 11 and 12. The shear stresses are increased, and normal stresses are decreased as a result of increase in the strip thickness. An optimum value of thickness is to be chosen for the strip such that the interfacial stresses are kept well within their limits without compromising on the strength it imparts to the beam to be retrofitted.

Fig. 9
figure 9

Effect of adhesive layer thickness on interfacial shear stresses at aP = 60 kN. bP = 80 kN

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Fig. 10
figure 10

Effect of adhesive layer thickness on interfacial normal stresses at aP = 60 kN. bP = 80 kN

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Fig. 11
figure 11

Effect of strip thickness on interfacial shear stresses at aP = 60 kN. bP = 80 kN

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Fig. 12
figure 12

Effect of strip thickness on interfacial normal stresses at aP = 60 kN. bP = 80 kN

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Figures 13 and 14 show the effect of length of the strips (Ls = 800, 1000 and 1100 mm, respectively) on the interfacial shear and normal stresses, respectively. The magnitudes of the interfacial stress decrease linearly with the increase in the length of the strips. A bond length equal to the effective length of the beam has been adopted in the experimental investigations for which the interfacial stresses are within the limits of the permissible values. In agreement with the model, for the adopted bond length, no cases of debonding were observed during the experimental investigations.

Fig. 13
figure 13

Effect of length of strip on the interfacial shear stresses

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Fig. 14
figure 14

Effect of length of strip on the interfacial normal stresses

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From all the above parametric studies, it was inferred that the likelihood of debonding initiating at the end of the strip is less for the beams subjected to the load under service conditions.

In the present work, the problem of intermediate crack-induced debonding is restricted to the constant bending moment zone based on Teng model [16].

The values of the parameters adopted in this study are as follows:

$$ \beta = \eta = 1,\;\delta_{f} = 0.34\,{\text{mm}},\delta_{1} = 0.062\,{\text{mm}}, \tau_{f} = 11.05\,{\text{MPa}}, G_{f} = 12.03\,N/{\text{mm}}. $$

A typical bond length of 100 mm representing a typical crack spacing in reinforced concrete beams along with other bond lengths of 10 mm and 20 mm representing closed crack spacing in UHPFRC has been adopted in the present study [25]. Figure 15 shows the interfacial load–slip behaviour of the strip for various bond lengths.

Fig. 15
figure 15

Interfacial load–displacement curves for various bond lengths

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It can be seen that since \( \beta = \eta = 1 \), for all three different bond lengths, the interface experiences the same sequence of interfacial shear stress distribution. But the corresponding load–displacement curves are very different in shape. Because both the geometry and the loading are symmetrical, the interfacial shear stress distribution is anti-symmetrical about \( x = L/2 \). Debonding starts when \( \tau \) reaches zero at both the ends and then propagates from both the ends towards the middle of the strip which is now in a debonding–softening–elastic–softening–debonding state. As both peak shear stresses \( \tau_{f} \) continue to move towards \( x = L/2 \) from both the sides, the load keeps increasing and the ultimate load becomes infinite if material failure of the strip is not considered. Thus, if the upper bound is taken as the material failure of the plate, it can be obviously inferred at the load corresponding to the material failure, debonding will not occur theoretically.

In Fig. 15, the green scatter points represent the elastic stage and red scatter points represent the softening stage. It can be observed from the above figure that the bonding lengths of 10 mm and 20 mm which depict the crack spacing in UHPFRC-strengthened beams take substantial loads to attain the limit of elastic and softening states. But in real conditions, the stresses in the plate at two adjacent cracks in an RC beam are always different; however, this difference may be small. So a value of \( \beta < 1 \) should be taken and the behaviour should be studied again in the future studies. As of now, the assumption taken for this study is reasonable as no debonding failure was observed for the specimens in between the adjacent cracks of constant bending moment and in fact anywhere else in the beam.

With this knowledge of the debonding behaviour of the strengthened beams under static conditions, the scope of problem has been extended to study the debonding behaviour of the strengthened beams under fatigue loading using the Diab model [17].

The values of non-dimensional parameters used in this model are as follows:

\( m_{1} = 1.2,\;n_{1} = 8.673 \), and the values of maximum bond slips are those taken for the previous model.

Figure 16 shows the evolution of the slip with number of cycles for a given loading condition. The maximum slip obtained from this model (3.21 mm) is normalized with respect to the other slip values. It can be observed from the above figure that the rate of debonding is high during the initial number of cycles which is followed by the stable propagation of the debonding. It can be clearly seen from Fig. 16 that the rapid propagation stage is missing. This can be viewed as the failure of the specimen without the debonding of the strips as the debonding slip increasing at a stable rate has not grown rapidly in the last few cycles. This can be attributed to the adequate provision of bonding length of the strips.

Fig. 16
figure 16

Normalized slip versus number of cycles

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From the results of the debonding models under both static and fatigue loadings discussed above, it can be inferred that the chances of debonding at the interface of beam–strip is relatively very less in comparison with other materials. Hence, UHPFRC-based strengthening scheme is efficient in making the beam to realize its original strength without any premature failure due to debonding.

Conclusion

Existing analytical models on the debonding behaviour of concrete beams strengthened with composite strips are studied which can be applied to UHPFRC strip-strengthened beams.

  1. 1.

    The bond–slip behaviour of the strips bonded to the concrete substrate at the strip and region between the adjacent cracks in the constant bending moment zone is studied through comprehensive review of the existing models on debonding behaviour.

  2. 2.

    The debonding behaviour of the strips due to the interfacial stresses at the strip end and between the adjacent cracks in the flexure region is studied through debonding models.

  3. 3.

    Debonding of the strips was not observed during the experimental investigations, and the same has been predicted in fair agreement by the analytical models for debonding.