Bond parameters for peeling and debonding in thin plated RC beams subjected to mixed mode loading

Abstract

Studies have primarily focussed on predicting mode-II debonding failure; whereas, in real-case-scenario, flexurally strengthened reinforced concrete (RC) beams observe premature failure mechanisms under mixed-mode loading conditions engaging geometrical and material variations. Peeling is a consequence of flexural crack as debonding is of interfacial shear crack. Under bending, peeling failure is considerably catastrophic over debonding due to the nature of crack formation; therefore, this needs to be distinguished in predictive analysis. In this paper, a new numerical modeling methodology is approached using eXtended finite element method (xFEM) for flexural cracks and Cohesive Zone Model (CZM) for shear cracks without predefining crack locations. The parameters of the constitutive models are identified through comparing finite element results with the experimental data. These parameters are related to key material properties. Based on proposed framework, the models provide a good estimation of plate strain distribution, cracks and failure type, in terms of mode and load of failure. Bilinear bond-slip curve is a closer match over exponential crack evolution at interface.

Keywords

peeling debonding bond parameters eXtended finite element method cohesive zone model

Introduction

External plating of reinforced concrete (RC) structures has been predominantly adopted as a retrofitting technique, reported from as early as 1967 (L’Hermite and Bresson, 1967), which is cost effective and brings about minimal aesthetic variations. To maintain the retrofitting procedures to be non-destructive, the plate is adhered using an epoxy adhesive. However, in most cases, the expected capacity of the retrofitted structure is adversely affected by the formation of premature cracks; thereby, leading to premature failure such as plate debonding (without or slightly damaging concrete surface) or peeling (ripping off concrete cover, exposing internal rebars) from soffit of the beam. Studies (Camanho et al., 2003; Khan et al., 2017c; Khan, 2021b; Li et al., 2005; Oh et al., 2003a) in general have indicated that adhesive can profoundly control the transfer of stresses between the plate and main structure, mainly determining debonding. Therefore, in spite of being a structurally insignificant component (to contribute towards the beam capacity), adhesive layer is a critical element of structural integrity. In pull-off shear tests, the interface is subjected to shear stresses in tangential direction to interface (called mode-II stresses); and in bending problems, the plate interface in shear span is also subjected to opening stresses in normal direction (called mode-I stresses) in combination to tangential stresses and this combined effect is mentioned as mixed-mode stresses (also refer Figure 1). Interfacial bond tests have been conducted in the past (Hussain et al., 1995; De Lorenzis et al., 2001; Oh et al., 2003a) under mixed-mode loading to study the bond behaviour between the plate and RC beam. The effectiveness of load transfer through this medium largely depends on adhesive stiffness, bond strength and fracture energy. Depending upon a failing medium as concrete (or adhesive only), the total fracture energy is a contribution of the initial fracture energy of adhesive and crack energy of concrete (or the fracture energy of adhesive only, respectively). However, this also depends on a failure mechanism to be through debonding or peeling; where in case of peeling, only concrete is involved as the two modes of failure can be differentiated (Campilho et al., 2008; Khan et al., 2017a). Peeling is rather highly catastrophic and ruptures the concrete cover and exposes the internal rebars. Due to this, the plate peeling loads can be lower than the associated ultimate (failure) loads of even unplated R.C. beams (Oehlers and Moran, 1990; Oh et al., 2003b; Raoof et al., 2000; Raoof and Zhang, 1997). Prediction of failure loads under premature modes of failure that involve failure of interface or separation of concrete cover is still under debate (Al-Mahaidi and Kalfat, 2018, Dar et al., 2019; Demir et al., 2018; Piątek et al., 2020; Tahar et al., 2019; Wu and Eamon, 2017).

Figure 1.

Schematic of a typical plated beam under 4-point bending showing failures due to unidirectional stresses and mixed-mode stresses.

Studies (Charif, 1983; Jones et al., 1985; Jumaat and Alam, 2008; Khan et al., 2017b; Macdonald and Calder, 1982; Oh et al., 2003b) signify on the fact that, including adhesive properties, a number of other beam parameters also directly influence debonding and peeling simultaneously. As discussed, although both are premature failure, but their impact on the main structure is different; peeling is relatively destructive to main structure than debonding, and accordingly requires different maintenance services. Therefore, it is necessary to predict debonding as a different failure type from peeling.

Bending tests types may include half-beam tests (Oh et al., 2003a) and full-beam tests; however, in addition to showing inconsistencies within the experimental data, the availability of literature on half-beam tests is considerably insufficient. However, many models have been conducted to describe the bond behaviour under interfacial stresses in pure shear debonding (mode-II direction). The parameters describing the models are either obtained by fitting the models through experimental data (Yang et al., 2007) or by fitting the numerical results with the experimental work (Lu et al., 2005; Obaidat et al., 2013). Lu et al. (2005) and Chen and Teng (2001) evaluated bond parameters of constitutive models defined specifically for mode-II loading problem, for predicting shear debonding, such as maximum shear stress and fracture energy of the interface. However, in a bending problem which is observing additional modes of failure, the behaviour of plate-to-concrete interface itself can be significantly different from that captured through traditional pull-off shear tests. For example, during experiments related to bending problem, the strain gauges are placed along the length (transverse direction) of the external plate to record longitudinal strains. It is noted that these strains are a consequence of mixed-mode loading along the length of plate and not purely mode-II loading; the fact also supported by Liu and Qiao (2017) for evaluating interface behaviour with and without mixed-mode loading. Therefore, a suitable procedure needs to be adopted that could yield to equivalent constitutive models. Studying a 3-point loading problem, De Lorenzis et al. (2001, 2013) developed closed form solutions considering geometrical and material properties (largely linear-elastic on a bilinear interface model). However, the current work intends to evaluate parameters describing constitutive failure models that can be suitably utilised to simulate peeling and debonding on a beam subjected to bending. This manuscript proposed a new numerical modeling method to predict mode-II debonding failure, using eXtended finite element method (xFEM) to model flexural cracks, and Cohesive Zone Model (CZM) to model interfacial cracks without predefining crack locations. The parameters of the constitutive model are determined by comparing the finite element results with experimental data, as explained next.

Rahimi and Hutchinson (2001) have shown that the choice of external material (GFRP/CFRP/Steel modulus) has a direct influence on the stiffness of the beam and thereby on the level of interfacial stresses. Therefore, this study would focus mainly on the case of beams retrofitted with steel plates, considered as being relatively more critical (to interfacial debonding) than other common FRPs in use, largely due to relatively higher initial modulus of steel (comparatively stiffer at interface).

Materials, models and method

In case of modeling a RC beam, there is no single approach; and the choice of modeling technique and minor changes in material non-linearities can directly impact the outcomes, for example, studies by Earij et al. (2017), Khan (2017) and Eliáš at al. (2015). Narayanamurthy et al. (2012) have summarised key models to signify the importance of interfacial stresses that are responsible for debonding, over the stresses responsible to cause peeling rip-off failure. Others (Chen et al, 2006, Khan, 2018; Khan et al., 2017b; Oehlers, 1992; Oehlers and Moran, 1990; Oehlers and Seracino, 2004) have identified the possible interaction between flexural cracks and interfacial cracks.

To differentiate peeling from debonding, the xFEM approach is utilised to observe peeling and other flexural cracks within a continuum model, while CZM is used for modeling debonding using discrete elements. Campilho et al. (2011) established that xFEM is better suited for modeling crack formation within a bulk material, while CZM is for modeling interfacial debonding. Elguedj et al. (2006) employed plastic enrichments in xFEM to model the singular fields; xFEM crack initiates and propagates orthogonally at regions experiencing maximum principal stresses/strains, that is, it is not mandatory in ABAQUS (2021) to define initial cracks. Khan (2017) has emphasised on the formation of plastic strains through Concrete Damage Plasticity (CDP) model in plated beams. CDP model is adopted for concrete to retain localised strains. With further loading, and depending on material definition, the localised strains may translate into plastic strains such as peeling.

Element type used on RC beam, external plate, support and load plates is a 4-node bilinear plane stress quadrilateral with reduced integration (CPS4R), for adhesive layer a 4-node two-dimensional cohesive element (COH2D4). Mesh size is 10 mm x 10 mm for RC beam, and 1 mm for other components to share the same nodes or connected by an interfacial link. As mentioned before, adhesive is not a main structural component but it is there to establish structural integrity and is located at critical path of stress transfer, COH2D4 elements do not represent any physical material rather have the peculiarity of generating bond forces. CDP model is used for concrete, elasto-plastic for steel and bi-linear/linear-exponential traction-separation for adhesive. Maxe-traction-separation xFEM model is defined on beam surface without predefining crack location to capture multiple cracks reaching plastic strains. Cohesive zone model is defined on adhesive layer to capture cracks reaching shear strength at concrete-adhesive interface. The formulations explaining the choice of xFEM and CZM models in our analysis and to relate their significate with the parameters responsible for peeling and debonding are described next. While, geometrical and material properties for 19 beam specimens (Charif, 1983; Heathcote, 2004; Jones et al., 1982) and five beams tested in our lab at AMU are summarised in Tables 1 and 2.

Table 1.

Specimen geometry range, load and mode of failure.

Specimen	Dimension, effective span/Shear span/width/height (mm) (Range: 800–2300/266.7–850/100155/120–300)	Effective depth (mm) (Range: 103–262)	Clear cover (mm) (Range: 15–50)	Tensile rebars (mm)/Layers (Percentage tensile steel Range: 0.47–2.70)	Compression rebars (mm) (Range: With or Without)	Plate, l/b/t (mm) (Range: 350–2200/80–150/1–10)	Adhesive thickness (mm) (Range: 1–3)
URB4	2250/750/100/150	130	15	2 @10/1	—	2150/80/5	3
URB5		130		2 @10/1		2150/80/10
ORB2		119		4 @16/2		2150/80/5
P1	2000/850/150/300	262	25	2 @10/1	2 @8/1	1700/150/5	1
P2A		252	35
P3A		237	50
P7		262	25
P8		252	35
P9		237	50
P10		262	25	3 @10/1
P11		252	35
P12		237	50
P13		247	25	4 @10/2
P14		237	35
P15		222	50
P19		247	25	6 @10/2
P20		237	35
P21		222	50
F31	2300/767/155/225	190	25	3 @20/1	—	2200/125/6	1-1.5
BES 1 × 350	800/266.7/100/120	103	20	2 @6/1	—	350/100/1
BES 1 × 430						450/100/1
BES 0.6 × 430						450/100/0.6
BES 1 × 700						700/100/1
BES 0.6 × 700						700/100/0.6

Beams with prefixes URB and ORB are from Jones et al. (1982), Beams with prefix P are from Heathcote (2004); Beam with prefix F is from Charif (1983).

Table 2.

Material properties range.

Properties	Specimen
Properties	Jones et al. (1982)	Heathcote (2004)	Charif (1983)	Test specimens
Concrete
Compressive cubic strength (MPa)^a	97.4–63.4	56.3–67.5	55.80	41.85
Tensile strength (MPa)^b	7.80–5.07	4.5–5.4	4.5	4.5
Elastic modulus (GPa)^c	18.11	16.68–20	36.74	32.3
Fracture energy (MPa-mm)^b	0.14	0.06–0.07	0.08	0.05
Plate
Elastic modulus (GPa)	200	202	200	200
Yield strength (MPa)	217.5–240.0	337	246	240
Tension rebar^d
Elastic modulus (GPa)	200	194–201	200	200
Yield strength (MPa)	487.5–530	488–563	430	415
Adhesive
Elastic modulus (GPa)	6.0	11.2	0.28	10
Shear modulus (GPa)	2.59^e	4.8^e	0.24	4.3^e
Tensile strength (MPa)	5.3	—	7.82	15

^aIf required, $f_{c} = 0.8 f_{c}^{'}$ , where $f_{c}$ is cylindrical compressive strength, $f_{c}^{'}$ is cubic compressive strength (BS:1881-120, 1983).

^bComputed using BS:8110 (1997).

^cComputed using material idealisation approach (Khan et al., 2017a).

^dTrilinear behaviour, as per BS:5400 (1990).

^eComputed with Poisson’s ratio of 0.16 (Ziraba and Baluch, 1995).

XFEM and discretization

eXtended Finite Element Modelling is an extension of FEM, whose fundamental features and approximations are presented by Belytschko and Black (1999). Melenk and Babuska (1996) have used the concept of partition of unity to compute the displacement field for the general point within an arbitrary domain $Ω$ discretized by $n$ – node finite elements on a vector of current coordinates $x$ as

u^{h} (x) = u^{F E} + u^{e n r}

(1)

u^{F E} = \sum_{i = 1}^{n} N_{i} (x) u_{i}

(2)

u^{e n r} = \sum_{h = 1}^{m} N_{h} (x) H (x) a_{h}

(3)

where

u^{h}

is the vector of nodal degrees of freedom (Dofs) for both the classical

u^{F E}

and enrichment

u^{e n r}

nodes. This uses additional Dofs

a_{h}

over the vector of the regular Dofs

u_{i}

of nodal displacements in the standard FEM to model crack or a discontinuous field without the need to define it explicitly in FE mesh.

$N (x)$ are the finite element shape functions. $H (x)$ is the generalised Heaviside function (Belytschko and Black, 1999) which is a discontinuous enrichment function that defines the nodal set ahead of the discontinuity within the support domain to simulate the displacement discontinuity of the crack faces, defined as

H (x) = {\begin{matrix} 1 & if (x - x^{*}) e_{n} \geq 0 \\ - 1 & otherwise \end{matrix}

(4)

where

x^{*}

is the nearest point to

x

, and

e_{n}

is the unit vector normal to the crack alignment. This means that

H (x)

has a value of +1 if

x

is above crack, and −1 otherwise.

To model crack surfaces and tips, Moës et al. (1999) established $u^{e n r}$ including the displacement enrichment as

u^{e n r} = \sum_{h = 1}^{m} N_{h} (x) H (x) a_{h} + \sum_{k = 1}^{t} N_{k} (x) F_{l} (x) b_{k}

(5)

where

m

is the set of nodes particularly belonging to the crack-face,

t

is the set of nodes associated with crack tip in their corresponding domain. For an enriched node located at the interior of element, the pertaining support domain is element surrounding the node; while for a node lying at the edges of elements, the domain corresponds to the surrounding elements connected to the enriched node.

b_{k}

is the vector of the additional Dofs corresponding to the crack tip.

Further, Asadpoure and Mohammadi (2007) discretized $u^{e n r}$ to include crack tip enrichment function $F_{l}^{j} (x)$ $(j = 1, 2)$ to demonstrate crack propagation as

u^{e n r} = \sum_{h = 1}^{m} N_{h} (x) H (x) a_{h} + \sum_{k = 1}^{t_{1}} N_{k} (x) [\sum_{l} F_{l}^{1} (x) b_{k}^{1}] + \sum_{k = 1}^{t_{2}} N_{k} (x) [\sum_{l} F_{l}^{2} (x) b_{k}^{2}]

(6)

According to studies (Asadpoure et al., 2006a, 2006b; Mohammadi, 2012), the crack tip enrichment function $F_{l}$ represents all possible displacement states, such that

{F_{l} (r, θ)}_{l = 1}^{4} = {\sqrt{r} \cos \frac{θ_{k}}{2} \sqrt{g_{k} (θ)}, \sqrt{r} \sin \frac{θ_{k}}{2} \sqrt{g_{k} (θ)}}, k = 1, 2

(7)

where (

r

θ

) are the local polar coordinates at the crack tip.

g_{k} (θ)

and

θ_{k}

are defined for all orthotropic composites in general form as

g_{k} (θ) = \sqrt{{(\cos θ + S_{k x} \sin θ)}^{2} + {(S_{k y} \sin θ)}^{2}}

(8)

θ_{k} = a r c \tan [\frac{S_{k y} \sin θ}{\cos θ + S_{k y} \sin θ}]

(9)

Multiple cracks can be treated through the above set(s) of functions by incorporating further enrichment of discontinuity and almost the entire crack front.

Since xFEM in ABAQUS is defined on linear elements (linear shape functions), a discrete system of linear equilibrium equations is given by

K u^{h} = f

(10)

The global stiffness matrix $K$ and vector of external forces $f$ are computed by assembling the corresponding matrix and vectors for each element.

Stiffness displacement matrix $K_{i j}^{r s}$ accommodating a crack formation over a domain $Ω$ is

K_{i j}^{r s} = \int_{r^{e}} {(B_{i}^{n})}^{T} D (B_{j}^{s}) d Ω, (r, s = u, a, b)

(11)

where,

B

is the strain displacement matrix which is a matrix of shape function derivatives.

After this, the progress of the arbitrary crack depends on enrichment function that can be chosen by applying appropriate analytical solutions. The choice of which depends on factors such as, the reproduction of the singular field around crack tip, independency of the strain field in two different sides of crack surface, continuity of the displacement field that connect the adjacent elements, and any additional features required by a given discontinuity problem. It is noteworthy, that multiple cracks can be treated in the above framework by incorporating additional enrichment of the discontinuity and almost the entire tip.

Formulations indicate that, as crack tip changes its position and path with loading conditions, the xFEM algorithm simulate discontinuities (such as cracks) as enriched features that allow discontinuities to grow around the crack path/tip as defined by associated displacement functions by enriching the Dofs of nearby nodes as recently demonstrated by Ostapska and Malo (2021) for modeling fracture in wood. In ABAQUS, $F_{l} (x)$ are used for stationary cracks; while in case of damage, crack propagation along an arbitrary path uses phantom nodes (superimposed on real nodes) that have same coordinates and constrains as original nodes up to damage initiation. That is, the behaviour and properties of phantom nodes is exactly the behaviour of original nodes before crack initiation. At the onset of crack initiation, two sub-elements are formed (real and phantom) that are now governed by independent displacement fields that are allowed to separate unconstrained from each other following a cohesive law up to failure, thereby simulating crack growth.

Observing crack using xFEM is independent of meshing rules as it is based on energetic criteria and hence, unlike conventional fracture mechanics, it does not require re-meshing (adaptive meshing) or finer meshing around the periphery of crack and crack tip for achieving accuracy; however, if damage is defined using damage initiation criteria, stresses/strains at concentration regions are mesh sensitive and some variations are expected. Therefore, this requires evaluation of corresponding cohesive parameters for xFEM model. The current implementation of xFEM in ABAQUS is restricted to either on the value of stress (using the maximum principal stress criteria) or strain (using the maximum principal strain criteria) to initiate damage. Since, the fracture process of thin layer (let us say adhesive at interface) is not consistent with that of bulk adherends; the use of xFEM is restricted onto bulk adherend(s); while for recording damage and cracks at interface, CZM is adopted and explained next.

Therefore, putting concisely, xFEM models singular fields using enriched Dofs that follows definition of material behaviour and failure criteria. Relating the parameters of CDP and CZM models with the xFEM procedure can simulate cracks without the need to predefine the initial crack location. eXtended finite element method algorithm regulates crack growth using principal stress/strain that can be associated with the parameters of CDP model.

CZM and discretization

In recent years, the use of CZMs has become frequent in fracture problems as these overcome the reported limitations by combining elements of strength and fracture mechanics to calculate fracture loads (Campilho et al., 2007; Cavalli and Thouless, 2001). Cohesive zone models are based on cohesive (Feraren and Jensen, 2004; Petrossian and Wisnom, 1998) or spring (Cui and Wisnom, 1993) elements, placed at regions where damage is prone to occur, or these are also used to define a cohesive continuum medium like an adhesive (Khan et al., 2017c) without the requirement of initial flaw. In an adherend, it is difficult to pre-identify regions or paths of damage, while in bonded assemblies the damage is largely restricted to common interfaces due to accumulation of stresses. Therefore, the use of CZM is more suitable to connect planes/faces of adherends by placing cohesive zone elements between a damage prone common interface (Campilho et al., 2008; Khan et al., 2017a).

Initiation of interfacial debonding is captured by a key parameter of strength introduced by CZM. The propagation of this crack-up to complete failure is regulated by the release of crack energy.

Cohesive zone model can utilise the concept of traction-separation laws to model crack through damage initiation and propagation, and retain damage. Elliptical crack initiation criterion is considered, and failure criterion to be energy based. The discretization of CZM is elaborated by Khan el al. (2017c), and is not repeated here due to space limitations.

Experimental programme

Five rectangular RC beams of dimensions 100 mm wide, 120 mm depth and 900 mm overall span with 800 mm effective span are tested for failure modes and loads, and to identify relation between shear and peeling cracks. The 43-grade ordinary Portland cement is used in the concrete. The beams are reinforced with 6 mm diameter bars at the tension side; and beam specimens are marked with 20 mm slots to capture actual location and propagation of cracks. Figure 2(a) shows detailing of a test specimen BES 1 × 350 for a 4-point bending problem. Stepwise gradual loading (that is, in increments of 10 kN) is applied to mark any crack formations at each pause between the load increments up to failure, for example, Figure 2(b) shows a beam specimen BES 0.6 × 450 to have initiated peeling that propagated into shear-peeling failure. Sikadur 30 LP (epoxy) used are extracted from manufacturer data sheet (Sika, 2020). Beams tested are listed in Table 1: BES 1 × 350, BES 1 × 430, BES 0.6 × 430, BES 1 × 700 and BES 0.6 × 700. Where the nomenclature of plated beams is: BES t × L = with specimen abbreviation from left to right: B = beam, E = epoxy-bonded plate, S = steel plate, t = plate thickness and L = plate length). The units for the numbers used in nomenclature are indicated in Table 1.

Figure 2.

Beam detailing, experimental setup and results (a) Schematic of test beam BES 1 × 350 (b) Beam BES 0.6 × 430 showing shear-peeling failure.

Arduini and Nanni (1997) and Klamer (2009) show that the specimen size (or prototyping) is shown to have no impact on the response of plated beam to loading. It is because the RC beam or section is to be designed based on design standard such as BS:5400 (1990) or IS: 456 (2000), etc. and specifications on minimum dimensions are adhered to. The beam size is selected to cover wide range of geometrical properties and designed to cover beams failing in shear-peeling (that is, a peeling crack developing into shear-block failure from plate-end). It is seen that shear-block and peeling failures can be closely related; and if the beam is without shear stirrups, the peeling stresses at plate-end may develop into shear-block failure (referred as, shear-peeling failure) as indicated by Khan (2021a) (refer, Figure 2(b)). However, standalone shear-block failure (which is not a premature failure) is out of the scope of the paper.

Results

Many studies have come to a common consensus for outlining the role of some key parameters (Oehlers and Moran, 1990; Oh et al., 2003b; Raoof et al., 2000; Raoof and Zhang, 1997), such as the increase in plate thickness or plate curtailment from beam support leads to peeling rather than debonding failure. However, the role of other parameters such as the depth of beam in general, percentage and distribution of tensile rebars showed no clear pattern on affecting the modes of failure (Heathcote, 2004). Therefore, it is imperative to develop accurate numerical model and evaluate required bond parameters. Figure 3 shows validation studies of tested specimen(s) on load-deflection behaviour and prediction of cracks.

Figure 3.

Validations and finite element model results (a) Load versus displacement validation (b) PE11 (covercrete at plate-end) and S11 (rebar above peeling location) developments in FE model of beam BES 0.6 x 430.

As peeling is highly catastrophic, the beams failing in peeling are to be simulated first to drive the corresponding predictive equations; following which, the cases of beams failing in debonding are covered; these are then compared with beams failing simultaneously in both modes of failure.

Initial stiffness of interface

Studies (Macdonald and Calder, 1982; Oh et al., 2003b) have suggested that the modulus of elasticity of adhesive has the direct impact on crack formations (both transverse and flexural), where Khan et al. (2017b) have specifically identified that the stiffness greater than 1 GPa/mm would not show noticeable changes in the rate of stress transfer at interface. The effect of stiffness can be directly reflected on the development of transvers strains in plate, for example, Figure 4. To relate elastic modulus and initial stiffness of CZM, the constitutive thickness has been considered unity, evaluating an equivalent initial stiffness for a given geometric thickness of adhesive (plate-concrete connecting medium) in terms of adhesive properties.

Figure 4.

Effect of stiffness on strain distribution, predicting best fit for equation, and comparison with equation.

Using the effect of Poisson’s ratio $ϑ$ , the general relation between shear stiffness $K_{I I}$ and normal stiffness $K_{I}$ is

K_{I} = 2 K_{I I} (1 + ϑ)

(16)

Figure 4 shows effect of interfacial stiffness on plate strain distribution for predicting best fit for beam P13 (obtained at a small initial load). The equation of the interface in mode-II direction is arrived

K_{I I} = 0.65 \frac{G_{a}}{t_{a}} + 182.7

(17)

Shear modulus $G_{a}$ is given in MPa and adhesive geometrical thickness $t_{a}$ in mm. The equation, thus developed, is further compared against actual properties (Figure 4).

Peeling

As per BS:8110 (1985) and IS: 456 (2000), in case of static loading, the initial modulus of concrete in tension remains same as for compression. Therefore, this step does not require evaluation of general equation for initial stiffness. For tensile strength $f_{c t}$ (responsible for crack initiation) and crack energy $G_{f c}$ (responsible for complete crack), the results are compared with experiments for the load of crack initiation and number of flexural cracks, respectively.

As discussed before, peeling being a direct consequence of flexural crack at plate-end, therefore, it depends on the properties of concrete in tension as demonstrated by Khan et al. (2017b). This required us to directly correlate the tension behaviour of CDP model with xFEM cohesive model. Figure 3(b) shows formation of plastic strain (PE11) at plate-end leading to strain localization in rebar; this causes rebar to yield and a drop in a corresponding load versus deflection plot (seen in Figure 3(a)). Now to correlate this with the simple case of elasto-plastic softening material models of xFEM and CDP, the fracture energy $G_{f c_X F E M}$ (after crack initiation) is equated to crack energy $G_{f c_C D P}$ (plastic strains, after crack initiation) of Concrete Damage Plasticity (CDP) model as

G_{f c_X F E M} = G_{f c_C D P} + 0.5 \in b f_{c t}

(18)

where

\in

is elastic strain,

b

is element characteristic width and

f_{c t}

is failure strength in tension. The second term in the above equation is the energy utilised to initiate a crack.

Simulations are performed on combinations of bond parameters ( $f_{c t_m a x}, G_{f c_C D P}$ ) falling within a practical range following a procedure described next.

Bond Strength varied

In case of static loading, BS:8110 (1997) has recommended that the tensile strength magnitude is up to 10% of the cubic strength $f_{c u}$ in compression, this is after considering factor of safety. Where IS: 456 (2000) has completely ignored the contribution of tensile strength. Studies (Khan et al., 2017b; Zhang et al., 1995) have identified that the tensile strength is a key parameter that effects the capacity and failure type of plated beam.

Hence, the load at crack initiation is compared against different values of tensile strength for available fracture energies. For instance, Figure 5 shows the effect of $f_{c t_m a x}$ on load ratio (FEM to experimental); beams (P1 and F31) are simulated for tensile strengths of 2 MPa, 3.5 MPa and 4 MPa to calibrate against the corresponding experimental peeling loads (of 78.9 kN and 182 kN, respectively). For these values of $f_{c t_m a x}$ , a total of nine combinations are performed on any given beam against $G_{f c_C D P}$ of 0.014, 0.042 and 0.15 MPa-mm.

Figure 5.

Effect on load of peeling (a) Beam P1 (b) Beam F31.

Estimated values of $f_{c t_m a x}$ for beam P1 and F31 are found to vary at calibration and are indicated on the corresponding figures.

Crack energy varied

Azad et al. (1989) have identified some key parameters that can affect the value of crack energy (in reinforced concrete), such as the area of tension steel, location of tensile rebars and further development of crack. For standard design as per BS:8110 (1985), the strain at failure for reinforced concrete can reach about 30 times that of elastic strain $\in$ . However, in experimental observations a crack is usually considered to occur at a mode-I strain of 0.001 (related with pure flexural crack) and is marked on the loaded beam as it becomes visible to a naked eye. Therefore, in our analysis, a complete crack is taken at 30 $\in$ or 0.001, whichever is lower. It is also noted that, in strain distribution, clear peaks are visibly the crack formations (that can be counted), while non-peaks are due to strain redistribution (effect of increased fracture energy).

For a given beam, number of flexural cracks are compared at peeling failure for the combinations of bond parameters ( $f_{c t_m a x}$ , $G_{f c_C D P}$ ) as suggested before. For chosen $f_{c t_m a x}$ , the effect of $G_{f c_C D P}$ is plotted for beams P1 and F31, respectively, in Figures 6 and 7; the boundary conditions and plate length are also indicated. For example, the experimental observations noted the crack numbers to be 7 and 15, respectively, for P1 and F31; they are matched with the numerical observations as follows.

Figure 6.

Beam P1, effect on flexural cracks (a) $f_{c t_\max}$ = 2.0 MPa (b) $f_{c t_\max}$ = 3.5 MPa (c) $f_{c t_\max}$ = 4.0 MPa.

Figure 7.

Beam F31, effect on flexural cracks (a) $f_{c t_\max}$ = 2.0 MPa (b) $f_{c t_\max}$ = 3.5 MPa (c) $f_{c t_\max}$ = 4.0 MPa.

The number of cracks captured through numerical model are indicated on the corresponding figures using matching colour codes. For example, in Figure 6(a) the crack numbers on half-beam are noted at 7, 3 and uniform distribution for $G_{f c_C D P}$ of 0.014, 0.042 and 0.15 MPa-mm, respectively.

Equations

The estimated values of $f_{c t_m a x}$ and $G_{f c_C D P}$ for all beams vary, signifying on the influence of the choice of material and geometrical parameters. For example, for beams P1 and F31, the estimated values (obtained from different combinations) are plotted as shown in corresponding Figures 8 and 9. The intersection (or through projection) is considered to be the best value that can fit well to generate accurate results for the corresponding beam.

Figure 8.

Obtaining best value at clear intersection.

Figure 9.

Obtaining best projected value.

The next step is to related the best values ( $f_{c t_m a x}, G_{f c_C D P}$ ) with the properties of concrete in tension. The crack in concrete is mainly dependent on its tensile strength $f_{c t}$ and crack energy $G_{c r}$ . The $f_{c t}$ is considered to be at 10% $f_{c u}$ and $G_{c r}$ is considered equivalent to 30 $\in$ so as to relate our observations with the assumptions adopted by design code BS:8110 (1985).

The equations relating $f_{c t_m a x}, G_{f c_C D P} and G_{f c_X F E M}$ , $f_{c t}$ and $G_{c r}$ are proposed

f_{c t_m a x} = a_{1} f_{c t}^{a_{2}} G_{c r}^{a_{3}}

(19)

G_{f c_C D P} = b_{1} f_{c t}^{b_{2}} G_{c r}^{b_{3}}

(20)

G_{f c_X F E M} = C_{1} f_{c t}^{c_{2}} G_{c r}^{c_{3}}

(21)

To determine the coefficients ( $a_{1} - a_{3}, b_{1} - b_{3} and c_{1} - c_{3}$ ), multiple regression analysis was used. The above proposed equations are rewritten in linear form by taking the natural logarithm on both sides. Then, the values of coefficients that give the best fit to the data available were obtained. The final equations are

f_{c t_m a x} = 0.02 f_{c t}^{2.26} G_{c r}^{- 0.85}

(22)

G_{f c_C D P} = 1.14 f_{c t}^{- 0.60} G_{c r}^{0.96}

(23)

G_{f c_X F E M} = 2.56 f_{c t}^{- 1.07} G_{c r}^{0.97}

(24)

A same correlation coefficient $R^{2}$ = 0.99 is obtained for all the above equations. Hence, $G_{f c_X F E M}$ is also obtained in terms of material properties.

Debonding

After optimising the beam for peeling, the parameters that would affect stress-strain distribution are interracial bond parameters, defining CZM. To evaluate parameters of debonding (dominated by failure in transverse shear), considering the beams with plates extended to supports are more suitable as the bending stresses reduce towards the ends; thereby, shear stresses dominate. With the formation of flexural crack, higher bending stresses would result into greater transverse strains on the fibre of plate (refer, Figure 10).

Figure 10.

Four-point loading, (a) axial stress in plate, (b) shear stress at interface.

Within elastic limits, the maximum transverse stress required to initiate debonding is shown to be directly proportional to the distance of plate-end from support, established by De Lorenzis and Zavaris (2009).

Bond strength and crack energy

To achieve parameters of debonding and taking $K_{I I}$ from equation (25), simulations are performed for the combinations of three values of $τ_{m a x}$ (1, 2, 4 MPa) and three values of $G_{f c_C Z M}$ (0.014, 0.042 and 0.15 MPa-mm); falling within a practical range (Charif, 1983; Heathcote, 2004; Jones et al., 1982, 1988; Sikadur, 2012). The parameters of concrete in tension are gathered from equations (27)–(28) for capturing flexural (and peeling) cracks; it is important to consider them at this stage as discussed before that the formation of flexural cracks are also responsible to the formation of interfacial stresses. It is noted that flexural crack opening is a mode-I opening, therefore, mode-I opening strength is retained for $f_{c t_m a x}$ from equation (27).

The effect of bond parameters at the interface are checked against load of debonding (Figure 11) and transverse strain distribution along the longitudinal length of the plate (Figure 12) or plate-end displacement as discussed later.

Figure 11.

Beam ORB2.

Figure 12.

Beam P13, transverse strain distribution (a) $G_{f c_C Z M}$ = 0.014 MPa-mm (b) $G_{f c_C Z M}$ = 0.042 MPa-mm (c) $G_{f c_C Z M}$ = 0.15 MPa-mm.

At this stage, the validation with transverse strain has been made as soon as the cohesive element(s) starts to damage or significant flexural cracks have developed, so that it can reflect the response of interface beyond elastic limit compared to what was done previously at a smaller load for finding the equation for initial stiffness. In support of this argument, for beam P13, Figure 13 indicates that the maximum load reached for $τ_{m a x}$ of 1 MPa and $G_{f c_C Z M}$ of 0.014 MPa-mm is 68.3 kN, which is lower than the experimental data of strain distribution available at 75 kN, yet greater than the data available at 50 kN. In this connection, it is seen that cohesive element starts to damage at 26.2 kN, signifying that the interface is now beyond its elastic stage (initial stiffness) and FE model can now be compared with experimental data for transverse strain distribution available at or beyond the load of 26.2 kN. This condition is required for the cohesive parameters that are being identified.Where the transverse strains are not available in literature for validation, displacement of plate-end is also found to be an important parameter that gets effected by the choice of interfacial bond parameters (Figure 14).

Figure 13.

Beam P13, load of interfacial crack.

Figure 14.

Beam ORB2, combinations effecting plate-end displacement.

The estimations of interfacial shear stresses are drawn next (Figure 15).

Figure 15.

Beam ORB2.

The relation of bond parameters with the displacement of plate-end may be attributed to the fact that, the interfacial shear stresses develop along the length of the plate available in the shear span; this is more prominent for beams having their plates extended towards the supports. In determining debonding, this length of the plate is the bond length.

Equation

Best fit values are obtained through a similar procedure as used before for peeling. For any beam that does not show a clear intersection of lines either side of the extremes, an average value is obtained towards the most logical end; for example, beam URB4 (Figure 16).

Figure 16.

Obtaining best logical average value.

Following this, the best values are related with key parameters that define the crack initiation of CZM, as debonding initiates. For practical reasons and ease of availability of experimental data, shear modulus of adhesive $G_{a}$ and tensile strength $f_{c t}$ of concrete are considered. The initial modulus of adhesive is directly related to stress redistribution at initial stages to cause failure through concrete (largely) or adhesive (may be rarely), and this fits the overall assumption made in related literature (Khan et al., 2017c; Obaidat et al., 2013) in relating the key parameters of CZM with the properties of concrete and adhesive at their common interface. As cited before, $f_{c t}$ is responsible for flexural cracks, which can develop into debonding crack.

The following equations are proposed that relate $τ_{m a x}$ and $G_{f c_C Z M}$ (best values obtained from plots) with the material defining factors $f_{c t}$ and $G_{a}$

τ_{m a x} = d_{1} f_{c t}^{d_{2}} G_{a}^{d_{3}}

(25)

G_{f c_C Z M} = e_{1} f_{c t}^{e_{2}} G_{a}^{e_{3}}

(26)

For base values of all beams, $f_{c t}$ is taken at 10% $f_{c u}$ as before, and shear modulus $G_{a}$ as available experimentally (or through adhesive manufacturer data sheet). As before, the equations are rewritten in linear form by taking the natural logarithm on both sides so as to evaluate for coefficients $(d_{1} - d_{3} and e_{1} - e_{3})$ through multiple regression analysis. This yields the following equations

τ_{m a x} = 0.18 f_{c t}^{1.18} G_{a}^{0.04}

(27)

G_{f c_C Z M} = 2.35 f_{c t}^{0.03} G_{a}^{- 0.63}

(28)

Corresponding $R^{2}$ values are 0.85 and 0.97, respectively.

It should be noted that, although the equations for debonding are in terms of basic parameters as for peeling; however, they should be used in conjunction with the equations for peeling in a series as generated. That is, a sequence should be maintained; it is because the model is arrived through a stepwise procedure. Likewise, the equations for peeling should not be used as standalone, rather equation for interfacial stiffness $K_{I I}$ should also be used with them.

Bond-slip curves and validation

A comparison of experimental results with the numerical results is made using parameters of bond action (for xFEM, CDP and CZM) thus obtained, is shown next in Figure 17. Comparison is made on bilinear and exponential bond-slip behaviours implemented on the interface.

Figure 17.

Beam P13, transverse strain distribution with different crack evolution laws at interface.

It is clear that the results obtained using bilinear behaviour are a closer match to experimental results. In addition, at a relatively high plate-end displacement, debonding is significantly delayed when exponential crack evolution is assumed. Due to this reason, with exponential bond-slip law, peeling is wrongly observed for beams that should first fail in debonding. Therefore, bilinear bond-slip behaviour (linear crack evolution) is assumed for further validation.

The onset of debonding is shown for beam P13 in Figure 18. It shows strain distribution with the progress of debonding, and indicates that at about 95% of the value of SDEG (the cohesive damage output indicator/variable in ABAQUS) a significant damage is already done to interface, as E11 starts to drop. As the interface at the plate-end damages, the plate strains elsewhere start to increase up to debonding at plate-end actually starts. After a substantial amount of debonding is observed, strains elsewhere also started to fall; indicating that the direction of interfacial damage is inwards.

Figure 18.

Progress of debonding in beam P13.

Number of flexural cracks also seem to be in close agreement with literature; for example, beam F31 (Figure 19). In Beam F31, the xFEM peeling crack is clearly seen to occur before debonding through CZM (Figure 20).

Figure 19.

Flexural cracks in Beam F31.

Figure 20.

SDEG output (a) Peeling crack (b) Peeling and debonding (c) Pure flexural cracks.

Figure 20(c) shows that multiple crack formations are achieved through both xFEM and CZM models. While Table 3 highlights on the accuracy of results in terms of load and mode of failure. It is clear that the calibrated values from equations established in this work yield more accurate results in terms of load and mode of failure.

Table 3.

Load and mode of failure.

Specimen	Experiment Load (kN)/Failure mode	FEM (with calibration) Load (kN)/Failure mode	FEM (without calibration) Load (kN)/Failure mode
URB4	55.9/D	54.3/D	50.0/D
URB5	49.6/D	48.8/D	42.0/D
ORB2	75.0/D	74.1/D	73.0/D
P1	78.9/P	78.0/P	77.0/P
P2A	74.8/P	73.2/P	75.6/D
P3A	75.7/P	74.0/P	72.5/P
P7	89.7/P	90.4/P	87.0/D
P8	94.9/P	92.0/P	89.0/D
P9	89.4/P	90.1/P	85.4/P
P10	105.4/P	107.0/P	99.3/D
P11	90.0/P	90.4/P	87.8/D
P12	90.0/P	89.0/P	87.4/D
P13	120.4/D	121.0/D	117.0/D
P14	100.3/P,D	102.8/P,D	96.3/P
P15	104.5/D	102.0/D	98.0/D
P19	109.4/P	109.7/P	100.2/P
P20	120.4/D,P	120.0/D,P	117.0/P
P21	119.6/P	118.0/P	111.6/P
F31	182.0/P	180.1/P	170.4/D
BES 1 × 350	28.9/P	26.7/P	21.0/P
BES 1 × 430	31.4/P	31.3/P	25.04/P
BES 0.6 × 430	22.56/P	20.6/P	16.01/D
BES 1 × 700	52.0/Compression, shear	51.9/P	45.51/D
BES 0.6 × 700	40.21/Compression, shear	41.1/P	35.51/D

D: debonding; P: peeling; FEM: from finite element model.

Effect of parameters

After the model is developed, the importance of the choice of bond parameters, such as stiffness, strength and fracture energy of interface, is indicated on affecting the capacity of the beam and mode of failure. These parameters also effect the factors that are directly related to the response of crack formation, such as plate strain and flexural strain are also studied.

The initial stiffness of the interface directly affects the stress transfer for a change in relative displacement at the interface; therefore, this impacts the load of initiation of debonding crack (at the interface). For example, the behaviour of the beam S41 [8] from wider literature is studied where required data (such as plate strain distribution) is not available to be considered for developing equations of bond parameters using the approach in this paper. The beam S41 is failing in debonding; the percentage change in the load of initiation of debonding is plotted against the percentage variation of $K_{I I}$ in Figure 21(a). The variation is made from 25% to 200% of reference value for the beam S41; the values fall within the range indicated in Table 2. The reference values for $K_{I I}$ and load of initiation/appearance of crack are 991.4 MPa/mm and 93.0 kN, respectively. It is evident that as the initial stiffness of the interface increases, load of crack initiation at the interface reduces almost linearly.

Figure 21.

Effect of key parameters on load and mode of failure, Beam S41 (a) Effect of transverse stiffness on load to initiate debonding crack (b) Effect of strength of interface on beam capacity and mode of failure (c) Effect of fracture energy on capacity, deflection and strains (d) Effect of fracture energy on brittleness of failures.

The effect of interfacial strength $τ_{m a x}$ is also studied towards its role in effecting the capacity of the beam and mode of failure. For example, referring Figure 21(b), for beam S41 with reference value of 2 MPa, it indicates that decreasing the strength by 50%, it drastically reduces the ultimate capacity of the beam as well as the load of first mode of failure (which remained to be debonding through plate-end). The ultimate capacity of the reference beam is 134.3 kN and the load at first failure (plate-end debonding) is 128.0 kN; and it is also possible that a first failure may occur at a load close or equal to the ultimate capacity of the beam. However, an increase in strength by 100% has shown an insignificant impact of the ultimate capacity as well as on the load of first mode of failure; however, a change in mode of first failure is observed to be peeling than debonding. This may be attributed to the fact that, with an increased interfacial strength, the required load to cause debonding would now increase; thereby, before a beam might fail in other mode of failure before a debonding capacity is reached.

The fracture energy at the interface $G_{f c_C Z M}$ would be critical to record propagation of cracks at interface. This energy is liberated due to formation of pure interfacial cracks and the localised interfacial cracks as a consequence of flexural cracks in beam. The formation of flexural cracks is directly reflected through the development of localised strains in plate. Therefore, in addition to measuring the effectiveness of this parameter on ultimate capacity, it is also judged through its effectiveness on the formation of flexural strains and plate strain with the deflection of the beam, all at midspan (refer Figure 21(c)). The impact of the variation of fracture energy is largely a straight line for most of the cases, except some fluctuations in case of flexural strains. Such that, for higher values of fracture energy, the fluctuation in flexural strain is directly reflected on plate strain; signifying the importance of strain localisation. However, no such effect is noted for lower fracture energies. This may be attributed to the fact, that for higher fracture energies the debonding is delayed up to higher strain levels (hence, causes strain localisation), while for lower fracture energies the debonding propagates earlier and thereby the strain localisation is avoided. The effect is also measured against Brittleness of failure ( $F_{1} / F_{u}$ × 100) for debonding and peeling (Figure 21(d)); where $F_{1}$ is the load at first mode of failure and $F_{u}$ is the ultimate capacity of the beam. Higher values indicate increased brittleness of that particular mode of failure from cracking to ultimate capacity, that is an increase in catastrophic nature. The values of Brittleness of failure for peeling are more than that for debonding, signifying on the fact that peeling is relatively more catastrophic than debonding. The empty dots in Figure 21(d) indicate first mode of failure type. Therefore, it is clear that an increase in fracture energy with 100%, although reducing the Brittleness of failures, has also caused peeling. In addition, for peeling being more catastrophic than debonding, there is also a sudden shift (increase) in Brittleness of failure by about 5%. This can be attributed to the fact that higher fracture energy at interface will maintain the bond, while the strain developments elsewhere in the beam would progress and lead to a corresponding failure type.

Conclusion

Recent studies have primarily focussed on numerically predicting mode-II debonding failure as a premature failure in RC beams strengthened in flexure; and in this paper, a numerical model is generated using xFEM for flexural cracks and CZM for shear cracks. Parameters of constitutive models are identified to model both possible premature failures: debonding as well as peeling; this allowed to conduct an all-inclusive study on plated beams. The method presented here demonstrates a simple procedure, for using xFEM model with CDP model for simulating peeling failure and CZM model for debonding mechanism, particularly to identify cracks that can be modeled using traction-separation constitutive laws. Covering a wide range of geometrical and material parameters, 19 steel-plated RC beams are considered from the literature. The parameters of constitutive modes are then obtained by fitting FEM results to experimental results for the corresponding beams either failing in peeling or debonding. For peeling, data considered consist of peeling load and the corresponding number of flexural cracks. For debonding, data consisted is debonding load, transverse strain distribution along the longitudinal direction of plate or plate-end displacement. Usually, this is available as basic information (data) extracted during the bending test of plated beams.

The parameters defining the material model describing peeling action are tensile strength and crack energy of concrete (for CDP model) and/or initial modulus, strength and fracture energy (for xFEM model). While the parameters defining the constitutive model for bond action between plate and RC beam are initial stiffness, strength and fracture energy of the composite interface. Together, the failure strength and crack energy define the cracking surface. The parameters of constitutive models are expressed in terms of material properties and free of geometrical factors:

The equations (22) and (24) correspond to traction-separation law of xFEM model, equations (22) and (23) to CDP model and equations (17), (27) and (28) to CZM. This study is first approach to develop constitutive models that deal with, (1) both premature failures (peeling and debonding) and, (2) on non-linear materials. Therefore, the research done in the manuscript is systematic and comprehensive. This procedure proposes a framework that can also be adopted for beams plated in flexure with materials other than steel, such as FRPs.

The accuracy of the simulations is demonstrated after adopting to these relations. The interface is also checked against bilinear and exponential bond-slip laws; where former law is found to be a closer match with experimental results on strain, cracks, failure mode and load. Comparing with test specimens, the use of exponential crack evolution law significantly increases load of debonding and plate-end displacement, and therefore is not recommended for use with plated RC beams.

The application of model is shown through parametric study on beam from outside the literature considered for developing equations of bond parameters. It is evident that the role of key bond parameters is significant in determining the capacity of the beam and the mode of failure.

Footnotes

Acknowledgements

The author is extremely thankful for the invaluable support from his parents (father and late mother). The author is thankful to the anonymous reviewers of this paper that helped improve the manuscript.

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: The work presented in this article is financially supported by DST-SERB project (Grant No. DST-SERB ECR/2017/000908), awarded to author Mohammad Arsalan Khan (PhD, MSc Engg., ACI (Faculty member, USA), IStructE (Graduate, UK), MIES (Chartered with Institution of Engineers Singapore), IACM (Spain), IET (UK)] as P. I.

Data availability statement

Data generated or analyzed during the study are available from the corresponding author by request.

ORCID iD

Mohammad Arsalan Khan

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Bond parameters for peeling and debonding in thin plated RC beams subjected to mixed mode loading – Framework

Abstract

Keywords

Introduction

Materials, models and method

XFEM and discretization

CZM and discretization

Experimental programme

Results

Initial stiffness of interface

Peeling

Bond Strength varied

Crack energy varied

Equations

Debonding

Bond strength and crack energy

Equation

Bond-slip curves and validation

Effect of parameters

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

Data availability statement

ORCID iD

References