Bond-stress and bar-strain profiles in RC tension members modelled via finite elements

Engineering Structures 194 (2019) 138–146

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Bond-stress and bar-strain profiles in RC tension members modelled via finite elements

T

⁎

Ronaldas Jakubovskisa,b, , Gintaris Kaklauskasa a b

Department of Reinforced Concrete Structures and Geotechnics, Vilnius Gediminas Technical, University/VGTU, Sauletekio 11, Vilnius 10223, Lithuania Laboratory of Innovative Building Structures, Vilnius Gediminas Technical, University/VGTU, Sauletekio 11, Vilnius 10223, Lithuania

A B S T R A C T

Concrete-reinforcement interaction is a complex issue in the mechanics of reinforced-concrete structures all the more because the reinforcement, the concrete layer at the interface and the undamaged concrete are involved. With reference to the modelling of short tension members embedded in the concrete, three alternative approaches are considered in this paper: a two-dimensional rib-scale approach (Model-1); a three-dimensional approach with the simplified representation of the ribs (Model-2); and a three-dimensional rib-scale approach (Model-3). The effectiveness of each approach is checked against the results yielded by a number of tests on RC prisms in tension reinforced with a single bar, where the strain distribution in the bar was measured. Special attention is devoted to bond deterioration close to the cracks. The three-dimensional rib-scale model is shown to be effective in describing the strain profiles in the reinforcement at different load levels, and rib height appears to be the most important parameter governing the calibration of the model. In addition, the role played by the net cover of the bar is thoroughly investigated.

1. Introduction Since its invention, reinforced concrete has established itself as a highly successful composite material for structural applications. The interaction between concrete and reinforcement bars, often referred to as bond, is essential in developing the required performance of the structure. Although the mechanics of bond behaviour is quite well understood by now, the various mechanisms still need to be properly quantified due to the inherently complex nature of this phenomenon [1,2]. The bond zone undergoes complex physical and mechanical actions such as the formation of secondary and splitting microcracks, local crushing and shearing-off the concrete in front of the ribs. It was experimentally shown [3,4] that cracks propagate around the deformed bar, forming a specific pattern of inclined concrete struts separated by the secondary cracks (Fig. 1). These struts transfer the force to concrete by the bearing between the face of the rib and uncracked concrete around the bar transmitting the longitudinal and radial force components. However, near the crack face, intense deterioration of concrete occurs, the struts are bent and crushed resulting in the reduction of forces transferred by the bond. Although the length of the weakened zone of the bond (also called the damage or bond deterioration zone) is relatively small, it may significantly influence the deformations of RC structures. The study by Bernardi et al. [5] has shown that the stiffness of tensile RC members decreases with increasing load because of the

expanding zone of bond deterioration. Gribniak et al. [6] have noted that due to the internal cracking, concrete undergoes intense deformation resulting in non-planar sections, especially in the areas close to the reinforcement. In a similar way, damage of bond also affects the formation of cracks due to the increase in the transfer length, crack spacing and crack width [7]. Length of the damage zone is generally related to the bar diameter. The extent of the deterioration is commonly related to the distance from the crack [8,9]. In a recent study, Jakubovskis [10] has shown that bond deterioration length is also dependent on the magnitude of the reinforcement strain in the crack. However, the mentioned models may be considered as intuitive, lacking the physical evidence of the phenomena. In fact, the length of the bond deterioration zone can hardly be measured experimentally due to the highly complex phenomena of internal cracking of concrete and a significant number of variables involved. Damage of concrete in front of the ribs may be considered as the main factor causing deterioration of the bond. In this respect, numerical methods can serve as an effective tool to model the formation of microcracks and to quantify the actual stress and strain state in the reinforcement and concrete. Several studies were performed to model the bond phenomenon at the structural level using finite element [11–13] or stress transfer algorithms [14,15]. Although these approaches well predict the distribution of bond stresses along the member, they are not able to capture the internal cracking or bond deterioration phenomena. More refined numerical rib-scale models were used to study the

⁎ Corresponding author at: Department of Reinforced Concrete Structures and Geotechnics, Vilnius Gediminas Technical, University/VGTU, Sauletekio 11, Vilnius 10223, Lithuania. E-mail addresses: [email protected] (R. Jakubovskis), [email protected] (G. Kaklauskas).

https://doi.org/10.1016/j.engstruct.2019.05.069 Received 25 January 2019; Received in revised form 16 April 2019; Accepted 22 May 2019 Available online 27 May 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.

Engineering Structures 194 (2019) 138–146

R. Jakubovskis and G. Kaklauskas

primary cracks

RC tensile member

secondary cracks

N

N

deterioration of bond

uncracked concrete

secondary cracks

internally cracked concrete N+ΔN circumferential tensile ring

N+ΔN

N reinforcement N force transmited by bond action

primary crack

secondary cracks

Fig. 1. Bond mechanics in tensile RC member.

Technical University. Three identical RC tensile specimens instrumented with electrical strain gauges were tested. Such a number of identical specimens was chosen to have a minimal statistical basis for reliable calibration of the numerical predictions. The sources of the experimental discrepancies are due to the internal defects of concrete, measurement errors or technological effects (different casting or compaction conditions). The detailed description of the experimental program is given in [21]. Concrete prisms 150 × 150 × 270 mm in size were reinforced with ribbed bars of 20 mm in diameter. The arrangement of test apparatus is shown in Fig. 2a. The Specimen geometry and strain gauges positions are shown in Fig. 2b. Tensometric strain gauges with 10 mm gauge length were spaced at 30 mm intervals resulting in 11 measurement points along the bar. The strain gauges were placed inside the reinforcement bar avoiding reduction of the bond performance. Following the technology used at Durham University [22], a reinforcement bar was longitudinally cut and glued after making a 2 × 10 mm groove in each half and placing the strain gauges and the required wiring inside the bar (Fig. 2c). The specimens were demoulded three days after casting and held in water to minimize the shrinkage effect.

reinforcement and concrete interaction at a micro-level. Several approaches were developed for two-dimensional representation of the problem [16,17]. It was shown that discretization of bars including ribs allows modelling the complex formation of secondary cracks. The main limitation of the two-dimensional approaches is their inability to take into account the circumferential tensile stresses in concrete around the reinforcing bar being responsible for the formation of splitting cracks that, in turn, result in a marked reduction of the bond stresses [18]. These effects can be modelled by three-dimensional approaches [19]. However, such refined models are highly computationally expensive due to the extremely fine finite element mesh around the ribs. In addition, the obtained results might be sensitive to the assumed simplification of the geometry of the reinforcement involving calibration of the model with representative experimental data. Thus, the selection of the modelling approach and adequate modelling of concrete damage in front of the ribs are the key aspects in achieving adequate numerical representation of the interaction between reinforcement bar and concrete. In the current study, three different numerical approaches are considered to model the bond in terms of reinforcement strain profile in the tensile RC member. A particular emphasis is made on the bond deterioration in the vicinity of cracks. Test results of three identical 270 mm long tensile RC members with recorded reinforcement strains along the member served as a reference to evaluate the numerical approaches. The study has shown that the three-dimensional rib-scale numerical model can adequately predict the bond stress distribution and formation of transverse and longitudinal microcracks, also quantifying the degree of bond deterioration and damage of concrete surrounding the reinforcement. The refined 3D model requires only a few empirical parameters and may be upscaled to increase the computational efficiency.

P

strain gauges

1 2 3 4 5 6 7 8

2. Experimental study 15 15

9

To study the reinforcement and concrete interaction, tensile tests of RC prisms instrumented with internal strain gauges were performed. Such a test set-up was chosen as an alternative to the classical pull-out test procedure due to a more realistic representation of stress and strain state in the concrete [20]. Moreover, the measured reinforcement strain profile allows to assess the bond stresses and slip along the reinforcement. The experimental program was performed at Vilnius Gediminas

10 11

270 mm

gauge No 30 30 30 30 30 30 30 30 30 30

15 15

strain gauges

P

Fig. 2. Experimental RC prisms: (a) test set-up; (b) location of strain gauges in the specimens; (c) internally instrumented bars. 139

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1200

strain, ×10-6 1 2

S-20-tz-1 S-20-tz-2 S-20-tz-3

gauge No

1000 1

3

10 11

4

8 5

3 1

7

6

7

10 11

3

200 1

2

30

30 kN

9

3 0

50 kN

9 8

5 4

0

6

4

2

400

70 kN

9

2

800

600

11

10

4 60

90

5

6

7

5

6

7

120

150

8 10 9

8 180

210

240

11

10 kN

270 x, mm

Fig. 3. The measured reinforcement strain profiles at four load levels.

points were used. The average element size was taken 8 mm with five times refinement at the reinforcement and concrete interface. For concrete, the SBETA model offered by ATENA was used, which is based on the smeared crack and damage approaches. Tensile strength (fct = 3.05 MPa), elasticity modulus (Ec = 35.7 GPa) and fracture energy of concrete (Gf = 76.4 N/m) were calculated from the experimentally established compressive strength of concrete (fc = 45.4 MPa) using Model Code [9] provisions. Reinforcement was modelled as a linear elastic material with experimentally determined elasticity modulus (203 GPa). Perfect bond of reinforcement to concrete was assumed in the rib front of reinforcement. The remaining contact area was modelled using the “normal“ bond conditions adopted from Mithou et al. [25]. Interface properties of the “normal“ bond are presented in Fig. 4 where ft and c are the tensile and cohesion strength of bond; Knn and Ktt are the normal and tangential stiffness of bond; φ is the friction coefficient. The typical calculated longitudinal strain profile of concrete is shown in Fig. 5a. Grey areas in this figure represent the strain fields of tensile concrete that are below the cracking strain. It may be observed that concrete strains localize in front of the ribs, showing the location of inclined internal cracks. The formation of cracks near the ribs well complies with the experimental evidence [3]. Due to internal cracking, concrete undergoes an intense deformation resulting in non-planar sections, especially in the areas close to the reinforcement. The strain profiles in the reinforcement obtained numerically for four load levels are plotted in Fig. 5b, together with the experimental values. Good agreement between the calculated and experimental strain profiles was obtained only at the initial load level. With an increase in load, the internal cracks spread and merged into the macro-crack that noticeably affected the shape of the reinforcement strain profile. Local strain peaks formed at the locations of such macro-cracks. However, the experimental data with rather smooth strain curves at all load levels does not support the numerical results. Most likely this is due to the limitation of the two-dimensional model to represent actual volumetric stress and strain state in the concrete. In the two-dimensional modelling approach, the growth of cracks might be overestimated as it is not arrested by the surrounding concrete. A more realistic three-dimensional representation is required to model the development of circumferential stresses and adequate internal cracking of concrete.

The specimens were tested 28 days after casting using a hydraulic machine with displacement control (Fig. 2c). The load capacity of the machine has limited the maximal strain in the reinforcement of 0.0012. The concrete compressive strength of 150 mm cubes and 300 mm cylinders was fc,150 = 45.4 MPa and fcyl = 40.4 MPa, respectively. The measured elasticity modulus of the reinforcement was 203 GPa. The strain profiles measured in the three specimens at four load levels are plotted in Fig. 3. As may be noticed, good agreement between the three tests was obtained what implied that accidental experimental errors were minimal. The test results are, therefore, reliable and may be used to assess the soundness of the numerical techniques. 3. Numerical modelling Three different numerical models were used to study reinforcement strain profile in the tensile RC prisms: two-dimensional rib-scale Model1, three-dimensional cylindrical Model-2 and three-dimensional ribscale Model-3. The two-dimensional Model-1 was generated simplifying concrete and reinforcement to the planar finite elements. To avoid the definition of the empirical bond stress-slip relationships, ribbed geometry of the bar was modelled with the different concrete–steel interface characteristics at the rib face and between the ribs. Model-2 was generated using 3D tetrahedral finite elements and applying plane symmetry conditions to the 1/8 part of the experimental RC prism. To simplify the model and reduce the number of finite elements, ribbed geometry of reinforcement was replaced by a series of cylinders of the same diameter but with different interface bond characteristics. Finally, Model-3 was generated using the same principles as Model-2, but taking into account the ribbed geometry of the reinforcement. The non-linear finite element (FE) software ATENA [23] was employed for the analysis. Further sub-sections discuss results by each of the models. 3.1. Two-dimensional Model-1 The FE Model-1 of the tested specimen is given in Fig. 4. Due to rectangular cross-section of the specimen, plane symmetry conditions were assumed formulating the problem in two dimensions and modelling a quarter of the member. The geometry of the ribbed bar was simplified in a way as to assume 1.2 mm rib height, 12 mm rib spacing and 55° rib face angle resulting in bond index fr = 0.1. Such bond index falls within the limits of standard values for steel reinforcement, which generally ranges between 0.05 and 0.1 [24]. Isoparametric triangular elements with 3 nodes and 4 integration

3.2. Three-dimensional Model-2 The illustration of Model-2 is given in Fig. 6a. Due to symmetry 140

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R. Jakubovskis and G. Kaklauskas

finite element model

RC tensile element

1

assumed rib geometry:

modeling bond:

8 mm

1.2 mm

ft=3 MPa c=3 MPa ij= 0.5 Ktt=2×108 MN/m3 Knn=2×108 MN/m3

rigid contact

2

3

Y

“normal bond”

4 mm

X

restrained horizontal displacement

restrained vertical displacement

applied load

Fig. 4. FE Model-1.

As may be noticed, concrete strain localizes near the regions with the “strong” bond characteristics. At the advanced loads, such strain concentration results in micro-cracks of concrete. The extent of microcracking is more clearly expressed at the loaded end and is gradually decreasing towards the mid-section of the specimen. As may be observed from the calculated reinforcement strain profile (Fig. 6e), the strain gradient representing bond stress significantly decreases in the vicinity of the loaded end. It can be also noticed that length of the debonding zone increases with an increase in load (or strain in the reinforcement). It can be kept in mind that the phenomenon of the loaded end in terms of the debonding effect well represents the behaviour of cracked sections in the bending or tensile members. In general, three-dimensional Model-2 permitted modelling a smooth reinforcement strain profile with the shape, in qualitative terms, rather similar to the experimental one. However, in quantitative terms, the reinforcement strain was adequately predicted only at the initial loading level. At high load levels, the discrepancy between the calculated and test strain profiles was rather remarkable with the clearest differences at the advanced load levels. The above discrepancies suggest that the transmission of the bond was inadequately modelled. Overestimation of bond stresses in Model-2 may be related to the simplified representation of the reinforcement ribs. If ribs are neglected, the bond action induces longitudinal force components only, whereas the radial forces are not captured properly. Considering a relatively large diameter of reinforcing bars used in the present study (20 mm), the radial force components may substantially affect the bond-transfer mechanism and, most importantly, the formation of splitting cracks. The results by Model-2 can be improved by means of tailoring the “normal“ and “strong“ bond parameters. Although such modelling approach allows transmitting the longitudinal component of the bond, it is not able to capture the radial pressure of bar to the surrounding concrete which, in turn, induces splitting cracks and contributes to the bond deterioration mechanism. To remedy this shortcoming, a threedimensional rib-scale model was generated representing the ribbed surface of the reinforcement.

8.500E-05 2.500E-04 5.000E-04 7.500E-04 1.000E-03 1.250E-03 1.500E-03 1.750E-03 2.000E-03 2.250E-03 2.500E-03 2.750E-03

Y

3.000E-03

X

X=0

1200

X=135

(a)

strain, ×10 -6 S-20-tz-1 S-20-tz-2 S-20-tz-3 Model-1

1000

X=0

800

600

400

200

0

0

15

30

45

60

75

(b)

90 105 120 135 distance from the midsection x, mm

Fig. 5. Calculation results of FE Model-1: (a) longitudinal concrete strain profile; (b) experimental and numerical reinforcement strain profiles.

conditions, 1/8 of the specimen was modelled. Isoparametric tetrahedral elements with 10 nodes and 4 integration points were used. The maximum element size of 8 mm was assumed in the analysis. To simplify rib modelling, the cylindrical bond model proposed in [25] was adopted. In this approach, the ribs of reinforcement are replaced by a series of cylinders with periodically ranging bond properties: the “strong” bond is adopted at the rib, whereas the “normal” bond is taken between the ribs. The schematic representation of the reinforcement with specific bond characteristics is shown in Fig. 6a. For concrete and reinforcement, the same material parameters were assumed as for Model-1. The strains in the concrete are indicated in Fig. 6c. This Figure uses the same colour pattern as described previously: grey areas represent strains that do not exceed the cracking strain, whereas sky-blue colour shows the areas with concrete strains over 0.003 as suggested in [26].

3.3. Three-dimensional Model-3 Earlier experimental studies [20] have demonstrated that almost 70–80% of the total bond force is transmitted by bearing action of the rib to concrete. As was shown in the analysis by two-dimensional Model-1, a specific internally cracked region forms around the ribs. The closer the section to the loaded end, the more pronounced the internal cracking around the ribs. To model the bond-transfer to the full extent of bond mechanics and to consider the deterioration of concrete at the loaded end, the three-dimensional rib-scale FE model (Model-3) was 141

Engineering Structures 194 (2019) 138–146

R. Jakubovskis and G. Kaklauskas

assumed simplification: 4 mm

assumed simplification:

8 mm

4 mm

4 mm

modelling bond

8 mm

0.8 mm

modelling bond “normal bond”: M = 0.5 8

3

Ktt=2×10 MN/m Knn=2×108 MN/m3

C=3 MPa ft=3 MPa

“normal bond”: M = 0.5 Ktt=2×108 MN/m3 Knn=2×108 MN/m3

“strong bond”: M = 0.5 Ktt=2×109 MN/m3 Knn=2×109 MN/m3

loaded end

C=3 MPa ft=3 MPa

C=175 MPa ft=175 MPa

loaded end loaded end

loaded end X=0

X=0

(b)

(a) strain

strain

loaded end loaded end X=0

X=135

X=0

X=135

(c)

(d)

strain, ×10-6 1200

1200 S-20-tz-1 S-20-tz-2 S-20-tz-3 Model-2

1000

800

600

600

400

400

200

200

0

15

S-20-tz-1 S-20-tz-2 S-20-tz-3 Model-3

1000

800

0

strain, ×10-6

30

60

45

(e)

75

90 105 120 135 distance from the midsection x, mm

0

0

15

30

60

45

(f)

75

90 105 120 135 distance from the midsection x, mm

Fig. 6. FE Model-2 and Model-3: (a) Model-2 and assumed bond characteristics; (b) Model-3 and the assumed bond characteristics; (c) longitudinal concrete strain profile calculated by Model-2; (d) longitudinal concrete strain profile calculated by Model-3; (e) and (f) experimental and numerical reinforcement strain profiles.

evident at the advanced loading stages. Such a difference in the numerical results is probably ascribable to a different bond transfer mechanism in Model-3 causing considerably larger deterioration of concrete around the reinforcing bar. These effects will be discussed in the next Section.

developed. Similarly to Model-2, 1/8 of the specimen was considered using isoparametric tetrahedral elements with 10 nodes and 4 integration points. The generated model with the simplified rib geometry is shown in Fig. 6b. For ribbed reinforcement, “normal” bond condition was assumed over the whole surface of reinforcement. It is important to note that the numerical results become sensitive to the assumed simplification of the real rib geometry, with specific reference to rib height. In present study, 0.8 mm rib height was assumed resulting in bond index equal to 0.07. In FE Model-3, rib height serves as the single most important parameter governing the calibration of the numerical outputs. The calculated longitudinal strain profile in concrete is shown in Fig. 6d. using the same colour field pattern as for Model-2. Similar to the latter model, localization of concrete strains in the vicinity of ribs show the location of secondary cracks forming around the bar. The calculated reinforcement strain profile is shown in Fig. 6f. Use of the ribbed surface of the reinforcement made it possible to adequately model reinforcement strain curves at different loading stages. A significant predictive improvement with regard to Model-2 was particularly

4. The interaction between reinforcement and concrete The previous Section has shown that three-dimensional rib-scale FE Model-3 can adequately represent reinforcement and concrete interaction at different loading stages. Moreover, some specific effects that are difficult to be measured and quantified experimentally can be studied numerically. Among such effects the formation of internal cracks, the phenomenon of bond deterioration, influence of concrete cover and bond in the post-yield regime can be mentioned. The FE Model-3 is further used to study the above aspects of the interaction between reinforcement and concrete.

142

Engineering Structures 194 (2019) 138–146

R. Jakubovskis and G. Kaklauskas 10 bond stress, MPa

4.1. Bond-slip relationships

3.5

maximal points

8 approximated

The bond stress and slip developing along a bar can be derived from the reinforcement strain profile. Slip at any point of the bar may be calculated through the integration of the reinforcement strain curve [8]:

∫ ε (x ) dx

ld,i

2.0 1.5

4

1.0

(1)

0

ED dε (x ) 4 dx

4.2. Bond deterioration As shown in Fig. 7b, moving away from the mid-section of the member towards its end, bond stresses are increasing with slip until the maximum point is reached. At this point, the bond stress curve starts descending, thus implying that the bond starts deteriorating. The length of the member between the loaded end and the point of maximal bond stress is often referred to as the bond deterioration zone or the damage zone. To evaluate the evolution of the damage zone with increasing load, the bond stress curves shown in Fig. 7b were approximated with polynomial functions. This allowed to determine the exact position of 10 9

500

2000

1500

1000

2500

(b)

10

bond stress, MPa

9

X=0

8

reinforcement strain, ×10-6

0

The interaction of reinforcement and concrete is generally studied in confined RC members having relatively large concrete covers (more than five bar diameters). Such a confinement prevents the formation of splitting cracks and facilitates the interpretation of the results. The current numerical Model-3 was also validated on the RC tensile specimen which can be considered as well-confined. However, in the

45 kN 40 kN 35 kN 30 kN

40 kN

X=0

0

4.3. The influence of concrete cover

bond stress, MPa

45 kN

2000

15

the stress peak at each loading stage and to relate the length of the damage zone to the maximal reinforcement strain, as schematically shown in Fig. 8a. As may be observed, length of the damage zone is not constant, but is dependant on the strain in the reinforcement: with load increase, the peak bond stress moves towards the centre of the member, i.e. length of the damage zone grows. A similar analysis was performed for the test member assuming different values of concrete grade ranging from 20 to 80 MPa. The results are summarized in Fig. 8b. As expected, length of the damage zone is also dependant on the strength of concrete: the higher is the concrete strength, the shorter is the damage zone. This effect can be clearly seen from the bond stress profile along the member assessed for four levels of concrete grade, see Fig. 9a–d. Quite naturally, the members with a higher compressive strength of concrete develop larger bond stresses. However, the bond damage for these members is more brittle as the descending part of bond stress profile curve is steeper compared to the members having smaller concrete strength. Fig. 9e–h show the same graphs like the ones from Fig. 9a–d with the difference that bond stresses were normalized with the tensile strength of concrete assessed by the Model Code [9]. As can be seen, rather close normalized values of maximal bond stresses were reached for different concrete strengths meaning that the established design practice of relating the ultimate bond stress to tensile strength is reasonable.

(2)

strain ×10-6

0

Fig. 8. Bond deterioration in tensile RC members: (a) location of peak bond stress as a function of load; (b) length of damage zone for different concrete grades.

where E is the elasticity modulus of reinforcement; D is the bar diameter. In Fig. 7a the strain profiles provided by FE Model-3 are plotted for various load levels below bar yielding. The bond stress variation along the member obtained by Eq. (2) is shown in Fig. 7b. Reinforcement slip was calculated by Eq. (1) and related to the bond stresses resulting in bond-slip relationships as shown in Fig. 7c. It may be observed that all the bond-slip curves exhibit a similar parabolic shape up to the peak stress that is ranging due to the difference in load level. After this peak, a short descending branch displaying the degradation of the bond was obtained. This form of bond-slip function allows for a more accurate representation of reinforcement and concrete interaction in the tensile members compared to the classical approach [9]. Fig. 7c shows the difference of the calculated bond-slip relations from the one suggested by the Model Code [9].

2400

0.5

x, mm 30 45 60 75 90 105 120 135

(a)

If concrete strain is neglected in the assessment of slip as it is in Eq. (1), the error generally does not exceed 10% [27]. Moreover, due to non-planar deformations of concrete close to the reinforcement, it is difficult to select the reference point for the assessment of concrete strains. Maekawa et al. [28] were among those who were in favour of excluding the concrete strain from the slip analysis. The bond stresses along the reinforcement bar are evaluated from the following equation [28]:

τ (x ) =

fc=20MPa fc=45MPa fc=60MPa fc=80MPa

2.5

bond stress

6

2

s (x ) =

normalized length of damage zone, ld/D

3.0

25 kN

Model Code 2010

8

35 kN

7

1600

6 25 kN

1200

20 kN

800

15 kN 10 kN

400

0

5 kN

distance from the midsection x, mm 0

15

30

45

60

(a)

75

90

105

120 135

7

20 kN

30 kN

6

15 kN

5

5 10 kN

4 3

3 5 kN

2

1 distance from the midsection x, mm 0

15

30

45

60

(b)

75

90

105

120 135

0

40 kN

45 kN

5 kN

slip,mm 0

0.05

0.1

0.15

(c)

Fig. 7. Analysis results of Model-3: (a) reinforcement strain profile; (b) bond stress profile; c) bond-slip relationships. 143

35 kN

2

1 0

25 kN 20 kN 15 kN 10 kN

4

30 kN

0.2

0.25

Engineering Structures 194 (2019) 138–146

R. Jakubovskis and G. Kaklauskas

14

bond stress, MPa

14

fc = 20 MPa

bond stress, MPa

14

fc = 45 MPa

12

12

12

10

10

10

45 kN 40 kN 35 kN

8 30 kN 25 kN 20 kN 15 kN

6

6

4

10 kN

2 0

5 kN

0

15

8

30

45

60

75

(a)

45 kN 40 kN 35 kN 30 kN 25 kN

4

10 kN

4

2

5 kN

15

30

45

60

75

(b)

bond stress, MPa

fc = 80 MPa

12

25 kN

20 kN 15 kN

10 kN

x, mm 0 90 105 120 135 0

15

30

45

60

75

(c)

25 kN

8

20 kN

6

15 kN 10 kN

4

5 kN

2

30 kN

45 kN 40 kN 35 kN

10

45 kN 40 kN 35 kN 30 kN

20 kN

6

14

fc = 60 MPa

8 15 kN

x, mm 0 90 105 120 135 0

bond stress, MPa

5 kN

2

x, mm 0 90 105 120 135 0

15

30

45

60

75

(d)

x, mm 90 105

3.5 normalized bond st ress, IJ/fct

3.5 normalized bond stress, IJ/fct

3.5 normalized bond stress, IJ/fc t

3.5 normalized bond st ress, IJ/fc t

3.0

3.0

3.0

3.0

fc = 20 MPa 45 kN 40 kN 35 kN 30 kN 25 kN

2.5

2.0

fc = 45 MPa

20 kN

15 kN 10 kN

1.5

2.5

5 kN

2.5

20 kN

45 kN 40 kN

2.0 35 kN 1.5

1.0

15 kN

30 kN 25 kN

2.5

45 kN 40 kN 35 kN 30 kN 25 kN

2.0

20 kN 15 kN

1.5

10 kN

10 kN

1.0

1.0 5 kN

0.5 0

15

30

45

60

75

x, mm 0 90 105 120 135 0

(e)

5 kN

0.5

0.5 0

fc = 80 MPa

fc = 60 MPa

x, mm 15

30

45

60

75

90 105 120 135

0

0

15

30

45

60

75

45 kN 40 kN 35 kN 30 kN 25 kN

2.0 1.5

20 kN 15 kN 10 kN

1.0

5 kN

0.5

x, mm 0 90 105 120 135 0

15

30

45

(g)

(f)

60

75

x, mm 90 105

(h)

Fig. 9. Bond stresses profiles (a–d) and normalized bond stresses (e–f) along the RC tensile members of different concrete grades.

strain below 500 × 10−6 when concrete reaches the cracking strain only in the vicinity of the reinforcement ribs. With an increase in load (reinforcement strain reaching 1000 × 10−6), a larger area of the internally cracked concrete forms in the member with a smaller cover. At a certain load (reinforcement strain around 1200 × 10−6), the secondary cracks merge into a macro-crack (sky-blue areas), which further progresses with an increase in load. Formation of internal cracks in the analysed RC prisms is shown in Fig. 11. As may be noticed, concrete strain localization areas (sky-blue regions) corresponds to the growth of micro-cracks. The calculated distribution of reinforcement strain and bond stresses in both members is shown in Fig. 12. Similar strain profiles for both members were obtained at the initial loading levels (up to 10 kN with respective reinforcement strain below 500 × 10−6). At loads between 15 and 20 kN (reinforcement strain below 1000 × 10−6), the internal cracking in the member of modified geometry has affected the capacity of concrete to transmit bond stresses, as may be seen in Fig. 12. At the load of 25 kN, the macro-crack has significantly re-shaped the bond stress profile in the RC element: the bond stress curve was having

design practice, much lower concrete covers are commonly used, reaching only one or two bar diameters. As a result, splitting cracks are generally observed at the surface of the concrete. The splitting cracks have an adverse effect on bond [18], however a quantitative evaluation of this effect can hardly be performed experimentally. Further, the numerical approach based on Model 3 was employed to study the bond deterioration mechanism in a RC member with a small concrete cover. For that an additional FE analysis was performed for a singly reinforced tension RC member having the same material, geometrical and FE model characteristics as the reference member with the exception of the geometry of the section: the member under consideration has a rectangular section (60 × 375 mm) and 20 mm concrete cover (one bar diameter) as to oppose to the square section (150 × 150 mm) of the test (reference) specimen having the cover of 65 mm. This geometry brings in identical cross-sections for the two members. The calculated longitudinal concrete strain profiles for both members are shown in Fig. 10. As may be observed, the models exhibit similar behaviour at the initial loading levels with the reinforcement Hs = 500 × 10-6

Hs = 1000 × 10-6

Hs = 1500 × 10-6

Hs = 2000 × 10-6

150 D20

150

60

D20 375

Fig. 10. Concrete strain profile in RC members with different concrete cover. 144

Hs = 2500 × 10-6

strain

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a local minimum in the location of the crack. Further load increase, contrary to the reference member, resulted in a more radical bond deterioration. The current analysis has shown that concrete cover may drastically affect the bond stresses developed in RC members. Moreover, the classical approach based on the bond-slip relationships derived from the members under well-confined conditions may be misleading as it may result in overestimation of the bond stresses. In this respect, the developed numerical modelling technique may be applied for analysis of anchorage or lap lengths of reinforcement in members with the small concrete cover.

secondary (Goto) cracks at the ribs

splitting crack transverse crack

formation of transverse crack

splitting cracks

4.4. Bond analysis in post-yield regime

Fig. 11. Internal cracking of the RC members with different concrete cover at final loading stage (average strain in reinforcement εs = 2500 × 10−6; crack filter > 0.01 mm). 2400

strain ×10 -6

2400

strain ×10 -6 45 kN

45 kN 40 kN

2000

40 kN

2000

35 kN

35 kN

1600

1600

30 kN 25 kN

1200

30 kN 25 kN

1200

20 kN

20 kN

800

800

15 kN

15 kN

10 kN

10 kN

400

The mechanism of steel-concrete interaction in the post-yield regime is essential to evaluate the lengths of plastic hinges, rotational capacity and ductility properties of RC members [29]. At the ultimate limit state, the tensile zone of RC beams and columns undergoes strain localization with reinforcement deformations significantly exceeding the strains from the elastic range. The length of this strain localization zone (or the length of the plastic hinge) is principally controlled by the bond of reinforcement and concrete. Several analytical models have been developed to describe the bond characteristics of yielded reinforcement and concrete [8,30]. However, generally, such models are applicable only for elements under specific loading conditions and bar arrangement (single bar, confined conditions, monotonic loading, etc.). In this respect, the numerical simulation is a powerful tool for modelling of bond and reinforcement strain profile at different concrete covers or loading conditions in the post-yield regime. The developed Model-3 was further used to study reinforcement strain and bond stress profile at the ultimate limit state. For steel, Von Misses plasticity model was assumed with a yield stress of 500 MPa, elasticity modulus of 200 GPa, hardening modulus of 5 GPa. Concrete and interface properties remained the same as was described in Section 3.3. Fig. 13 shows the simulated reinforcement strain and bond stress profile at the advanced loading stages. As may be noticed, loads over 50 kN induced plastic strains at the loaded end of the reinforcement that caused a sudden drop of the bond stresses. With the increase in load, plastic zone of reinforcement spread towards the mid-section of the RC member changing the profile of calculated bond stresses. It should be noted, that the bond stresses in the plastic zone were calculated assuming bilinear stress-strain diagram of steel. The more precise results may be obtained if the experimental stress-strain curve of reinforcement is available. Nevertheless, the numerical analysis allowed

400

5 kN

5 kN

x, mm x, mm 0 0 15 30 45 60 75 90 105 120 135 0 15 30 45 60 75 90 105 120 135 bond stress, MPa 10 bond stress, MPa 10 0

8

8

45 kN 40 kN

45 kN

15 kN

6 40 kN

10 kN

4 25 kN

6 35 kN

30 kN 25 kN 4 20 kN

20 kN

5 kN

2

15 kN 10 kN

35 kN 30 kN

5 kN

2

x, mm 0 0 15 30 45 60 75 90 105 120 135

0

x, mm 0 15 30 45 60 75 90 105 120 135

(a)

(b)

Fig. 12. Reinforcement strain and bond stress profiles in the reference member (a) and the member with a smaller concrete cover (b).

strain

0.02

10

60 kN

60 kN

reinforcement strain εx = 0.00175

0.018

bond stress, MPa

9 50 kN 59 kN

0.016

8

58 kN 0.014 0.012

59 kN

6

εx = 0.0025 εx > 0.0025

55 kN

60 kN

7

58 kN

57 kN

58 kN

0.01

5 57 kN

56 kN

0.008

53 kN

54 kN

0.004

0

56 kN

3

55 kN

55 kN

0.006 x=135 mm

x=0 elastic limit

0.0025 0.002

4

54 kN

2

53 kN 50 kN

53 kN

1

0

15

30

45

75

60

90

105

120

x, mm 135

(a)

0

x, mm 0

15

30

45

60

75

(b)

Fig. 13. Reinforcement strain (a) and bond stress (b) profiles in the post-yield regime. 145

90

105

120

135

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evaluating the exact length of plastic zones and reinforcement strain profile at different load levels. Such modelling technique does not require any analytical bond stress-slip relationships and, therefore, it can be used to determine length of strain localization in any type of structure under monotonic or cyclic loading conditions.

the activity ‘Improvement of researchers qualification by implementing world-class R&D projects’ of Measure No. 09.3.3-LMT-K-712 (Project No. 09.3.3-LMT-K-712-01-0145).

5. Conclusions

[1] Gambarova PG. Bond in reinforced concrete: where do we stand today. In: Proc 4th int symposium “Bond in Concrete 2012: bond, anchorage, detailing; 2012. p. 1–13. [2] Balázs GL. Connecting reinforcement to concrete by bond. Beton-und Stahlbetonbau 2007;102(S1):46–50. [3] Goto Y. Cracks formed in concrete around deformed tension bars. J Proc 1971;68(4): 244–51. [4] Borosnyói A, Snóbli I. Crack width variation within the concrete cover of reinforced concrete members. Építõanyag. 2010;62(3):70–4. [5] Bernardi P, Cerioni R, Ferretti D, Michelini E. Role of multiaxial state of stress on cracking of RC ties. Eng Fract Mech 2014;1(123):21–33. [6] Gribniak V, Rimkus A, Torres L, Hui D. An experimental study on cracking and deformations of tensile concrete elements reinforced with multiple gfrp bars. Compos Struct 2018. [7] Kaklauskas G. Crack model for RC members based on compatibility of stress-transfer and mean-strain approaches. J Struct Eng 2017;143(9):04017105. [8] Ruiz MF, Muttoni A, Gambarova PG. Analytical modeling of the pre-and postyield behavior of bond in reinforced concrete. J Struct Eng 2007;133(10):1364–72. 8. [9] Beverly P, editor. fib model code for concrete structures 2010. Ernst & Sohn; 2013. [10] Jakubovskis R. Compatible modeling of cracking, deformation and bond in reinforced concrete members: doctoral dissertation. Vilnius: Technika; 2015. in Lithuanian. [11] Jendele L, Cervenka J. Finite element modelling of reinforcement with bond. Comput Struct 2006;84(28):1780–91. [12] Fernández Ruiz M, Muttoni A, Gambarova P. A re-evaluation of test data on bond in R/C by means of FEM modeling. Studi e ricerche, Starrylink, pub. 2007;27(EPFLARTICLE-116119):113–34. [13] Santos J, Henriques AA. New finite element to model bond–slip with steel strain effect for the analysis of reinforced concrete structures. Eng Struct 2015;1(86):72–83. [14] Yankelevsky DZ, Jabareen M, Abutbul AD. One-dimensional analysis of tension stiffening in reinforced concrete with discrete cracks. Eng Struct 2008;30(1):206–17. [15] Casanova A, Jason L, Davenne L. Bond slip model for the simulation of reinforced concrete structures. Eng Struct 2012;1(39):66–78. [16] Salem HM, Maekawa K. Pre-and postyield finite element method simulation of bond of ribbed reinforcing bars. J Struct Eng 2004;130(4):671–80. [17] Daoud A, Maurel O, Laborderie C. 2D mesoscopic modelling of bar–concrete bond. Eng Struct 2013;1(49):696–706. [18] Kanazawa T, Sato Y, Mikawa T. Modeling of bond deterioration between steel bar and concrete subjected to freeze-thaw action. J Adv Concr Technol 2017;15(8):397–406. [19] Shang F, An X, Mishima T, Maekawa K. Three-dimensional nonlinear bond model incorporating transverse action in corroded RC members. J Adv Concr Technol 2011;9(1):89–102. [20] du Béton FI. Bond of reinforcement in concrete: state-of-art report. Bulletin. 2000;10:160–7. [21] Kaklauskas G, Sokolov A, Ramanauskas R, Jakubovskis R. Reinforcement strains in reinforced concrete tensile members recorded by strain gauges and FBG sensors: experimental and numerical analysis. Sensors. 2019;19(1):200. [22] Scott RH, Gill PA. Short-term distributions of strain and bond stress along tension reinforcement. The Structural Engineer. 1987;65(2):39–43. [23] Cervenka V, Jendele L, Cervenka J. ATENA program documentation, part 1: theory. Cervenka Consulting, Prague; 2007 Aug 23;231. [24] Metelli G, Plizzari GA. Influence of the relative rib area on bond behaviour. Mag Concr Res 2014;66(6):277–94. [25] Michou A, Hilaire A, Benboudjema F, Nahas G, Wyniecki P, Berthaud Y. Reinforcement–concrete bond behavior: experimentation in drying conditions and meso-scale modeling. Eng Struct 2015;15(101):570–82. [26] Gribniak V, Jakubovskis R, Rimkus A, Ng PL, Hui D. Experimental and numerical analysis of strain gradient in tensile concrete prisms reinforced with multiple bars. Constr Build Mater 2018;30(187):572–83. [27] Kankam CK. Relationship of bond stress, steel stress, and slip in reinforced concrete. J Struct Eng 1997;123(1):79–85. [28] Maekawa K, Okamura H, Pimanmas A. Non-linear mechanics of reinforced concrete. CRC Press; 2014. [29] Zhao XM, Wu YF, Leung AY. Analyses of plastic hinge regions in reinforced concrete beams under monotonic loading. Eng Struct 2012 Jan;1(34):466–82. [30] Manfredi G, Pecce M. Behaviour of bond between concrete and steel in a large postyelding field. Mater Struct 1996;29(8):506–13.

References

Three different models were used in this research project to investigate the bond between the concrete and the reinforcement: a twodimensional rib-scale model (Model-1), a three-dimensional model (Model-2) with the simplified representation of the ribs, and a threedimensional rib-scale model (Model-3). The effectiveness of the models in describing bond behaviour and specifically the strain profiles of the reinforcing bar were assessed by fitting the results of a series of tests on reinforced prisms, where the strains in the bar were measured. Comparing the experimental and numerical results makes it possible to draw the following conclusions: The two-dimensional approach (Model-1) allows to describe localization of concrete strains in front of the ribs as well as the formation of secondary cracks and deterioration of bond at the loaded end of the reinforcement. However, with an increase in load, the internal cracks rapidly spread and merge into a macro-crack, thus noticeably affecting the shape of the reinforcement strain profile. Most likely, this occurs due to the limitation of the two-dimensional model to represent the actual volumetric stress and strain state in the concrete. In the two-dimensional modelling approach, the crack growth is not arrested by the surrounding concrete and, therefore, has a tendency to be overestimated. The three-dimensional models (both Model-2 and Model-3) were able to capture the reinforcement strain profiles that were similar to the experimental evidence. However, in the quantitative terms, the prediction results by the Model 2 were not acceptable, particularly at the advanced load levels. Although Model-2 is able to represent the longitudinal bond force component, due to the absence of ribs, it cannot capture the radial forces properly. The radial force component may substantially affect the bond-transfer mechanism and, most importantly, the formation of splitting cracks. The three-dimensional rib-scale approach (Model-3) was capable to adequately predict reinforcement strain curves at different loading stages. In this approach, rib height serves as the single most important parameter governing the numerical outputs. The considered well-calibrated bond modelling technique may be used as a powerful tool to study various aspects of reinforcement and concrete interaction, such as the effect of bond deterioration near the crack, the influence of concrete cover or material parameters on bond as well as bond behaviour under the post-yield regime. Such modelling technique can be used in design situations requiring a high accuracy of the results: precise analysis of anchorage or lap lengths of reinforcement in members with different concrete covers or evaluation of rotational capacity and length of plastic hinges in RC members. This study demonstrated that effects, which can hardly be measured experimentally, can be studied numerically. Among such effects are the influence of concrete shrinkage and multiple bar arrangement (as in RC beams and frames) on bond properties. Future studies are required to quantify these effects numerically. Funding This research was funded by the European Social Fund according to

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