Analytical models to predict structural behaviour of reinforced concrete beams bonded with prestressed fibre-reinforced polymer laminates

Abstract

The strengthening of reinforced concrete members with prestressed fibre-reinforced polymer laminates has been investigated by researchers due to major improvements in member serviceability characteristics. Currently, analytical models generally employ mostly empirical procedures in predicting member behaviour, and as a result, the analytical results exhibit poor correlation to experimental investigations. In this article, an analytical model is developed using new and existing theoretical techniques to critically analyse strengthened reinforced concrete beams for a range of loading scenarios to generate moment–rotation and load–deflection relationships. The prestress level and the intermediate crack debonding strain of the prestressed fibre-reinforced polymer laminate with the inclusion of mechanical end anchorage were highlighted as key parameters within the model. The proposed model adopts closed-form solutions to allow for a wide range of beams with varying steel and fibre-reinforced polymer reinforcement ratios and dimensions. The model incorporates calibrated crack spacing theory to predict the crack width and spacing as well as the length of the cracked region in the beam. The models have good correlation with collected experimental data and thus can be used for the analysis of reinforced concrete beams strengthened with prestressed fibre-reinforced polymer, throughout all stages of loading from serviceability to failure.

Keywords

bond–slip fibre-reinforced polymer plate debonding moment–rotation prestressed reinforced concrete beam shear-friction

Introduction

Due to the ongoing need for reinforced concrete (RC) structures to maintain their serviceability requirements, retrofitting is commonly used as an alternative to replacing structural members. A common technique is the use of externally bonded (EB) or near surface mounted (NSM) fibre-reinforced polymer (FRP) laminates. The bonding of FRP laminates to the tension face of a concrete member results in an increase in flexural strength and reduction in deflections and crack widths. However, these plated beams are prone to fail prematurely due to various forms of debonding due to which the strength of FRP laminate is not fully exploited. A common method used to overcome this issue is the use of end-anchored prestressed FRP laminates (El-Hacha et al., 2004). The application of a prestressed laminate alters the beams structural characteristics throughout loading, thus increasing the serviceability behaviour of the member. Furthermore, it substantially increases the member’s ultimate load while delaying the onset of the concrete cracking and yielding of the internal steel reinforcement. There are many experimental investigations reported by various researchers with a range of test variables, namely a variation of concrete strength between 20 and 53 MPa, with steel reinforcement ratios between 0.8% and 2.8% of beams strengthened with 1–5 prestress laminates between 0.2 and 1.4 mm thick and 20–300 mm wide, with tensile a strength between 1500 and 3800 MPa and prestressing force between 5% and 60% of ultimate strength (Correia et al., 2015; El-Hacha et al., 2004; Garcez et al., 2008; Hong and Park, 2013; Kim et al., 2008; Motavalli et al., 2011; Mukherjee and Rai, 2008; Nordin and Taljsten, 2006; Sena-Cruz et al., 2015; Wight et al., 2001; Xue et al., 2009, 2010; Yang et al., 2008; Yu et al., 2008). Despite the availability of vast amount of experimental investigations, current analytical techniques generally employ procedures that are reliant on empirically derived components, notably in the post-cracking range. It is worth mentioning that in the analytical procedures presented in this article, it is assumed that the mechanical end anchorages will not fail prematurely before the desired flexural is achieved; however, there are a variety of anchorages used in practice for prestressing with FRP laminates and their details and performance are described by Michels et al. (2016). Moreover, there are several analytical and numerical models such as reported by Bensaid et al. (2015) and Rabahi et al. (2016) that focus only on some aspects of the strengthened beam. It is standard practice to quantify the behaviour of a prestressed concrete member (PC) member in terms of flexural rigidity which is estimated by performing a strain-based moment–curvature (M/χ) analysis. This moment–curvature analysis is based around the assumption of full interaction (FI) between concrete–rebar and concrete–FRP plate interfaces. Being strain based and applied at a discrete section in a two-dimensional section, the FI M/χ analysis cannot directly incorporate slip between the reinforcement and adjacent concrete due to crack widening, and therefore cannot directly quantify the laminate’s contribution. The M/χ analysis therefore requires semi-empirical values post-cracking stage which restricts the application of the approach to within the bounds of which the empirical values are derived. Therefore, the aim of this research article is to develop a rational structural mechanics–based model that can accurately categorise and quantify the structural properties corresponding to various failure mechanisms of a prestressed FRP laminated RC member. A full parametric study is undertaken using the model to determine the optimum set of beam parameters.

Research outline

In order to formulate an analytical model which can accurately analyse the flexural behaviour of an RC beam strengthened with prestressed anchored EB or NSM FRP laminates, a combination of partial interaction (PI) and FI theory is utilised. Prior to the occurrence of cracks, the FI analysis is mechanically accurate and thus can be used. In the post-cracking stage, a slip between the reinforcement and concrete exists due to crack widening resulting in the need for a PI analysis to accurately predict the flexural capacity and structural characteristics. In this research, a generic structural mechanics–based method is developed for determining the behaviour of RC members strengthened with EB prestressed FRP laminates. This method attempts to simulate the initial application of prestress force, formation of cracks and the associated discrete rotation and the formation of concrete wedges associated with softening due to crushing of concrete in flexural compressive region. The slip between the reinforcement and concrete is accounted by adopting a PI analysis whereas the standard shear-friction theory (Mohr–Coloumbs theory) is used to directly simulate concrete wedge formation due to onset of concrete crushing. A complete bending moment–roation (M/θ) relationship is established and effectively converted to a member load–deflection response through using standard methods. The models are finally validated and calibrated by comparing with a database compiled from published beam test results. Finally, a parametric study is also carried out to identify the variables that have major influence on the optimal structural performance of RC beams strengthened with prestressed FRP laminates.

Model development

Pre-cracking

The first stage of analysis of the prestressed RC beam is the pre-cracking analysis using a FI moment–curvature (M/χ) approach. This procedure yields reliable results up to the cracking moment (M_cr) of the member; in the post-cracking stage, it becomes reliant on empirical values. The pre-cracking analysis of the beam can be broken down into three specific stages, as illustrated in Figure 1. Before the beam section reaches its cracking capacity, the model applies the FI approach. The cracking moment is the upper limit for this stage of the analysis and is found through using a transformed section analysis. There are three cases for the FI analysis up until cracking as shown in Figure 1. Initially, the act of prestressing the FRP laminate induces a tensile strain in the top fibre of the RC beam (Figure 1(a)). The second case shown in Figure 1(b) is when the strain at the bottom fibre of the beam is equal to zero. This point is known as the decompression moment as any additional load acting on the beam causes the bottom of the beam to move into tension. Further application of a moment increases the bottom fibre tensile strain until cracking occurs as shown in Figure 1(c).

Figure 1.

(a) Initial strain profile due to prestressing (b) strain profile at the decompression moment (c) strain profile at cracking.

The force in the concrete can be determined for any given strain in the topmost compressive fibre, at any stage throughout loading using a standard compressive stress–strain relationship for concrete such as proposed by Hognestad (1951). The force in the steel reinforcement and prestressed FRP laminate can be determined based on the linear strain profile over the section, assuming a linear elastic stress–strain relationship. An iterative process is adopted such that the top fibre strain is incrementally varied to achieve the force equilibrium ( $\sum F = 0$ ) at which point the corresponding curvature and moment are determined; this generates the section’s moment -curvature response until cracking. Once cracking occurs and due to crack widening, a slip between the reinforcement and concrete exists resulting in strain differentials and the requirement of a PI analysis to determine the reinforcement (internal reinforcing bars or external FRP) behaviour. The PI is used to determine the reinforcement load–slip relationship and is dependent on the bond–slip characteristics of the reinforcement/concrete interfaces.

Bond–slip characteristics

The local bond stress-slip characteristics influence the global load-slip relationship between the concrete to rebar and concrete to FRP interfaces. As this research is focused on both the strength and serviceability characteristics of an anchored prestressed EB or NSM FRP laminated RC beam, a bi-linear bond characteristic is adopted (Figure 2); the initial ascending line is characterised by a peak bond stress of τ_max at a corresponding local slip of Δ₁ and the descending branch is characterised by complete loss of bond (τ = 0) at a corresponding local slip value of Δ_max.

Figure 2.

Idealised local slip (Δ)–bond stress (τ) characteristics.

Closed-form solutions for the bi-linear relationship are adopted to determine the maximum bond stress and deformation of both the FRP laminate and internal steel reinforcement. The closed-form solutions for the bi-linear relationship as given by Seracino et al. (2007) and Muhamad et al. (2011, 2012) can be used to determine the maximum bond stress (equation (1)) and deformation (equation (2)) of both the FRP laminate and internal reinforcement. The bi-linear model requires a Δ₁ value, which is taken as 1.5 mm for steel reinforcement (Muhamad et al., 2011); however, for FRP laminates, it is not defined. Therefore, in this research, a value as specified by Knight and Callary (2010) will be adopted where Δ_1-FRP = 0.1 Δ_max

τ_{m a x - F R P} = (0.802 + 0.078 φ_{f}) {f'}_{c}^{0.6}

(1)

τ_{\max - rebar} = 1.25 \sqrt{f'_{c}}

(2)

Δ_{\max - FRP} = \frac{0.976 {φ_{f}}^{0.526}}{0.802 + 0.078 φ_{f}}; φ_{f} = \frac{d_{f}}{b_{f}}

(3)

Δ_{\max - rebar} = 15 mm

(4)

where d_f is the length of the failure plane perpendicular to the concrete (in the case of EB plate, it is the thickness of the plate and for NSM plate, it equals to the width of the plate); b_f is the length of the failure plane parallel to the concrete surface (in the case of EB plate, it is the width of the plate and for NSM plate, it equals to twice the depth of the plate) and $f'_{c}$ is the compressive strength of concrete. A crucial aspect of the moment-rotation approach is the force-slip (P-Δ) relationship for the reinforcement, as the crack width (twice the length of slip) and the corresponding force in the steel reinforcement for a given slip () can be determined by knowing the local slip-bond stress characteristics as depicted in Figure 2; the profile of debonding failure planes of rebars and FRP plates is also needed to estimate P-Δ values (Refer to Figure 3 where Lper is the perimeter of the failure plane). The slip at the concrete–reinforcement interface and hence the location and the widths of cracks are dependent on the bond–slip characteristics at the concrete–reinforcement interface, as discussed later.

Figure 3.

Intermediate crack debonding failure planes.

Post-cracking

Closed formed solutions for the force in the steel reinforcement for a given slip relative to the adjacent concrete based on a bi-linear bond-slip () characteristics, as illustrated in Figure 2, can be determined from the equations developed by Mohamed Ali et al. (2012) and Muhamad et al. (2011). Equation (5) determines the slip in the steel reinforcement when yielding occurs, while equations (6) to (9) determine the load in the internal tensile steel reinforcement for varying slips within the $τ / Δ$ relationship

Δ_{y} = \frac{f_{y} β_{0 - el} \coth (λ_{1 - el} L)}{λ_{1 - el}}

(5)

where

λ_{1 - el} = \sqrt{\frac{β_{2 - el} τ_{\max}}{Δ_{1}}}

β_{2 - el} = β_{1} β_{0 - el}

β_{0 - el} = (\frac{1}{E_{r - el}} + \frac{A_{r}}{E_{c} A_{c}})

β_{1} = \frac{L_{p}}{A_{r}}

For $0 < Δ_{rb} < {Δ_{1 -}}_{rb}$ and $Δ_{rb} < Δ_{y}$

P_{rb} = \frac{Δ_{rb} λ_{1 - el} A_{r}}{β_{0 - el} \coth (λ_{1 - el} L)}

(6)

For $0 < Δ_{rb} < {Δ_{1 -}}_{rb}$ and $Δ_{rb} > Δ_{y}$

P_{rb} = \frac{(Δ_{rb} - Δ_{y}) λ_{1 - sh} A_{r}}{β_{0 - sh} \coth (λ_{1 - sh} L)} + f_{y} A_{rb}

(7)

For $Δ_{rb} > {Δ_{1 -}}_{rb}$ and $Δ_{rb} < Δ_{y}$

P_{rb} = \sqrt{\frac{L_{p} A_{r} E_{r - el} τ_{\max}}{(Δ_{\max - rb} - Δ_{1})}} (\frac{(Δ_{rb} - Δ_{\max - rb}) \sqrt{1 - {k_{s}}^{2}} + Δ_{\max - rb} - Δ_{1}}{k_{s}})

(8)

For $Δ_{rb} > {Δ_{1 -}}_{rb}$ and $Δ_{rb} > Δ_{y}$

P_{rb} = \sqrt{\frac{L_{p} A_{r} E_{r - sh} τ_{\max}}{(Δ_{\max - rb} - Δ_{1})}} (\frac{(Δ_{rb} - Δ_{\max - rb}) \sqrt{1 - {k_{s}}^{2}} + Δ_{\max - rb} - Δ_{1}}{k_{s}}) + f_{y} A_{rb}

(9)

where

k_{S} = 0.97 \sqrt{\frac{Δ_{rb} - Δ_{1}}{Δ_{1}}}

L is the reinforcement length; $τ_{\max}$ is the peak bond stress as shown in Figure 2; L_p and A_rb are the perimeter and cross-sectional area of the reinforcement, respectively; $Δ_{\max}$ is the bond–slip capacity of the reinforcing; $Δ_{1}$ is the point at which the bond–slip relationship transitions from the ascending branch to the descending branch as illustrated in Figure 2 and $Δ_{rb}$ is the slip in the reinforcement for a given rotation as illustrated in Figure 4.

Figure 4.

Analytical stage between cracking and compression softening of concrete.

Closed-form PI equations (Muhamad et al., 2011) are modified in order to incorporate a residual prestress force in the FRP laminate for various stages throughout the τ/δ relationship. This additional force can be calculated through knowing the additional strain in the laminate due to prestressing (Xue et al., 2009). As previously done for steel reinforcement, the load–slip relationship for the prestressed FRP laminate is idealised through adoption of closed-form equations (equations (10) and (11)) based on specific slip criteria based on the bi-linear $τ / δ$ characteristic idealised in Figure 2.

For $Δ_{0} < Δ_{lam} < Δ_{1 - lam}$

P_{lam} = \frac{Δ_{lam} λ_{1 - el} A_{r}}{β_{0 - lam} \coth (λ_{1 - lam} L)} + A_{r} (E_{lam} ε_{pr})

(10)

For $Δ_{lam} ⩾ Δ_{1 - lam}$

P_{lam} = \sqrt{\frac{L_{p} A_{lam} E_{r - el} τ_{\max - lam}}{(Δ_{\max - lam} - {Δ_{1}}_{- lam})}} (\frac{(Δ_{lam} - Δ_{\max - lam}) \sqrt{1 - {k_{s}}^{2}} + Δ_{\max - lam} - {Δ_{1}}_{- lam}}{k_{s}}) + A_{lam} (E_{lam} ε_{pr})

(11)

where

λ_{1 - el} = \sqrt{\frac{β_{2 - l am} τ_{\max - lam}}{Δ_{1}}}

β_{2 - el} = β_{1} β_{0 - lam}

β_{0 - el} = (\frac{1}{E_{r - el}} + \frac{A_{lam}}{E_{c} A_{c}})

β_{1} = \frac{L_{p}}{A_{lam}}

k_{S} = 0.97 \sqrt{\frac{Δ_{lam} - Δ_{1 - lam}}{Δ_{1 - lam}}}

In Equations (10) and (11), is the peak bond stress as given by Equation (1); L_p and A are the perimeter and cross sectional area of the reinforcement, respectively; is slip at which the bond stress is zero (Equation (3)); is the initial strain in the laminate due to prestressing; is the point at which the bond slip relationship transitions from the ascending branch to the descending branch as indicated in Figure 2; is the slip in the FRP laminate and A_lam is the cross-sectional area of the laminate

The load–slip relationship for the prestressed FRP laminate illustrated in Figure 5 highlights the additional load that is gained through prestressing the laminate. This increase in load is dependent on the amount of prestressing; however, there is a limit to the amount of load which can be gained from the laminate, denoted by P_capacity where $σ_{frac}$ is the fracture stress of the laminate. Therefore, excessively prestressing of the laminate results in the maximum force P_max to rise above the material capacity resulting in laminate rupture prior to debonding. The load–slip relationship illustrated that once the slip reaches $Δ_{\max - lam}$ , the force remains constant at P_max such that P_IC is denoted by the first term in equation (12), such that the first component is equal to the IC debonding load denoted P_IC in Figure 5

P_{\max} = \sqrt{\frac{L_{p} A_{lam} E_{lam} τ_{\max - lam}}{{(Δ_{\max} - Δ_{1})}_{lam}}} (\frac{{(Δ_{\max} - Δ_{1})}_{lam}}{k_{s}}) + A_{lam} (E_{lam} ε_{pr})

(12)

Figure 5.

Force–slip relationship for an un-prestressed and prestressed FRP laminate.

Concrete compression zone

The rigid body analysis of an RC beam allows for the beam shown in Figure 6 to be split into three specific regions: the non-softened compression zone, softened region and the cracked/uncracked tension region. Within the non-softened compression region of depth d_asc in Figure 6, a concrete material stress–strain relationship as defined by Hognestad (1951) (as depicted in Figure 7 for the ascending branch) can be applied, where the peak stress is $f'_{c}$ corresponding to peak strain ε_pk as depicted in Figure 7. In order to model the member behaviour, two cases are considered. The first case is when there is no concrete softening due to the onset of concrete crushing and hence, no wedge forms. This occurs when the maximum compressive strain in the concrete (ε_c.max) is less than the peak strain (ε_pk) (Figure 6). As a result, it is assumed that a linear strain profile is applicable in the compression zone and hence, using a stress–strain relationship for the concrete under uni-axial loading, the forces in the concrete can be determined. Equation (13) yields the force in the compressive concrete for the case when the maximum strain in the extreme compressive concrete fibre (ε_c.max) is less than the peak strain (ε_pk) in Figure 7

F o r ε_{c . m a x} < ε_{p k} : P_{c c} = \frac{f_{c} b ε_{c . m a x}}{ε_{p k}} [1 - \frac{ε_{c . m a x}}{3 ε_{p k}}] [\frac{ε_{c . m a x} d_{a s c}}{(ε_{c . m a x} + ε_{r b})}]

(13)

here ε_rb is the strain in the internal tensile reinforcing bar and b is the width of the beam.

Figure 6.

Analytical stage beyond compression softening of concrete.

Figure 7.

Stress–strain relationship of concrete under uni-axial compression (Mohamed Ali et al., 2012).

The second case occurs when the compressive strain in the concrete exceeds the peak strain resulting in the formation of a softening region of depth (d_soft) and length (L_soft) (Figure 6); this is due to the formation of a wedge due to the progressive crushing of the concrete under compression once the maximum strain exceeds ε_pk. To determine the force in the softening region, a shear-friction analysis must be applied assuming the stress at softening (σ_soft) is known (Oehlers et al., 2008). The stress at softening is dependent on the lateral confinement (σ_lat) and the shear-friction material properties, m and c (Oehlers et al., 2009). The behaviour of the concrete wedges in Figures 6 and 10 can be derived from Mattock and Hawkins’ (1972) shear-friction theory. For a given depth of wedge d_soft in Figs. 6 and 10, the force resisted by the wedge is derived by using shear friction theory and it is given by the following equation.

P_{soft} = w_{b} d_{soft} [\frac{c + σ_{lat} \cos α (\sin α + m \cos α)}{\sin α (\cos α - m \sin α)}]

(14)

where w_b is the width of the wedge which is generally the width of the beam, σ_lat is the lateral confinement which can be induced by the stirrups or any additional confinement provided by external retrofitting, m and c are the shear-friction material properties and α is the angle of the weakest plane and it can be assumed as 27° as a conservative value. Here, m can be taken as 0.73 and c = 0.17f_c. Therefore, the compressive force in concrete P_cc has now two components: the first one is the compression in the uncrushed region (P_asc) and the second one is due to the softening wedge (P_soft) from equation (14). Equation (15) yields the force in the compressive concrete for the case when the maximum strain in the extreme compressive concrete fibre (ε_c.max) is less than the peak strain (ε_pk) in Figure 7

F o r ε_{c . m a x} ⩾ ε_{p k} : P_{c c =} P_{a s c} + P_{s o f t}; P_{a s c} = \frac{2}{3} f_{c} b d_{a s c}

(15)

Moment–rotation limits

A complete moment–rotation relationship for a FRP strengthened RC beam is produced using data from the pre-cracking, post-cracking and post-softening stages of analysis. Slight discrepancies observed in force lever arms result in kinks occurring when the relationship changes between stages. However, the effect of this was seen to be negligible when comparisons were made against experimental data. Throughout loading, there exist rotational limits that need to be considered within the newly formed model; these are due to concrete wedge sliding between the interface between the crushed and uncrushed regions of compressive concrete (Figure 8), reinforcement rupture and reinforcement debonding. To determine the slip that will result in debonding of the laminate, To determine the slip corresponding to the debonding of the laminate Equation (16) as derived by Haskett et al. (2009, 2011) can be applied.

Δ_{debonding} = δ_{\max} + \frac{ε_{δ \max} d}{2}

(16)

where $ε_{δ \max}$ is the strain in the FRP laminate at the point of debonding and d is the effective depth of the flexural member.

Figure 8.

Wedge formation and sliding failure in eccentrically loaded prism.

Once the formation of a concrete wedge occurs, the wedge displaces along a wedge sliding failure plane as shown in Figure 8. Significant deformation in the concrete wedge can lead to sliding failure as defined by Haskett et al. (2009) in equation (17) as s_slide

s_{slide} = 2.51 \frac{σ_{lat}}{f_{c}} + 0.42

(17)

where σ_lat is the lateral confinement provided by the stirrups.

While debonding of the laminate is highly likely as reported in the literature (Xue et al., 2009), it is worth noting that laminate rupture may also occur depending on the material and bond properties of the laminate. The slip to cause FRP rupture is denoted by Δ_rupture and it is deduced from equations (10) and (11)

Δ_{rupture} = \frac{\frac{[A_{p} σ_{frac} - A_{p} (E_{r - el} ε_{pr}]}{\sqrt{\frac{L_{p} A_{p} E_{r - el} τ_{\max}}{(δ_{\max} - δ_{1})}}} k_{s} - δ_{\max} + δ_{1}}{\sqrt{1 - {k_{s}}^{2}}}

(18)

Typically, FRP laminate debonding has three primary modes: critical diagonal crack (CDC) debonding, intermediate crack (IC) debonding and laminate end peeling. However, within this research, IC debonding will be considered the dominant mode of debonding as it affects flexural behaviour and overall ductility of the member. Furthermore, experimental investigations describe the possibility of the laminate debonding occurring up to the end anchorages at which point the FRP laminate acts as unbonded reinforcement. Ceroni et al. (2008) suggested that the addition of end anchorages to prestressed EB FRP laminates can increase the capacity of the section as well as the debonding capacity by up to 25%.

Crack spacing

A key aspect of this model is the accurate closed-form crack spacing equations incorporated into the PI analysis. To accurately determine the deflection of each beam, the spacing between each crack must be known. The rotation is then found for each ‘cracked region’, as shown in Figure 9. Equation (19) as proposed by Pecce and Ceroni (2004) accounts for both un-strengthened and FRP strengthened RC beams and hence can be used for a wide range of applications. The model allows for the imposition of multiple cracks on either side of the initial (central) crack in order to fully analyse the RC beam throughout all stages of loading. Therefore, both crack spacing and number of cracks should be calculated accurately.

S_{cr} = 1.58 \times (20 + 4 \frac{A_{c . eff}^{γ} ϕ}{A_{s}^{δ} + {(\frac{A_{f} E_{f}}{E_{s}})}^{β}})

(19)

where S_cr is the mean crack spacing; β is a characteristic design factor taken as 1.7; $ϕ$ is the reinforcement steel bar diameter; A_s is the area of tensile reinforcement steel; A_f is the area of FRP reinforcement; E_s and E_f are Young’s moduli of steel and FRP, respectively, and A_c.eff is the effective area of concrete in tension taken as the lesser of (2.5 × B × c) and (B × (H – NA)/3), where B, H, c and NA are the width, height, concrete cover and neutral axis of the cracked section, respectively. It was suggested by Pecce and Ceroni (2004) that the crack spacing be multiplied by a calibration factor of 1.58 after a comparison against experimental data producing the characteristic crack spacing, S_rm.th.k. This value was tested against experimental crack spacing data and the theoretical maximum crack spacing, S_r.max and was found to produce more accurate values compared to those predicted without the calibration factor, and hence, this was incorporated into the model. Previous models used for the analysis of FRP strengthened RC beams suggest that the crack spacing is to be calculated using equations proposed by Muhamad et al. (2013) (equation (19)). The Knight and Callary (2010) model suggests a crack spacing of 2S should be applied for FRP strengthened beams, where S is the crack spacing calculated using equation (19). Knight and Callary (2010) suggested that doubling the crack spacing found through Equation (19) is needed to incorporate the additional effects due to prestressing the FRP laminate. This relationship of 2S is supported through tests by Xue et al. (2009), which suggested that RC beams strengthened with prestressed FRP laminates experience fewer cracks. It was found however that this method provided high variation and an overall poor correlation for beams with multiple cracks in the experimental crack space data and hence was only used as a comparison in the final model

S = \frac{δ_{1}^{0.286}}{1.2 \times β_{2}^{0.285} τ_{m a x}^{0.715}} \times {(\frac{A_{r} E_{r} \times 0.000151 E_{c} - 1.04}{E_{c} \times (0.517 A_{r} E_{r} β_{2} + 0.22 L_{p}})}^{0.43}

(20)

where

β_{2} = β_{1} β_{0 - el}

β_{0 - el} = (\frac{1}{E_{r - el}} + \frac{A_{r}}{E_{c} A_{c}})

β_{1} = \frac{L_{p}}{A_{r}}

Figure 9.

Cracking spacing and rotation of each ‘cracked section’.

Prior to crushing, the concrete remains in the ascending region of its stress–strain relationship as shown in Figure 10. This region occurs prior to the maximum strain (ε_c.max) in the concrete reaching the peak concrete strain (ε_pk) and hence, crushing of the concrete is yet to occur. Once the strain in the concrete reaches and exceeds the concrete peak strain, concrete crushing occurs above the primary crack and a concrete softening wedge forms also shown in Figure 10. In this case, the compression force in concrete comprises of two components: the first one due to the ascending branch of stress -strain curve where P_asc of depth d_asc can be estimated as discussed before and the second component due to the softening of concrete P_soft with a corresponding depth d_soft associated with a wedge slip s_soft which is derived as follows. Complete M-θ relationships are formed through a combination of FI and PI theories. The deflections in the non-cracked region of the beam are determined through the integration of the FI moment–curvature relationship. At the onset of cracking, the rotation is determined through an iterative procedure which incorporates PI theory to accommodate slip in the reinforcements relative to the adjacent concrete. Finally, when the maximum strain in the concrete reaches the peak strain (ε_pk), the rotation is determined based on both the PI tension and compression regions.

Figure 10.

Parameters at the crack face of the FRP strengthened beam.

Unbonded FRP laminates and mechanical end anchorage

This article primarily focuses on the analytical models for fully bonded prestressed FRP-plated RC beams. However, FRP laminates used for the strengthening of RC members can be bonded or unbonded depending on the beams flexural and serviceability requirements. Unbonded systems using laminates or external prestressed tendons are commonly used for rehabilitation of RC beams due to their ease of installation and the ability to adjust the prestress force post-application in order to align with the structural needs of the member. In unprestressed and unanchored EB FRP systems, a common failure mechanism is laminate end peeling and premature laminate debonding; however, these are eliminated due to the presence and effects of end anchorage. A key focus of this research was to incorporate the presence and effects of mechanical end anchorage on the flexural capacity and structural behaviour of a prestressed RC beam. As this is the case, the mechanical end anchorage is assumed to be infinitely strong and hence, anchorage failure will not be observed at any stage. The presence of anchorage allows for the FRP laminate to act as an unbonded system in certain beams once debonding occurs. To determine the stress in the unbonded FRP laminates, a modification of the equation presented by Naaman and Alkhairi (1991) can be used as in equation (21)

f_{pn} = f_{pe} + Ω_{u} E_{pr} ε_{cu} (\frac{d_{p}}{c} - 1) (\frac{L_{1}}{L_{2}})

(21)

where $Ω_{u} = 3 / (L / d_{p})$ ); E_pr is the elastic modulus of the prestressing steel; L₁ is the sum of the length of the loaded spans containing the laminate being analysed; L₂ is the total length of the laminate between anchorages; ε_cu is the failure strain of the concrete; Ω_u is the bond reduction coefficient; d_p is the distance from the extreme compression fibres to the midpoint of the steel reinforcement and f_pn is the stress in the prestressing laminate at nominal bending resistance. As the laminate is not bonded to the concrete, a uniform strain over the length of the laminate between the mechanical anchors is observed. As a result, a member analysis method must be adopted to calculate the strain in the FRP laminate as opposed to a sectional analysis.

For the specimens strengthened with bonded FRP, the strains vary linearly over the beam depth and hence, the steel strain (ε_s) and the FRP strain (ε_f) can be calculated using equations (22) and (23) (El-Maaddawy and Soudki, 2008), respectively

ε_{s} = \frac{ε_{c} (d - c)}{c}

(22)

ε_{f} = \frac{ε_{c} (h - c)}{c}

(23)

where ε_c is the concrete compressive strain at top face of the beam, d is depth of the steel reinforcement, h is the height of the beam and c is depth of the neutral axis.

For mechanically anchored unbonded specimens, the strain in the external unbonded FRP (ε_f) does not vary linearly with concrete compressive strain. Consequently, a bond factor (ξ) is introduced to account for the strain incompatibility between the unbonded FRP laminate/tendons and the concrete surface, and hence, equation (24) can be rewritten as follows

ε_{f} = (ξ) \frac{ε_{c} (h - c)}{c}

(24)

The value of the bond factor is dependent on the number of anchors and whether or not the laminate is bonded and is inversely proportional to the number of anchors. Values for the bond factor based on FRP strain values measured experimentally by El-Maaddawy and Soudki (2008) can easily be incorporated into the proposed model, allowing for changes in anchorage/bond type.

Model comparison and validation

The presented mechanics approach has been used to determine the load–deflection responses of bonded prestressed FRP laminated RC members as tested by many researchers (El-Hacha et al., 2004; Garcez et al., 2008; Kim et al., 2008; Mukherjee and Rai, 2008; Nordin and Taljsten, 2006; Wight et al., 2001; Xue et al., 2009, 2010) while major milestones of members in the collected experimental tests by Knight and Callary (2010) are compared with predicted responses. It was found that the approach well-predicted the concrete cracking with a mean accuracy of 9%, while yielding of the internal steel reinforcement was predicted to be within 11% of the collected 33 test specimens. It should be noted that in this article, the IC debonding strain is the limiting failure mode; however, the laminate once deboned continues to act as unbonded but anchored reinforcement for the concrete.

A rectangular RC beam (203 mm wide and 279 mm deep). 16 mm steel bars in tension and 2 No. 10 mm steel bars in compression EB with a Carbon FRP (CFRP) laminate at prestressed 15% of the 3790 MPa ultimate tensile strength (from test database compiled by Knight and Callary, 2010) is now used to illustrate the accuracy of the presented model. It is reported that the member undergoes ultimate failure due to laminate rupture. As illustrated in Figure 11, it is observed that the theoretical approach predicts well the behaviour of the member throughout serviceability loading; however, a noticeable loss in accuracy occurs towards ultimate limit state. As discussed earlier, the limits of the prestress laminate are that of IC debonding or FRP rupture. In this case, however, in order to prevent laminate end peeling, the test specimens incorporate mechanical end anchorage such that the laminate effectively commences debonding prior to rupture occurring. The presented approach, however, determines ultimate failure based on a singular failure mode, namely IC debonding, rather than a combination as reported for member B28 (Knight and Callary, 2010). Despite a good general trend of the predicted response, this IC debonding failure mode results in observable discrepancies between the theoretical and experimental results. An increase in IC debonding strength due to end anchorages as suggested by Ceroni et al. (2008) can be seen in Figure 11 as the predicted trend which provide better correlation to the test results as IC debonding failure is postponed, allowing the member to obtain a greater load-carrying capacity and larger resulting mid-span deflection.

Figure 11.

Member comparison specimen B28 and cross-sectional dimensions.

Parametric study

The purpose of the parametric study is to apply the developed model to determine the optimal beam parameters to maximise the flexural characteristics and structural properties of a RC beam prestressed with an EB FRP laminate; it is worth noting that prestressed NSM plated beams were not considered in this parametric study. One parameter was varied while all others were held constant. The effects on the ductility and flexural capacity of the beam were then critically analysed to determine the most influential input parameters. The ductility of a beam is a measure of its ability to absorb energy and fail in a non-brittle manner. In this research, member ductility was defined as the ratio of deflection at ultimate to deflection at yield.

Laminate aspect ratio

The laminate aspect ratio, defined as the ratio of the laminates width to thickness, is a primary parameter to the overall structural properties of the strengthened RC beam. Both the laminate debonding strain and rupture strain are controlled by the FRP aspect ratio as it affects the primary bond characteristics. By increasing the width of the FRP, τ_max-FRP is reduced due to the inversely proportional relationship between the width of the FRP and Δ_max-FRP. Due to the change in aspect ratio, there was found to be little to no change in Δ_debonding (equation (16)); however, the slip at rupture decreased with the increase of the aspect ratio. Based on both ductility and flexural strength, it can be determined that the optimal aspect ratio is approximately 35 (Figure 12). Here, member ductility was defined as the ratio of deflection at ultimate to deflection at yield.

Figure 12.

Sectional ductility for variation in aspect ratios.

Number of FRP laminates

Depending on the requirements of the strengthened RC beam, the number of FRP laminates used may vary. This affects the total area of external laminate used and in turn the bond characteristics, force in steel, crack spacing and the debonding and rupture strains. Varying the number of sheets used is the most common method of changing the area of FRP laminate. A proportional relationship between the number of FRP laminate and sectional ductility was observed (Figure 13).

Figure 13.

Sectional ductility for variation in number of FRP laminates.

Similarly to the changes in ductility of the section, varying the number of FRP laminates has predictable affects on the strength and flexural capacity of the beam. A relatively uniform increase in load was observed for similar deflections at cracking, steel yield and debonding of the FRP laminate (Figure 14). The slip at which debonding and rupture of the FRP laminate occurs is unchanged due to the bond characteristics being unaffected for the laminates. The overall flexural capacity of the beam, however, is greatly increased.

Figure 14.

Load deflections of number of FRP laminates.

Laminate prestress level

The effective prestress level applied to the FRP laminate is another critical parameter affecting the structural characteristics and flexural properties of a strengthened RC beam. The prestress level of the FRP laminate directly affects the IC debonding load (P_IC) and hence can greatly impact on a strengthened RC beams performance under loading. The optimal value was found when there was no further increase in the ultimate load capacity for a respective increase in prestressing level. It is illustrated in Figure 15 that maximum sectional ductility is achieved when the FRP laminate is prestressed to approximately 30% of the ultimate tensile strength of the laminate.

Figure 15.

Sectional ductility for variation in prestress level.

Figure 16 shows relatively uniform increases in the flexural capacity of the section with increases in the laminate prestress level. The deflections at cracking, steel yielding and debonding of the FRP laminate remain comparatively constant for significant increases in the applied load. At rupture of the FRP, however, lower prestressing levels resulted in substantially higher deflections being observed for lower loads. This implies that higher prestress levels are desirable as they allow for a higher load-carrying capacity for lower deflections.

Figure 16.

Load deflections of prestress levels.

Debonding load

IC debonding is considered as the dominant debonding failure mode due to its affect on both the beams flexural strength and ductility. Previous analytical models tend to underestimate the IC debonding load at the ultimate limit state by 25% as the contribution of of mechanical end anchorage was ignored (Ceroni et al. 2008). Using equation (25), the IC debonding load was increased for a single beam with a prestressed EB FRP laminate in order to observe the change in the ultimate load

P_{IC} = \sqrt{\frac{L_{p} A_{r} E_{r - el} τ_{\max}}{(δ_{\max} - δ_{1})}} (\frac{δ_{\max} - δ_{1}}{k_{s}}) + A_{r} (E_{r - el} ε_{pr})

(25)

It can be seen that there is little increase in ultimate load between a P_IC factor of 1.26 and 1.3 (Figure 17). An increase of 26% in the IC debonding load is the optimal value to accurately predict the load at the ultimate limit state (Ceroni et al., 2008).

Figure 17.

Load deflections for debonding load factors.

Conclusion

The main objective of this research paper was to develop an analytical model that can be generically applied to reinforced concrete (RC) beams retrofitted with prestressed externally bonded (EB) or near surface mounted (NSM), FRP laminates to determine their flexural characteristics. The developed model incorporates new analytical methods allowing for a multiple crack analysis. The presence and effects of mechanical end anchorage have been considered, thus increasing the accuracy of the model when compared to current analytical techniques. The model was developed by combining a partial interaction based rigid body moment rotation analysis for the post-cracking behaviour with a full interaction analysis for pre-cracking behaviour.

The model can estimate a range of outputs including moment-curvature, moment-rotation and load-deflection relationships for strengthened beams; this assisted in comparing the analytical results with the test results of retrofitted RC members as reported in literature. Theoretical data produced using the newly developed model were compared to data from existing experimental investigations to highlight any model inaccuracies. Model calibration was then carried out over multiple stages of loading to ensure the maximum accuracy was achieved. Comparing the output of the calibrated model to that from experimental investigations proved the accuracy of the model in predicting the flexural behaviour of RC beams strengthened with prestressed EB or NSM FRP laminates. The results proved that analytical models developed in this study had strong correlation with experimental results and in particular they yielded more accurate values of deflections throughout all stages of loading. The incorporation of rigid body displacement (RBD) analysis and shear-friction theory, as developed at the University of Adelaide, allows for model output to be in the form of moment–rotation and load–deflection relationships. Various governing failure modes are considered, namely, FRP rupture and debonding, showing good agreement with experimental findings and therefore validating the approach. A parametric investigation was undertaken to study the member sectional behaviour and derive optimal sectional characteristics. Parameters investigated include the number of prestressed FRP laminates, laminate aspect ratio and prestress level.

The significant findings from this study include (a) the identification of the parameters that greatly affect the flexural behaviour of a prestressed FRP laminated beam and (b) the determination of the recommended FRP laminate prestress level such that sectional strength is increased, while suitable ductility is maintained. A key finding of the parametric study was the optimal debonding load factor of the FRP laminate. The optimal prestress level of the FRP laminate was found to be approximately 30% of the laminate’s ultimate tensile strength.

Footnotes

Acknowledgements

The paper is based on the honours research project report compiled by Knight et al. (2012) and by D. Knight and D. Callary (2010) under the supervision of the author at the University of Adelaide. The contributions and suggestions made by Dr Daniel Knight for the preparation of this article are gratefully acknowledged.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

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