Concrete cover separation of reinforced concrete beams strengthened with near-surface mounted method: Mechanics based design approach

Abstract

The primary mode of premature failure for near-surface mounted strengthened beams is the concrete cover separation. Due to its complexity, most of the prediction methods for concrete cover separation tend to be empirical based, which can limit their usage to specific near-surface mounted strengthening configurations. In response to that, this article presents a mechanics-based design which uses the moment-rotation approach and the global energy balance approach which is less reliant on empirical formulations, as the mechanics of reinforced concrete beam such as tension stiffening and propagation of concrete cover separation debonding crack are directly simulated rather than empirically derived. The proposed design procedure was validated against published experimental results of reinforced concrete beams strengthened with near-surface mounted carbon fibre–reinforced polymer bars, near-surface mounted carbon fibre–reinforced polymer strips or side-near-surface mounted carbon fibre–reinforced polymer bars and show good accuracy. As it is less reliant on empirical formulations, the proposed design procedure should be applicable to various near-surface mounted reinforcement configurations and materials.

Keywords

fibre-reinforced polymer fracture mechanics moment-rotation near-surface mounted partial interaction reinforced concrete

Introduction

Structural reinforced concrete (RC) members can require strengthening to compensate for deficiencies in either flexural or shear strength. The reasons for structural strengthening are varied; most structures require strengthening to compensate for strength loss due to ageing, while some structures were damaged in some ways that result in loss of strength. The focus of this article is the near-surface mounted (NSM) method (Badawi and Soudki, 2009; Bilotta et al., 2011; Capozucca and Bossoletti, 2014; Galati and De Lorenzis, 2009; Sharaky et al., 2017; Wu et al., 2014), which is a type of flexural strengthening for RC beams or slabs. The NSM method involves drilling grooves within the concrete cover and placing an NSM reinforcement within it, after which the groove is filled with epoxy adhesive. Currently, there are also several new derivative methods based on the NSM method, such as the side-NSM (SNSM) method (Sharaky et al., 2017; Shukri et al., 2016a) where the NSM reinforcement is placed at the sides of the beam to allow NSM strengthening to be applied on beams with small width or with beam soffits which are inaccessible and the partially bonded NSM (Choi et al., 2011; Seo et al., 2016) where the high moment area of the NSM is left unbonded to increase ductility.

The main problem of the NSM method is its vulnerability to concrete cover separation (CCS) failure. The CCS involves a debonding crack which appears at the location of NSM reinforcement curtailment, which then propagates towards the higher moment area of the beam. It is a type of premature mode of failure, which means that NSM-strengthened beams that failed by CCS will have failed well below the design strength. There has not been much research done on CCS failure on NSM-strengthened beams, likely due to the presence of a large number of parameters involved. Several methods to predict or simulate CCS have been proposed using the finite element method (Al-Mahmoud et al., 2010; Zhang and Teng, 2014) or using the concrete tooth model (De Lorenzis and Nanni, 2003). Recently, Teng et al. (2016) proposed a strength model for NSM carbon fibre–reinforced polymer (CFRP) strips derived using finite element study while an analytical design approach was proposed by Rezazadeh et al. (2016), which was derived using concrete fracture mechanic. Most of these methods can be highly empirical, such as in terms of predicting crack spacing. Empirical methods that are formulated around a specific shape or material type of NSM reinforcement are only accurate within the regime of testing used to formulate them, which can limit their usage.

In response to all these problems, this article proposes a mechanics-based approach to design, which can prevent CCS failure while being less reliant on empirical means. An example of this can be seen in the work of Shukri and Jumaat (2016), where the moment-rotation (M/θ) approach (Knight et al., 2014; Mo et al., 2016; Oehlers et al., 2012; Shukri et al., 2015, 2016b; Visintin et al., 2013; Visintin and Oehlers, 2016, 2017) was used in conjunction with the global energy balance approach (GEBA) (Achintha and Burgoyne, 2013, 2011; Guan and Burgoyne, 2014) to simulate CCS failure. The M/θ approach is a mechanics-based method that applies the partial interaction theory (Gupta and Maestrini, 1990; Haskett et al., 2008; Muhamad et al., 2012; Visintin et al., 2013) to directly simulate concrete cracking, crack widening, and tension stiffening of RC beams. The GEBA, on the other hand, applies the fracture mechanics of concrete to predict whether the CCS debonding crack will propagate and cause failure. The proposed method by Shukri and Jumaat (2016) shows that the use of these two mechanics-based methods allows the CCS failure of NSM-strengthened beams to be predicted with good accuracy and applicable to various types of NSM reinforcement material and shape due to lower reliance on empirical means. However, it was not made for general design purpose and as such is difficult to use. Hence, in this article, closed-form solutions for crack spacing and load–slip relationships will be used to formulate a simpler design procedure based on the M/θ-GEBA method, which is then validated against published experimental results.

Fundamental theories

In this section, the fundamental theories used in the M/θ approach and GEBA will be presented. The aim of this article is to introduce a design procedure, hence only a brief description of the M/θ approach and GEBA that is relevant to this article are given while references to the original research are provided.

M/θ approach

The M/θ approach used is a segmental simulation focused on the behaviour of cracked RC beam, as shown in Figure 1(a), which feature an NSM-strengthened RC beam section with cross-section as shown in Figure 1(b). The spacing between these primary cracks is designated S_cr, and due to symmetry of forces within the RC beam segment, it is possible to only consider half the crack spacing, L_def, for analysis purpose.

Figure 1.

RC beam: (a) NSM-strengthened beam segment and (b) cross-section of NSM-strengthened beam.

Slip occurs along L_def between the steel and NSM reinforcements and adjacent concrete, causing a cumulative slip of δ_r and δ_f at the crack face, respectively; this occurrence can be idealized as prism with a reinforcement and its adjacent concrete as shown in Figure 2, where slip of steel reinforcement (δ_r) is shown as example. The size of the prisms is as shown in Figure 1(b), where c_r refers to the distance from the centroid of the steel reinforcement to beam soffit and c_f refers to the distance from the centroid of the NSM reinforcement to the beam soffit.

Figure 2.

Partial interaction tension stiffening analysis of steel reinforcement.

The bond stress, τ, gradually transfers load P_r onto the adjacent concrete, causing both P_r and δ_r to reduce the further it gets from the crack face. Using the boundary condition of δ_r = 0 at length L_def, the stresses and strains within the prism can be solved to determine the force P_r which causes the slip δ_r, which allows the load–slip (P-δ) relationship to be obtained as shown in Figure 3. This procedure is called the partial interaction tension stiffening analysis (Haskett et al., 2008; Muhamad et al., 2011, 2012); the benefit of this analysis is the resulting load–slip relationship has directly accounted for the effects of tension stiffening through the use of bond stress–slip relationship, such that other empirical indirect methods, such as the commonly used Branson’s equation (Branson, 1968), are not needed.

Figure 3.

Load–slip relationship of steel or NSM reinforcement.

From Figure 3, it can be seen that there is an initial linear portion OA with stiffness K (Zhang et al., 2017), such that the P_r-δ_r and P_f-δ_r relationship can be written as P_r = K_rδ_r and P_f = K_fδ_f, respectively. This assumption should be correct for steel reinforcements prior to steel yielding and for NSM FRP reinforcements, which do not yield. Closed-form solutions based on this theory have been proposed (Visintin and Oehlers, 2016; Zhang et al., 2017) and will be used extensively in this article.

GEBA

The GEBA is a fracture mechanics-based method where the primary assumption is that CCS debonding cracks will always appear on strengthened RC beams, causing a debonded length L_d as shown in Figure 4. It is then only a matter of determining whether there is enough energy for the debonding crack to propagate and cause failure. When the GEBA is applied on beams strengthened with FRP sheet (Achintha and Burgoyne, 2011) or NSM (Shukri et al., 2018; Shukri and Jumaat, 2016), it is usually assumed that the debonding crack starts to propagate at an angle of 45° to the beam axis until it reaches the shear link; with this assumption, L_d is considered equal to the concrete cover’s depth as shown in Figure 4(a). It was stated by Achintha and Burgoyne (2013) that the actual direction of the crack may be slightly varied from 45°, but it should not have a significant effect on the results. On the other hand, for SNSM-strengthened beams, the experimental studies on SNSM-strengthened beams (Hosen et al., 2015; Sharaky et al., 2017; Shukri et al., 2016a) had shown that the shear crack starts to propagate horizontally as it reaches the SNSM reinforcement, such that L_d = 0 as shown in Figure 4(b).

Figure 4.

Initial debonded length of NSM- and SNSM-strengthened beams: (a) NSM strengthened beam; (b) SNSM strengthened beam.

As the GEBA is concerned with the start of the fracture process, mode II effects of concrete fracture such as aggregate interlock are not relevant; hence, the focus of the GEBA is mainly on mode I (Achintha and Burgoyne, 2008). For the fundamental procedure of the GEBA, consider the NSM-strengthened beam section as shown in Figure 4, with a CCS debonding crack already present at location L_a where moment M_a is acting. It is assumed that the debonding crack will propagate instantaneously, such that M_a remains the same as the change from strengthened to unstrengthened section due to CCS debonding occurs, as shown in Figure 5. The strain energy available at L_a that can cause the debonding crack to propagate, designated as W_a, can be determined from the difference between the moment–curvature (M/χ) relationship of the NSM-strengthened section (M/χ)_s and the unstrengthened section (M/χ)_u. The method to determine fracture energy available for debonding (G_a) is presented in section ‘CCS prediction’.

Figure 5.

Moment–curvature relationship of strengthened and unstrengthened beam section.

Design procedure for CCS

The proposed mechanics-based design procedure will be presented in this section, while a flowchart is given in Figure 6 as an overview. A preliminary design for the NSM strengthening is first made. Assuming the maximum moment that the beam needs to withstand is known, the maximum moment at L_a, M_a is known. The rotation at the strengthened and unstrengthened section due to moment M_a is then determined and the GEBA is used to determine if the debonding crack will propagate. If the beam is predicted to fail by CCS, the design for the NSM strengthening is changed and the procedure is repeated. Once a suitable design for NSM strengthening is determined, the flexural strength of the NSM-strengthened beam is then determined.

Figure 6.

Design procedure for NSM-strengthened beams.

The proposed design procedure is applicable for both virgin and cracked RC beams. It has been shown that the crack spacing near the support of the beams is usually similar regardless of whether the beam is virgin or cracked prior to NSM strengthening (Shukri et al., 2018), which allows the method for CCS prediction presented in Figure 6 to be used as it is for both types of beams. Furthermore, it was shown that the difference in ultimate load between virgin and cracked NSM-strengthened RC beam to be negligible when the failure is by flexure rather than CCS (Shukri et al., 2016a, 2018).

Primary crack spacing and load–slip stiffness

The primary crack spacing, S_cr, and the length of deformation, L_def, can be defined through the mechanically derived equation by Sturm et al. (2018)

S_{cr} = α {(\frac{2^{α} (1 + α)}{λ_{2} {(1 - α)}^{1 + α}})}^{\frac{1}{1 + α}} {(\frac{f_{ct}}{E_{c}} (\frac{E_{c} A_{c}}{E_{r} A_{r}} + 1))}^{\frac{1 - α}{1 + α}}

(1)

where

λ_{2} = \frac{τ_{\max}}{δ_{1}^{α}} β

(2)

β = L_{per} (\frac{1}{E_{r} A_{r}} + \frac{1}{E_{c} A_{c}})

(3)

L_{def} = \frac{S_{cr}}{2}

(4)

where f_ct is the tensile strength of concrete, L_per is the total perimeter of a single tensile steel reinforcement of area A_r contained within the tension stiffening prism as shown in Figure 2. A_cr is the area of adjacent concrete in the tension stiffening prism, which can be determined using Figure 1(b). The variables α, δ₁ and τ_max are the ascending branch of the non-linear bond stress–slip relationship by CEB-FIP (1993) where α = 0.4, δ₁ = 1 mm and τ_max = 1.25√f_c.

Having determined the S_cr, the load–slip stiffness parameter for the steel reinforcement, K_r can be determined as

K_{r} = \frac{2 E_{r} A_{r}}{S_{cr} c_{2}}

(5)

where E_r is the elastic modulus of steel and c₂

c_{2} = 1.08 {(\frac{A_{r}}{A_{cr}})}^{0.105}

(6)

The coefficient c₂ allows for the effect of bond and was determined using semimechanical means by Zhang et al. (2017) using a numerical tension stiffening analysis shown in Figure 2 to perform parametric study, from which c₂ was extracted. Since it was derived for steel reinforcement, it is unsuitable for NSM reinforcements. Hence, a parametric study was performed in this research to extract c₂ for NSM CFRP bar and NSM CFRP strip.

Parametric study for c₂

The objective of this brief parametric study is to obtain the relationship between A_f/A_cf and c₂; A_f is the area of NSM reinforcement and A_cf is the area of adjacent concrete for the tension stiffening prism containing the NSM reinforcement. Three types of NSM reinforcement configuration will be used:

NSM CFRP bar with ratio of groove to bar diameter of 2;

NSM CFRP bar with ratio of groove to bar diameter of 1.5;

NSM CFRP strip with ratio of groove height to width of 2.75.

The NSM reinforcement configurations above were chosen as they are commonly used in the literature. The bond stress–slip relationship by De Lorenzis (2004) and Zhang et al. (2013) were used for NSM CFRP bar and NSM CFRP strip, respectively. The material properties used in the parametric study were fixed and are shown in Table 1. The S_cr of a NSM-strengthened beam is assumed to be controlled by the steel reinforcement and not the NSM reinforcement (Shukri et al., 2015; Shukri and Jumaat, 2016); hence, for this parametric study of NSM reinforcements, the S_cr is fixed to the value given in Table 1.

Table 1.

Fixed properties for parametric study.

Properties	Value
Concrete compressive strength, f_c (N/mm²)	35
Concrete elastic modulus, E_c (N/mm²)	27,800
Concrete tensile strength, f_t (N/mm²)	3.2
Epoxy adhesive tensile strength, f_et (N/mm²)	27
Yield strength of steel reinforcement, σ_y (N/mm²)	500
Elastic modulus of steel reinforcement, E_r (N/mm²)	200,000
Elastic modulus of NSM reinforcement, E_f (N/mm²)	165,000
Primary crack spacing, S_cr (mm)	142.9

NSM: near-surface mounted.

The full numerical procedure for the partial interaction tension stiffening analysis, as illustrated in Figure 2, has been shown in multiple published research papers (Haskett et al., 2008; Muhamad et al., 2011; Shukri et al., 2015, 2016b; Shukri and Jumaat, 2016) and so will not be repeated again here. The results of the parametric study are shown in Figures 7 to 9, where c_2b2 is the coefficient of bond for NSM CFRP bar with ratio of groove to bar diameter of 2, c_2b1.5 is the coefficient of bond for NSM CFRP bar with ratio of groove to bar diameter of 1.5 and c_2s is the coefficient for NSM CFRP strip. From the results, it can be seen that the bond effect coefficient c₂ decreases when A_f/A_cf increases.

Figure 7.

Parametric study for NSM CFRP bars with ratio of groove to bar diameter of 2.

Figure 8.

Parametric study for NSM CFRP bars with ratio of groove to bar diameter of 1.5.

Figure 9.

Parametric study for NSM CFRP strips with ratio of groove height to width of 2.75.

Extracting the coefficients from the parametric study results using a linear relationship yield

c_{2 b 2} = - 0.529 (\frac{A_{f}}{A_{cf}}) + 0.884

(7)

c_{2 b 1.5} = - 0.586 (\frac{A_{f}}{A_{cf}}) + 0.862

(8)

c_{2 s} = - 1.645 (\frac{A_{f}}{A_{cf}}) + 1.367

(9)

The stiffness parameter for NSM CFRP bar can then be determined as

K_{f} = \frac{2 E_{f} A_{f}}{S_{cr} c_{2 b 2}}

(10)

where E_f, A_f and A_cf are the elastic modulus of the NSM reinforcement, area of a single NSM reinforcement and area of adjacent concrete area within the tension stiffening prism of the NSM reinforcement, which can be determined using Figure 1(b). The coefficient c_2b2 in equation (10) can be changed to either c_2b1.5 or c_2bs according to the NSM reinforcement configuration used. If some other NSM material or NSM configuration is used, to obtain the relevant coefficient c₂ is only a matter of performing a numerical parametric study similar to what is shown here.

Depth of neutral axis for unstrengthened RC beam section

Having determined the K_r and K_f, the load acting on a single steel and NSM reinforcement for a given slip can be determined as

P_{r} = K_{r} δ_{r}

(11)

P_{f} = K_{f} δ_{f}

(12)

Let n_r be the number of total steel reinforcement used in the RC beam. From equation (11), the sum of forces acting on all the steel reinforcements can be simplified as

P_{r} n_{r} = δ_{r} α_{1}

(13)

where

α_{1} = K_{f} n_{r}

(14)

Now consider the equation of equilibrium of the unstrengthened section

P_{c} = P_{r} n_{r}

(15)

where P_c is the compressive force of concrete. Assuming a triangular-shaped concrete stress and inserting equation (13) into equation (15) leads to

E_{c} (\frac{δ_{T}}{L_{def}}) (\frac{b d_{na - u}}{2}) = δ_{r} α_{1}

(16)

where E_c is the elastic modulus of concrete, δ_T is the deformation of the topmost section of the beam and d_na-u is the depth of neutral axis for the unstrengthened section. From Figure 10(a), the relationship between the rotation and the slips and deformations of a strengthened beam is

\tan θ = \frac{δ_{r}}{h - d_{na} - c_{r}} = \frac{δ_{f}}{h - d_{na} - c_{f}} = \frac{δ_{T}}{d_{na}}

(17)

Figure 10.

Moment-rotation of NSM-strengthened beam section: (a) beam section of length L_def and deformation profile, (b) strain profile, (c) stress profile and (d) forces acting on the beam section.

where h is the height of the beam and d_na is the depth of neutral axis. From equation (17), the relationship between the slip of reinforcement and concrete deformation at the topmost section of the beam for the unstrengthened section can be written as

δ_{r} = \frac{δ_{T}}{d_{na - u}} (h - d_{na - u} - c_{r})

(18)

Inserting equation (18) into equation (16) leads to

E_{c} (\frac{δ_{T}}{L_{def}}) (\frac{b d_{na - u}}{2}) = \frac{δ_{T}}{d_{na - u}} (h - d_{na - u} - c_{r}) α_{1} n_{r}

(19)

To simplify equation (19), let

α_{2} = \frac{E_{c} b}{2 L_{def}}

(20)

Replacing equation (20) into equation (19) yields

δ_{T} d_{na - u} α_{2} = \frac{δ_{T}}{d_{na - u}} (h - d_{na - u} - c_{r}) α_{1}

(21)

Solving equation (21) for d_na-u gives

d_{na - u} = \sqrt{\frac{α_{1}^{2}}{4 α_{2}^{2}} - \frac{c_{r} α_{1} - h α_{1}}{α_{2}}} - \frac{α_{1}}{2 α_{2}}

(22)

Rotation of unstrengthened RC beam section

Since a triangular-shaped concrete stress was assumed, the lever arm for the steel reinforcement in the unstrengthened section, Z_ru, is

Z_{ru} = h - \frac{d_{na - u}}{3} - c_{r}

(23)

The equation of moment for the unstrengthened beam section is

M_{a} = P_{r} n_{r} Z_{ru}

(24)

Inserting equation (13) into equation (24) and rearranging leads to

δ_{ru} = \frac{M_{a}}{α_{1} Z_{ru}}

(25)

where δ_ru is the slip of steel reinforcement for the unstrengthened beam section. From equation (17), the rotation of the unstrengthened section can be determined as

θ_{u} = \tan^{- 1} \frac{δ_{ru}}{h - d_{na - u} - c_{r}}

(26)

Depth of neutral axis for strengthened beam section

From equation (17), the following relationships can be obtained

δ_{rs} = \frac{δ_{T}}{d_{na - s}} (h - d_{na - s} - c_{r})

(27)

δ_{f} = \frac{δ_{T}}{d_{na - s}} (h - d_{na - s} - c_{f})

(28)

where d_na-s is the depth of neutral axis for the strengthened section of the RC beam. The equilibrium equation for the strengthened beam section is

P_{c} = P_{r} n_{r} + P_{f} n_{f}

(29)

where P_f is the force acting on the NSM reinforcement and n_f is the number of NSM reinforcement. Similar to equation (13), total NSM FRP force P_fn_f can be written as

P_{f} n_{f} = δ_{f} α_{3}

(30)

where

α_{3} = K_{f} n_{f}

(31)

Assuming a triangular shape for the concrete stress and inserting equations (13), (27), (28) and (30) into equation (29) gives

\begin{matrix} E_{c} (\frac{δ_{T}}{L_{def}}) (\frac{b d_{na - s}}{2}) \\ = (\frac{δ_{T}}{d_{na - s}}) (h - d_{na - s} - c_{r}) α_{1} \\ + (\frac{δ_{Ts}}{d_{na - s}}) (h - d_{na - s} - c_{f}) α_{3} \end{matrix}

(32)

Replacing equation (20) in equation (32) and simplifying it gives

α_{2} d_{na - s}^{2} = (h - d_{na - s} - c_{r}) α_{1} + (h - d_{na - s} - c_{f}) α_{3}

(33)

Solving equation (33) for d_na-s gives

d_{na - s} = \sqrt{\frac{{(α_{1} + α_{3})}^{2}}{4 {α_{2}}^{2}} - \frac{c_{r} α_{1} - h α_{1} + c_{f} α_{3} - h α_{3}}{α_{2}}} - \frac{α_{1} + α_{3}}{2 α_{2}}

(34)

Rotation of strengthened beam section

Since a triangular-shaped concrete stress was assumed, the lever arms for the steel and NSM reinforcements in the strengthened section can be written as

Z_{rs} = h - \frac{d_{na - s}}{3} - c_{r}

(35)

Z_{fs} = h - \frac{d_{na - f}}{3} - c_{f}

(36)

where Z_rs and Z_fs are the lever arm for the steel and NSM reinforcement in the strengthened beam section, respectively. The equation for moment in the strengthened beam section is

M_{a} = P_{r} n_{r} Z_{rs} + P_{f} n_{f} Z_{fs}

(37)

From equation (17), the following relationship can be obtained

δ_{f} = \frac{δ_{rs}}{h - d_{na} - c_{r}} (h - d_{na} - c_{f})

(38)

where δ_f and δ_rs are the slips of NSM and steel reinforcement in the strengthened section, respectively. Replacing equations (13), (30) and (38) into equation (37) and solving for δ_rs leads to

δ_{rs} = \frac{M_{a}}{α_{1} Z_{rs} + (\frac{h - d_{na} - c_{f}}{h - d_{na} - c_{r}}) α_{3} Z_{fs}}

(39)

From equation (17), the rotation of the strengthened section can be determined as

θ_{s} = \tan^{- 1} \frac{δ_{rs}}{h - d_{na} - c_{r}}

(40)

CCS prediction

The fracture strength of concrete can be determined using any appropriate model. Here, the CEB-FIP model (CEB-FIP, 1993) will be used

G_{\max} = G_{fo} {(\frac{S_{a}}{10})}^{0.7}

(41)

where S_a is maximum aggregate size used in the concrete in millimetres and G_fo is the base value for concrete fracture strength and can be taken as 0.037 for aggregate of size 20 mm. The fracture energy of the NSM-strengthened beam due to moment M_a can be determined as

G_{a} = \frac{W_{a}}{b_{f} Δ_{L}}

(42)

where Δ_L is the propagation of CCS debonding crack, b_f is the total width of the tension stiffening prism for NSM reinforcements, which as shown in Figure 1(b) can be taken as the width of the beam (b) if two or more NSM reinforcement is used, or half the width of beam if only one NSM reinforcement is used (Shukri et al., 2015) and

W_{a} = \frac{M_{a}}{(χ_{s} - χ_{u})}

(43)

χ_{s} = \frac{θ_{s}}{L_{def}}

(44)

χ_{u} = \frac{θ_{u}}{L_{def}}

(45)

If G_a > G_max, the beam is predicted to fail by CCS. Assuming this occurs, to redesign the NSM-strengthened beam is only a matter of changing the properties and configuration of the NSM strengthening, such as diameter of the NSM reinforcement (which results in a smaller A_f) and number of NSM reinforcement (n_f). The design procedure is then repeated starting from section ‘Depth of neutral axis for strengthened beam section’.

With regard to the value of Δ_L, as the GEBA is concerned with the start of the fracture process, the value Δ_L used is usually a very small value. Previous researchers had used Δ_L = 1 mm in their research (Achintha and Burgoyne, 2008; Shukri et al., 2018; Shukri and Jumaat, 2016). It was also noted that values of Δ_L <1 mm can cause numerical convergence problem (Achintha and Burgoyne, 2008). In this article, Δ_L = 1 mm will be used based on a sensitivity analysis of Δ_L that will be presented in section ‘Sensitivity analysis for Δ_L’

Existing slip due to dead load

The dead load on RC beams prior to strengthening can affect the serviceability condition of the beam. Assuming that the moment due to existing permanent action (dead load), M_p, can be determined by the design engineer, equations derived for the unstrengthened beam section can be used to determine the slip of steel reinforcement due to permanent action

δ_{p} = \frac{M_{p}}{α_{1} Z_{ru}}

(46)

The lever arm should remain the same, as the equation used to obtain it and to obtain the depth of neutral axis is independent of applied load. Note that the equation above assumes a triangular shape for concrete stress. The existing crack width prior to strengthening would be equal to 2δ_p.

Design flexural strength of NSM-strengthened beam

The design flexural strength of NSM-strengthened beam will be limited by the concrete crushing. For simplicity, several assumptions are used:

A rectangular concrete stress block is assumed.

The concrete strain will be limited to 0.0035 to adhere to Eurocode 2 requirement.

The strain hardening of the steel reinforcements will be ignored.

Compression bars will be ignored.

The assumptions used are common in design practice. Since the concrete strain will be limited to 0.0035, the following relationship is obtained

\frac{δ_{T}}{L_{def}} = 0.0035

(47)

The equilibrium equation at ultimate limit state is

P_{c} = P_{y} n_{r} + P_{f} n_{f}

(48)

where P_y is the load at which the steel reinforcement yields. From equation (17)

δ_{fd} = \frac{δ_{T}}{d_{na - d}} (h - d_{na - d} - c_{f})

(49)

where d_na-d is the depth of neutral axis for the design strength of the NSM-strengthened beam and δ_fd is the slip of the NSM reinforcement at ultimate limit state. Inserting equation (47) into equation (49) yields

δ_{fd} = \frac{0.0035 L_{def}}{d_{na - d}} (h - d_{na - d} - c_{f})

(50)

Replacing equations (30) and (50) into equation (48) and expanding yields

(α_{c} f_{c}) (β_{c} b d_{na - d}) = σ_{r} E_{r} n_{r} + (\frac{δ_{T}}{d_{na - d}}) (h - d_{na - d} - c_{f}) α_{3}

(51)

where concrete stress is assumed to be equal to f_c at concrete strain of 0.0035, while α_c and β_c are ratio of equivalent concrete stress and equivalent concrete stress block height, respectively. From Eurocode 2, α_c = 0.85 and β_c = 0.8 for f_c≤50 MPa. Equation (51) can be simplified into

α_{4} d_{na - d}^{2} = α_{5} d_{na - d} + (h - d_{na - d} - c_{f}) α_{6}

(52)

where

α_{4} = f_{c} b

(53)

α_{5} = σ_{r} E_{r} n_{r}

(54)

α_{6} = 0.0035 L_{def} α_{3}

(55)

Solving equation (52) for d_na-d

d_{na - d} = \sqrt{\frac{{(α_{6} - α_{5})}^{2}}{4 {α_{4}}^{2}} - \frac{c_{f} α_{6} - h α_{6}}{α_{4}}} - \frac{α_{6} - α_{5}}{2 α_{4}}

(56)

With d_na-d known, the slip of the NSM FRP bar, δ_f, can be determined using equation (49). The design flexural moment of the NSM-strengthened beam can be determined as

M_{d} = σ_{r} E_{r} n_{r} Z_{rd} + K_{f} δ_{fd} Z_{fd}

(57)

where Z_rd and Z_fd are the lever arm for steel and NSM reinforcements at ultimate limit state, respectively, and are determined as

Z_{rd} = h - \frac{β_{c} d_{na - d}}{2} - c_{r}

(58)

Z_{fd} = h - \frac{β_{c} d_{na - d}}{2} - c_{f}

(59)

It is also possible to determine the crack width of the beam at ultimate limit state. From equation (17)

δ_{rd} = \frac{δ_{T}}{d_{na - d}} (h - d_{na - d} - c_{r})

(60)

Inserting equation (47) into equation (60) yields

δ_{fd} = \frac{0.0035 L_{def}}{d_{na - d}} (h - d_{na - d} - c_{f})

(61)

where δ_rd is the slip of steel reinforcement due to applied load. The crack width, Δ_r, of the beam is then determined as twice the slip due to permanent action and applied load

Δ_{r} = 2 (δ_{p} + δ_{rd})

(62)

Validation

The proposed design procedure was validated using published experimental results (Al-Mahmoud et al., 2010; Barros et al., 2007; Ceroni, 2010; Sharaky et al., 2015; Shukri et al., 2015, 2016a; Teng et al., 2006) that failed by either CCS or flexure. In the validation process, the experimental and predicted failure mode will first be compared by determining the available fracture energy (G_a) and the fracture strength (G_max). Comparisons between predicted and experimental moment of resistance at failure will also be given for beams that failed by flexure.

The geometric and material properties of the beam that are necessary to apply the design procedure are given in Tables 2 and 3. The beams were strengthened using NSM CFRP bars (Al-Mahmoud et al., 2010; Ceroni, 2010; Sharaky et al., 2015; Shukri et al., 2015), NSM CFRP strips (Barros et al., 2007; Teng et al., 2006) or with side-NSM CFRP bars (Shukri et al., 2016a). As the use of SNSM method only changes the lever arm of the NSM reinforcement, the proposed design procedure can be used without any changes. The maximum size of aggregates in concrete, S_a, was assumed to be 20 mm if not specified in the original research paper. Referring to Table 2, the moment M_a is the moment at the length L_a corresponding to the reported experimental failure load, P_max, of each beam.

Table 2.

Geometric and loading properties of beams.

Reference	Beam	h (mm)	b (mm)	n_r	c_r (mm)	n_f	c_f (mm)	L_a (mm)	M_a (kNm)
Shukri et al. (2015)	A2	250	125	2	41	1	12	77	5.13
Shukri et al. (2015)	A1	250	125	2	41	1	12	127	8.31
Ceroni (2010)	A10	180	100	2	30	2	7.5	215	5.45
Sharaky et al. (2015)	F2C1	280	160	2	44	2	8	230	13.48
Al-Mahmoud et al. (2010)	S-C (FPT) (270)	280	150	2	42	2	6	80	5.34
Al-Mahmoud et al. (2010)	S-C (FPT) (210)	280	150	2	42	2	6	380	20.82
Teng et al. (2006)	B2900	300	150	2	44	1	11	80	3.99
Teng et al. (2006)	B1800	300	150	2	44	1	11	622	28.52
Teng et al. (2006)	B1200	300	150	2	44	1	11	922	29.09
Teng et al. (2006)	B500	300	150	2	44	1	11	1272	30.40
Barros et al. (2007)	NSM S1	170	120	2	38.5	1	7.5	80	3.16
Barros et al. (2007)	NSM S2	170	120	2	39.5	2	7.5	80	3.71
Shukri et al. (2016a)	SNC8	250	125	2	39	2	39	50	3.55
Shukri et al. (2016a)	SNC10	250	125	2	39	2	39	50	4.42
Shukri et al. (2016a)	SNC12	250	125	2	39	2	39	50	4.33
Shukri et al. (2016a)	PSNC8	250	125	2	39	2	39	50	3.54
Shukri et al. (2016a)	PSNC10	250	125	2	39	2	39	50	4.28
Shukri et al. (2016a)	PSNC12	250	125	2	39	2	39	50	4.24

h: beam height; b: beam width; n_r: number of tensile reinforcement bars; n_f: number of FRP bars/strips; c_r: distance from beam soffit to centre of tensile reinforcement bars; c_f: distance from beam soffit to centre of FRP bars/strips; L_a: distance from end of beam to the location of curtailment of FRP bars/strips.

Table 3.

Material properties.

Reference	Beam	f_c (N/mm²)	σ_y (N/mm²)	E_r (kN/mm²)	E_f (kN/mm²)
Shukri et al. (2015)	A2	35.63	520	200	165
Shukri et al. (2015)	A1	35.63	520	200	165
Ceroni (2010)	A10	26.88	441	200	109
Sharaky et al. (2015)	F2C1	30.5	455	200	158
Al-Mahmoud et al. (2010)	S-C (FPT) (270)	36.1	600	200	146
Al-Mahmoud et al. (2010)	S-C (FPT) (210)	36.1	600	200	146
Teng et al. (2006)	B2900	44	532	200	131
Teng et al. (2006)	B1800	44	532	200	131
Teng et al. (2006)	B1200	44	532	200	131
Teng et al. (2006)	B500	44	532	200	131
Barros et al. (2007)	NSM S1	44.2	788	200	159
Barros et al. (2007)	NSM S2	44.2	788	200	159
Shukri et al. (2016a)	SNC8	40 (cube)	520	200	124
Shukri et al. (2016a)	SNC10	40 (cube)	520	200	124
Shukri et al. (2016a)	SNC12	40 (cube)	520	200	124
Shukri et al. (2016a)	PSNC8	40 (cube)	520	200	124
Shukri et al. (2016a)	PSNC10	40 (cube)	520	200	124
Shukri et al. (2016a)	PSNC12	40 (cube)	520	200	124

f_c: concrete strength; σ_y: yield strength of steel; E_r: elastic modulus of steel; E_f: elastic modulus of FRP bar/strip.

The result of the validation is given in Table 4, where the calculated G_max, G_a, failure mode, experimental maximum moment (M_e) and maximum moment obtained using the design procedure (M_d) are presented. It should be noted that M_d is only given when it is predicted that the beam fails by flexure. From Table 4, it can be seen that the proposed design procedure was able to correctly predict the CCS failure mode of the beams apart from beam A2, SNC10 and PSNC10. In the case of beam A2, the predicted failure mode is flexure, whereas the actual failure mode is CCS. The ratio of M_d/M_e, however, is very close, at 0.99, which shows that while the beam was reported to fail by CCS, the loss of strength due to the premature debonding is negligible. For the case of beams SNC10 and PSNC10, the predicted failure mode was CCS, while the actual failure mode was flexure. The ratio of G_a/G_max for beam SNC10 shows that the fracture strength was only very slightly exceeded, hence the incorrect predicted failure mode can be due to difference in the calculated and actual fracture energy, G_max, which is not uncommon due to the variable nature of concrete.

Table 4.

Predicted and experimental result comparison.

Reference	Beam	G_a/G_max	FM_e	FM_d	M_e (kNm)	M_d (kNM)	M_d/M_e
Shukri et al. (2015)	A2	0.82	CCS	F	50.0	49.3	0.99
Shukri et al. (2015)	A1	2.15	CCS	CCS	49.1	–	–
Ceroni (2010)	A10	2.39	CCS	CCS	22.3	–	–
Sharaky et al. (2015)	F2C1	1.70	CCS	CCS	515.7	–	–
Al-Mahmoud et al. (2010)	S-C (FPT) (270)	0.17	F	F	53.4	55.4	1.04
Al-Mahmoud et al. (2010)	S-C (FPT) (210)	2.52	CCS	CCS	32.9	–	–
Teng et al. (2006)	B2900	0.13	F	F	59.9	57.3	0.96
Teng et al. (2006)	B1800	6.61	CCS	CCS	55.0	–	n/a
Teng et al. (2006)	B1200	6.88	CCS	CCS	37.9	–	n/a
Teng et al. (2006)	B500	7.51	CCS	CCS	28.7	–	n/a
Barros et al. (2007)	NSM S1	1.69	CCS	CCS	11.8	–	n/a
Barros et al. (2007)	NSM S2	1.47	CCS	CCS	13.9	–	n/a
Shukri et al. (2016a)	SNC8	0.60	F	F	46.2	37.7	0.82
Shukri et al. (2016a)	SNC10	1.08	F	CCS	57.5	–	n/a
Shukri et al. (2016a)	SNC12	1.14	CCS	CCS	56.2	–	–
Shukri et al. (2016a)	PSNC8	0.59	F	F	46.0	37.7	0.82
Shukri et al. (2016a)	PSNC10	1.01	F	CCS	55.6	–	n/a
Shukri et al. (2016a)	PSNC12	1.09	CCS	CCS	55.1	–	n/a

G_a: predicted fracture energy; G_max: fracture strength; FM_e: experimental failure mode; FM_d: predicted failure mode; M_e: experimental maximum moment at failure; M_d: predicted maximum moment at failure; CCS: concrete cover separation failure mode; F: flexural failure mode α; n/a: not applicable.

Sensitivity analysis for Δ_L

A sensitivity analysis was conducted using the properties of nearly all the beams in the used in the validation process that failed by CCS with the result as shown in Figure 11. As all the beams had experimentally failed by CCS, the G_a/G_max should be more than 1. Where G_a/G_max is less than 1, it shows that the value of Δ_L used failed to provide a correct prediction of CCS failure for the beam.

Figure 11.

Sensitivity analysis for Δ_L.

It can be seen from Figure 11 that Δ_L = 1 mm was able to give an accurate assessment of CCS failure for all beams. Values of Δ_L <1 mm were also able to correctly predict the CCS failure; however, it should be noted that when Δ_L <1 mm, the resulting G_a can be considerably higher. This can cause a very conservative design, as the design engineer may have to greatly reduce the amount of NSM reinforcement provided in order to ensure that G_a < G_max and prevent CCS failure. Hence, the value of Δ_L = 1 mm is suggested as it gives a reasonable balance between accuracy and conservativeness.

Conclusion

In this research article, a mechanics-based design procedure was proposed. The proposed design procedure uses the M/θ approach and the GEBA to predict the behaviour of NSM-strengthened RC beams and the CCS failure mode. Several conclusions can be made based on this study:

Published experimental results of beams strengthened with NSM CFRP bars, NSM CFRP strips or SNSM CFRP bars were used to validate the proposed design procedure and good correlation was found between the experimental and predicted results.

The proposed design approach should be more versatile compared to other existing design approach as it is less reliant on empirical formulations. Hence, it can easily be applicable to most types of NSM reinforcement material and configurations. The design approach can also accommodate any new innovations in terms of the NSM strengthening material or configurations. The coefficient of bond c₂ for those new NSM material or configuration can be determined using a similar approach described in this article.

Footnotes

Appendix 1 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) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: Financial support from the Fundamental Research Grant Scheme (FRGS), Ministry of Education Malaysia (project number FP004-2014B) is gratefully acknowledged.

ORCID iD

Ahmad Azim Shukri

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Concrete cover separation of reinforced concrete beams strengthened with near-surface mounted method: Mechanics based design approach

Abstract

Keywords

Introduction

Fundamental theories

M/θ approach

GEBA

Design procedure for CCS

Primary crack spacing and load–slip stiffness

Parametric study for c2

Depth of neutral axis for unstrengthened RC beam section

Rotation of unstrengthened RC beam section

Depth of neutral axis for strengthened beam section

Rotation of strengthened beam section

CCS prediction

Existing slip due to dead load

Design flexural strength of NSM-strengthened beam

Validation

Sensitivity analysis for Δ L

Conclusion

Footnotes

Appendix 1

Declaration of Conflicting Interests

Funding

ORCID iD

References

Parametric study for c₂

Sensitivity analysis for Δ_L