Serviceability analysis of flexural reinforced concrete members/Lenkiamuju gelzbetoniniu elementu tinkamumo analize.

Kaklauskas, Gintaris ; Gribniak, Viktor ; Jakubovskis, Ronaldas 等

1. Introduction

Adequate modelling of reinforced concrete (RC) cracking, particularly post-cracking behaviour as one of the major sources of nonlinearity, is the most important and difficult task of serviceability analysis. Post-cracking deformation response of RC members is a process including a wide range of effects such as different strength and deformation properties of steel and concrete, the shrinkage and creep of concrete, bond slip between reinforcement and concrete. Due to this complexity, deflection predictions using different techniques may vary in the range of 20-37%, whereas the variability of predictions for crack width is of much higher order (Kaklauskas 2004; Gribniak 2009; Juozapaitis et al. 2010).

Fig. 1 presents a typical load-strain curve of RC members subjected to tension. The load-displacement diagram points to four stages of deformation behaviour. The first stage represents elastic deformations of the member up to the start of cracking (part OA). The second stage covers the region between the first and the final primary cracks (part AB). At the end of this stage (Final cracking point in Fig. 1), the RC member becomes separated by the developed cracks into a number of concrete blocks. The length of each block falls into specific interval [l.sub.tr] [less than or equal to] [l.sub.cr] [less than or equal to] 2[l.sub.tr], where [l.sub.tr] is transfer length. It is reported (Bigaj 1999) that the average block length (crack spacing) could be in the range of 1.3 [l.sub.tr]-1.5 [l.sub.tr]. The behaviour of the member with fully developed cracks (part BC) corresponds to the third stage. The fourth stage starts with the yielding of reinforcement. The problems of serviceability are mainly related to the third stage usually covering the service loading region. Similar stages of behaviour can be observed in bending members.

In the vicinity of cracks, reinforcement slip occurs and bond stress develops between the reinforcing bar and surrounding concrete, transmitting tensile force from the bar to concrete. Many theoretical models to predict deformations and/or the crack width of RC members have been proposed. Generally, the models may be divided into four groups:

--Semi-empirical: the earliest approaches were developed based on test data. Such simplified models are broadly presented in design codes but are not universal due to specific constitutive experiments;

--Fracture mechanics: such approaches use the principles of the fracture mechanics of concrete. They are generally applied in combination with other approaches analyzing RC structures;

--Average stress-average strain: simple approaches based on the smeared crack model. Such models are extensively used for numerical analysis and can evaluate the average response of a member; however, they are not able to predict the cracking character;

--Discrete crack: these approaches are suitable to assess the opening of each individual crack.

[FIGURE 1 OMITTED]

The latter models are capable of reflecting the realistic behaviour of RC members (Kwak, Ha 2006; Wu, Gilbert 2009; Ng et al. 2010); nevertheless, the prediction results become dependent on the assumed bond stress-slip law. Different bond-slip relationships were proposed by Eligehausen et al. (1983), Shima et al. (1987), Uijl den, Bigaj (1996) and other researchers. However, the application of these relationships is based on sophisticated calculation procedures. Marti et al. (1998) have suggested a simple model of RC tie assuming a rigid-plastic bond stress-slip law. This assumption resulted in idealized load-displacement behaviour as shown in Fig. 1 by the dashed line. This study extends such simplified approach for the serviceability analysis of RC flexural members.

2. Simplified discrete crack model

The model is based on the following assumptions:

1. Linear-elastic properties were assumed for reinforcement and concrete, both in tension and compression.

2. All cracks appear under cracking load dividing the beam into a number of blocks. Strain and stress distribution in a concrete block are symmetrical about its centre (Fig. 2).

3. Transfer length [l.sub.tr] is calculated using the chosen technique. This study has assumed (EN 1992-1-1:2004) that [l.sup.tr] = [S.sub.r,max]/2 where [S.sub.r,max] is maximum crack spacing. For analyzing deformation and average crack width, block length (crack spacing) [l.sub.cr] is taken 1.5 [ll.sub.tr], whereas 2[l.sub.tr is used for calculating maximum crack width.

4. Linear strain distribution is assumed for reinforcement both in tension and compression and compressive concrete. As shown in Figs. 3a and 3b, strain distribution in tensile concrete is also linear, but may have an individual shape (Fantilli et al. 1998; Kobielak et al. 2010).

5. Tensile concrete response in a cracked section is ignored.

6. Strain distribution of tensile reinforcement is assumed to be linear in a concrete block (Kankam 2003).

[FIGURE 2 OMITTED]

Strain in tensile reinforcement is related to bond stress by the following equation:

d[[epsilon].sub.s]/dx = [4/[D.sub.s][E.sub.s]] [tau](x), (1)

where [D.sub.s] and [E.sub.s] are the diameter and modulus of the elasticity of the steel bar respectively. Based on assumption 6 and relationship (1), we obtain [tau](x) = const.

The equilibrium conditions of all forces and moments in the section result in the following equations:

[N.sub.cc] + [N.sub.sc] + [N.sub.ct] + [N.sub.st] = 0;

[M.sub.cc] + [M.sub.sc] + [M.sub.ct] + [M.sub.st] = [M.sub.ext], (2)

where [M.sub.ext] is the external bending moment; N and M are internal forces and moments respectively. The first subscript corresponds to either c for concrete or s for steel and the second subscript refers to compression (c) or tension (t) as shown in Fig. 3d.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

The curvature and crack width of the RC member can be calculated by the following algorithm:

1. Calculate the second moments of the inertia of elastic and fully cracked sections [I.sub.el] [I.sub.cr].

2. Calculate the cracking moment:

[M.sub.cr] = [f.sub.ct][W.sub.0], (3)

where [f.sub.ct] is the strength of concrete in tension; [W.sub.0] is the elastic section modulus taken in respect to the tension edge of the transformed section.

3. At cracking load [M.sub.cr], calculate steel strains [[epsilon].sub.s,cr] and [[epsilon].sub.s,el] for fully cracked and uncracked sections respectively (see Fig. 2).

4. Define transfer length [l.sub.tr] by EN 1992-1-1:2004 (2004) (see assumption 3).

5. Determine the strain distribution law of tensile reinforcement (see Fig. 2):

[[epsilon].sub.s](x) = [[epsilon].sub.s,cr] - [chi] x ([[epsilon].sub.s,cr] - [[epsilon].sub.s,el])/[l.sub.tr]. (4)

6. Based on equation (1) and Fig. 4, for cracking load [M.sub.cr], calculate bond stress [[tau].sub.a]. The latter value is applied at all successive load stages.

7. Divide the concrete block into a number of sections n as shown in Fig. 5 and calculate strain distribution along the block using the results of step 6 and equilibrium equations (2).

8. Calculate curvature at each section [[kappa].sub.i] (i = 1 ... n) as shown in Fig. 2 where [[epsilon].sub.s,i] and [[epsilon].sub.cc,i] are steel and compressive concrete strain respectively.

9. Calculate the average curvature of the member:

[kappa] = [1/n] [[summation].sup.n.sub.i=1] [[kappa].sub.i]. (5)

10. Calculate crack width w using the strains of steel [[epsilon].sub.s](x) and tensile concrete [[epsilon].sub.cl](x) obtained from Fig. 5:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[FIGURE 5 OMITTED]

3. Modelling flexural members: a comparison with test results

The discussed model has been applied to simulate RC beams tested by Clark and Speirs (1978) and Kaklauskas et al. (2008). Fig. 6 shows the experimental and calculated moment-curvature diagrams for six RC sections with a different ratio of tensile reinforcement [rho]. Beams S2-3 and S2-3R were twin specimens, but had different reinforcement in the compression zone (2[empty set]6 and 3[empty set]14 respectively). The assumed rigid-plastic bond laws are shown in Fig. 7 using solid lines. Bond stresses [[tau].sub.a] ranged from 1.1 [f.sub.ct] to 1.8 [f.sub.ct] with the tendency that the beams with a lower reinforcement ratio and fewer bars possessed higher stresses. It should be noted that for tie members Marti et al. (1998) have assumed [[tau].sub.a] = 2 [f.sub.ct]. The present study has shown that such assumption might result in unrealistically small crack width and spacing.

The accuracy analysis of curvature predictions at service load [M.sub.ser] = 0.6 [M.sub.u] ([M.sub.u] is the ultimate moment) was performed. The obtained results are given in Table 1. The differences between the measured and calculated curvatures ranged from 2 to 5%, whereas, the average crack widths for beams S2-2, S2-3 and S2-3R (Kaklauskas et al. 2008) were overestimated by about 30% (see Table 1). Such agreement can be considered as satisfactory: crack width analysis dealing with a single section, due to a stochastic nature of cracking, generally results in larger errors in regard to deflections. The latter represent the global response of structure with smeared out local effects such as the cracking, slippage and degradation of bond stresses.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

Numerous studies have shown that bond-stresses degrade with an increase in load (Torres et al. 2004; Kaklauskas et al. 2009, 2011a, b; Wu, Gilbert 2009; Gribniak et al. 2010; Kala et al. 2010; Ng et al. 2010; Zanuy 2010; Ho, Peng 2011). This also can be observed from the moment-curvature diagrams shown in Fig. 6. The modelled curvature response was too stiff for all beams due to the assumption of constant bond stress.

To overcome the above deficiency, the authors have developed an inverse procedure for deriving a rigid-plastic bond-slip law based on the test curvatures of RC beams. The concept of the procedure is similar to that described in Kaklauskas and Gribniak (2011). Fig. 7 shows the relationships between bond stress [[tau].sub.a] and the bending moment. The calculated and experimental moment-curvature diagrams will coincide at each loading level assuming respective bond stress. It should be noted that bond stresses calculated by the inverse technique were in a good agreement at service load with [[tau].sub.a] assumed in the discrete crack model (see Fig. 7 and Table 1).

4. Conclusions

The paper discusses the discrete crack model based on bond stress-slip relationship and considers its applicability for deformation and crack widths analyses of reinforced concrete flexural members. The performed serviceability analysis has indicated that the model becomes a useful tool for predicting crack widths and deformations. Based on the obtained results, the following conclusions can be drawn:

1. The simplified model based on the rigid-plastic bond stress-slip law was able to predict curvature and crack width. Inaccuracies in deflection predictions at service load varied from 2 to 5%. A satisfactory agreement was achieved for crack widths.

2. Bond stresses [[tau].sub.a] assumed in the discrete crack model ranged from 1.1 [f.sub.ct] to 1.8 [f.sub.ct] with the tendency that the beams with a lower reinforcement ratio and fewer bars possessed higher stresses. The application of [[tau].sub.a] = 2 [f.sub.ct], commonly taken for tie members, may result in underestimating crack width.

3. The potentials of the discussed technique should be further investigated regarding application for the modern types of reinforcement such as fibers and/or nonmetallic bars. Moreover, the influence of various cases of distribution reinforcement bars in a section on cracking behaviour should be investigated.

doi.org/10.3846/13923730.2011.643553

Acknowledgements

The authors gratefully acknowledge financial support provided by the Research Council of Lithuania (research project No. MIP-126/2010). Viktor Gribniak also wishes to acknowledge support provided by the Research Council cil Lithuania for the Postdoctoral fellowship granted within the framework of the EU Structural Funds (project "Postdoctoral Fellowship Implementation in Lithuania").

References

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Gintaris Kaklauskas (1), Viktor Gribniak (2), Ronaldas Jakubovskis (3), Eugenijus Gudonis (4), Donatas Salys (5), Rimantas Kupliauskas (6)

(1,2,3,4,5) Department of Bridges and Special Structures, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania

(2) Department of Strength of Materials, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania

(6) UAB Grinda, Eiguliug. 7, LT-03150 Vilnius, Lithuania

E-mails: (1) Gintaris.Kaklauskas@vgtu.lt (corresponding author); (2) Viktor.Gribniak@vgtu.lt; (3) 3Ronaldas.Jakubovskis@vgtu.lt; (4) Eugenijus.Gudonis@vgtu.lt; (5) sdonat@gmail.com; (6) R.Kupliauskas@grinda.lt

Received 18 Jul. 2011; accepted 08 Sept. 2011

Gintaris KAKLAUSKAS. Professor and the head of the Department of Bridges and Special Structures at VGTU, Lithuania. He received his engineering degree, PhD and Dr Habil (Dr Sc.) from VGTU. Research interests include various topics of reinforced concrete, particularly constitutive modeling and numerical simulation of reinforced concrete structures. He was a recipient of Fulbright Fellowship (for research work at the University of Illinois, Urbana-Champaign, 1996) and Marie Curie Fellowship (experienced researchers category, 2002-2003). Prof. G. Kaklauskas is a member of FIB Task Group 4.1 "Serviceability Models".

Viktor GRIBNIAK. Researcher at the Department of Bridges and Special Structures, VGTU, Lithuania. He received his engineering degree, PhD (2009) from VGTU. A study visit to Polytechnico di Torino (2003). The author of a number of papers. Research interests: mathematical modeling, a statistical analysis and numerical simulation of concrete structures.

Ronaldas JAKUBOVSKIS. PhD student at the Department of Bridges and Special Structures, VGTU, Lithuania. BSc (2008, Construction Engineering) and MSc (2010, Construction Engineering) from VGTU. Research interests: a bond of reinforcement in concrete.

Eugenijus GUDONIS. PhD student at the Department of Bridges and Special Structures, VGTU, Lithuania. BSc (2008, Construction Engineering) and MSc (2010, Construction Engineering) from VGTU. Research interests: mathematical modeling, a nonlinear numerical analysis of reinforced concrete structures, a bond of reinforcement in concrete.

Donatas SALYS. PhD student at the Department of Bridges and Special Structures, VGTU, Lithuania. BSc (2005, Engineering Informatics) and MSc (2007, Engineering Informatics) from VGTU. Research interests: mathematical modeling, a nonlinear numerical analysis of reinforced concrete structures.

Rimantas KUPLIAUSIAS. Head of the Department of Storm-Water Service Development and Exploitation at Joint-Stock Company Grinda, Vilnius, Lithuania. Civil Engineer (1978) from VGTU (former VISI). Research interests: the durability of concrete and reinforced concrete buildings, cracking and long-term effects in concrete structures.

Table l. The curvatures and average crack widths of the
beams at service load

Beam      [rho][%]   [[tau].sub.a]/     [[kappa].       [[kappa].
                       [f.sub.ct]       sub.exp]         sub.an]
                                      [[km.sup.-1]]   [[km.sup.-1]]

B3          0.79          1.54            3.12            3.22
B7          0.62          1.76            2.96            3.04
B4          0.44          1.78            2.09            2.15
S2-2        0.62          1.08            6.59            6.46
S2-3        0.61          1.43            6.40            6.09
S2-3R       0.61          1.49            6.06            6.01

Beam      [[kappa].sub.an]/   [w.sub.exp]   [w.sub.an]
          [[kappa].sub.exp]      [mm]          [mm]
               - 1 [%]

B3               3.2               -            -
B7               2.7               -            -
B4               2.9               -            -
S2-2            -2.0             0.11         0.145
S2-3            -1.8             0.12         0.151
S2-3R           -0.8             0.12         0.147

Beam       [w.sub.an]/
          [w.sub.exp] -
              1 [%]

B3              -
B7              -
B4              -
S2-2          31.8
S2-3          25.8
S2-3R         22.5