Numerical stochastic analysis of RC tension bar cracking due to restrained thermal loading

Theoretical and Applied Fracture Mechanics 74 (2014) 39–47

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Numerical stochastic analysis of RC tension bar cracking due to restrained thermal loading Jens U. Hartig ⇑, Ulrich Häussler-Combe Institute of Concrete Structures, Faculty of Civil Engineering, Technische Universität Dresden, 01062 Dresden, Germany

a r t i c l e

i n f o

Article history: Available online 1 July 2014 Keywords: Reinforced concrete Cracking Thermal loading Numerical model Random field Statistics

a b s t r a c t Cracking of concrete has deteriorating effects on the behavior of reinforced concrete structures, which results for instance in stiffness reduction and increase of permeability. In general, it is assumed that the deterioration increases with increasing crack width and, thus, maximum crack width might be most critical. However, models and recommendations in design codes regarding the serviceability limit state are usually based on mean crack width. In this contribution, a model for tensile members based on the Finite Element Method is presented, which takes spatially scattering material properties into account. As a special loading case restrained thermal loading is considered. The model is validated based on comparison with experimental results and with an analytical model. As an example of application, the model is embedded in a Monte-Carlo simulation to predict the statistics of crack widths due to spatial scatter of concrete tensile strength and bond strength. Results are crack width distributions and characteristic statistical values like mean values, standard deviations and quantile values over the entire loading range. Besides the presentation of the methodology of determining statistical properties of crack widths, it is shown that mean and maximum crack widths might be underestimated by analytical models as recommended in current design codes. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Imposed deformations, e.g., due to temperature changes and concrete shrinkage, often result in concrete cracks in restrained reinforced concrete (RC) structural members, which might impair serviceability and structural integrity. Available design codes and recommendations as, e.g., CEB-FIP Model Code 90 [1], ‘‘Eurocode 2: Design of concrete structures’’ [2] and ‘‘ACI Building Code Requirements for Structural Concrete’’ [3], are rather vague in predicting cracking properties especially concerning restrained imposed loading. A more detailed description of this issue is given by the authors in a recent paper [4] where also an extension of the analytical model for the estimation of crack width as provided by [1,2] regarding imposed deformations is presented. While this analytical model is an efficient means for the estimation of mean crack widths, it does not account for stochastic variations of crack widths due to the scattering material properties. For design purposes, especially the maximum crack width values are of interest as they ⇑ Corresponding author. Present address: Institute of Steel and Timber Construction, Faculty of Civil Engineering, Technische Universität Dresden, 01062 Dresden, Germany. Tel.: +49 351 463 35784. E-mail addresses: [email protected] (J.U. Hartig), ulrich.haeussler-combe@ tu-dresden.de (U. Häussler-Combe). http://dx.doi.org/10.1016/j.tafmec.2014.06.004 0167-8442/Ó 2014 Elsevier Ltd. All rights reserved.

might have the most detrimental influence on the performance of tensile structural members. In the present contribution, the statistics of crack widths in RC bars exposed to direct tension resulting from restrained imposed deformations are analyzed. The investigations are focused on the case of thermal loading caused e.g. by insolation or other environmental temperature changes. For the determination of statistical properties of crack widths, Monte-Carlo-type simulations are performed. This is based on a deterministic Finite Element model representing the load-bearing behavior of RC tension bars. The model consists of one-dimensional bar elements and zero-thickness bond elements. It incorporates nonlinear constitutive relations for concrete and reinforcement as well as the bond in between. Such a type of model was used in [5] for the analysis of the tensile behavior of Textile Reinforced Concrete. However, the statistical properties of crack widths were not analyzed. A similar model extended to 2D was applied by [6] for the analysis of continuously reinforced concrete pavements. Another 2D model, where the reinforcement contribution is taken into account in a so-called embedded crack FE model, was developed in [7]. Both models provide only mean values of crack width. For the validation of the numerical model, experimental results from literature are used. Experimental results regarding cracking

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J.U. Hartig, U. Häussler-Combe / Theoretical and Applied Fracture Mechanics 74 (2014) 39–47

Nomenclature Ac,eff As E Ec Es Fimp L P S T DT Y ds fct fst fsy lt lcorr n nrv nset s sf smax sr W x

effective cross-sectional area of concrete cross-sectional area of reinforcement Young’s modulus Young’s modulus of concrete Young’s modulus of reinforcement force due to restrained imposed loading length of bar autocorrelation matrix realization of random field temperature temperature change random variable diameter of reinforcement bar tensile strength of concrete tensile strength of reinforcement yield strength of reinforcement stress transfer length correlation length number of cracks number of random variables number of simulations in MCS slip slip corresponding to sf slip corresponding to smax crack spacing crack width longitudinal coordinate ratio between Young’s moduli of reinforcement and concrete thermal expansion coefficient

ae aT

of RC tension members due to thermal loading were performed in [8–11]. 2. Modeling 2.1. Revision of analytical model in common design codes In this section, the analytical model presented in [4] based on [2,1] is briefly summarized. The results of the numerical stochastic model will be subsequently compared to the prediction of this model. The model distinguishes between single cracks and stabilized cracking. At a crack, differences in concrete strain ec and reinforcement strain es occur along the stress transfer lengths lt, see Fig. 1. The crack width w is given as the integral of the differences between ec and es along longitudinal direction x where the crack is situated at x = 0

w¼2

Z

shape coefficient of reinforcement stress course strain concrete strain imposed concrete strain mean concrete strain imposed strain reinforcement strain imposed reinforcement strain mean reinforcement strain eigenvalue of P standard Gaussian random number effective reinforcement ratio autocorrelation coefficient stress concrete stress mean concrete stress concrete stress at the crack eigenstress reinforcement stress mean reinforcement stress reinforcement stress at the crack difference between minimum and maximum reinforcement stress bond stress residual/frictional bond stress mean bond stress bond strength eigenvector of P

bt

e ec ec,imp ecm eE es es,imp esm k n

qeff qY,Y r rc rcm rcr rimp rs rsm rsr Drs 0

s sf sm smax w

where bt is an empirical factor (0 < bt < 1) depending on the shape of the reinforcement stress distribution along lt. Values of bt are provided for typical loading conditions in [2,1]. Forces are transferred between concrete and reinforcement by means of bond stresses s leading to increasing tensile stresses in the concrete with increasing distance to the crack. The bond stress s is controlled by the slip s, which is the relative displacement between the concrete and the reinforcement. The respective relation s(s) is also called bond law. At a position x0 , the slip s is given as the integral

sðx0 Þ ¼

Z

lt

½es ðxÞ  ec ðxÞdx:

σ σsr

lt

½es ðxÞ  ec ðxÞdx ¼ 2lt ðesm  ecm Þ;

ð3Þ

x0

ð1Þ

0

which can be simplified using the mean strains of the concrete ecm and the reinforcement esm in the stress transfer length lt. At a crack, the reinforcement stress has its maximum value, which is denoted with rsr. Neglecting the post-cracking resistance of the concrete, the concrete stress is zero at a crack. The value of rsr can be determined based on the normal force applied to the RC bar. The difference in the reinforcement stress between the maximum value at the crack rsr and the minimum value distant to the crack is denoted with Drs. The mean value of the reinforcement stress along lt is given with

rsm ¼ rsr  bt Drs

ð2Þ

σs(x)

σsm σs(lt)

Δσs

σc(x) τ x

lt

lt

Fig. 1. Stress distribution in the stress transfer length.

J.U. Hartig, U. Häussler-Combe / Theoretical and Applied Fracture Mechanics 74 (2014) 39–47

The end of the stress transfer length x = lt can be defined by the position where the concrete stress gains a maximum value and the reinforcement stress a minimum value. Along lt, a mean bond stress sm can be established. The value of sm is defined in [2,1] depending on the concrete tensile strength fct. Thus, the size of lt can be determined by means of equilibrium considerations as

2lt ¼

ds Dr s 2sm

ð4Þ

where ds is the diameter of the reinforcement steel bar. Furthermore, the mean concrete stress along lt is given with

rcm ¼ qeff bt Drs

ð5Þ

where qeff = As/Ac,eff is the effective reinforcement ratio incorporating the cross-sectional area of the steel As and the effective crosssectional area of the concrete Ac,eff. The strains are related to the stresses by Hooke’s law, which results for the reinforcement and the concrete in

esm ¼

rsm Es

þ es;imp and

ecm ¼

rcm Ec

þ ec;imp ;

ð6Þ

respectively. Es and Ec are the Young’s moduli of reinforcement and concrete, while es,imp and ec,imp are imposed strains in reinforcement and concrete. The imposed strains might result from temperature changes or shrinkage of concrete. For the determination of crack width w according to Eq. (1), Drs remains to be determined. For single cracks, concrete and reinforcement strains are equal at x = lt. This results in

Dr s ¼

rsr þ rimp 1 þ qeff ae

ð7Þ

where ae = Es/Ec and rimp = Es (ec,imp–es,imp), which is also referred to as eigenstress. A crack spacing cannot be determined with this approach. For stabilized cracking, the strains cannot be assumed equal at x = lt. However, the concrete stresses are limited by the concrete tensile strength fct and equilibrium considerations lead to

Dr s ¼

rc ðlt Þ : qeff

ð8Þ

With rc(lt) = fct, crack spacing sr is equal to

sr ¼ 2lt ¼

fct ds 2sm qeff

ð9Þ

because at the end of a stress transfer length the stress transfer length of the next crack follows immediately. The crack width for stabilized cracking is given with



1 W ¼ sr Es



rsr  bt

fct

qeff



ð1 þ qeff ae Þ þ



 

es;imp  ec;imp :

ð10Þ

Although the presented analytical model for the estimation of crack width is useful for engineering practice, it has several shortcomings. It neglects for instance spatial scatter of material parameters as, e.g., concrete tensile strength and Young’s modulus, which, in turn, will result also in scattering crack widths. Thus, maximum crack width will be, in general, underestimated with this model. Moreover, the model considers only mean values of stresses and strains along the stress transfer length, which might mask certain effects resulting from load transfer between concrete and reinforcement. The detailed nonlinear behavior of materials and bond cannot be regarded with this model. The quantification of the influence of these properties requires more detailed investigations. On the other hand, an enhancement of the model considering the mentioned properties is difficult because of its analytical character. A larger variability is given with

41

numerical models. Such a model is presented in the following for the case of uniaxial tensile loading. 2.2. Numerical model 2.2.1. Applied finite elements For the simulation of the uniaxial tensile load-bearing behavior of RC bars, a model based on the Finite Element Method (FEM) is used. The model, which was already applied in a more general way in [12–14], consists of one-dimensional bar elements representing the uniaxial load-bearing behavior of either concrete or reinforcement and zero-thickness bond elements for the load transfer in between. Thus, transverse concrete stresses and strains are neglected in a first approach. Hence, differences in the crack width close and distant to the reinforcement as shown experimentally, e.g., by [15,16] are not taken into account in the model. Moreover, cracking due to thermal gradients over the cross section is not covered by the model. Nevertheless, these effects can be considered using plane stress or plane strain elements for the two-dimensional case or continuum volume elements for the three-dimensional case. However, this leads to a considerably increasing numerical effort especially because stochastic analysis with a series of calculations has to be carried out. Thus, these cases are not investigated in this contribution. The model consists of two bar element chains representing matrix and reinforcement. At corresponding nodes these bar elements are coupled with bond elements, see Fig. 2. The boundary conditions are given with constraints for the end nodes of the bar element chains. 2.2.2. Deterministic constitutive behavior The material behavior of the concrete is assumed as linear elastic with a Young’s modulus Ec until reaching the tensile strength fct. Subsequently, concrete failure occurs:

(

rc ¼

Ec 0



ec  ec;imp



for for

ec  ec;imp 6 fct =Ec : ec  ec;imp > fct =Ec

ð11Þ

The post-cracking resistance of the concrete is neglected as it has only small influence on maximum crack widths. For the reinforcement steel, a bilinear elastic–plastic constitutive relation considering hardening is applied: 8   > E e  es;imp for es  es;imp 6 fsy =Es > < s s   rs ¼ fsy =Efsysfestðfst Þ es  es;imp  fsy =Es þ fsy for fsy =Es < es  es;imp 6 eðfst Þ: > > : 0 for es  es;imp > eðfst Þ

ð12Þ

Eq. (12) contains the Young’s modulus Es of the reinforcement, the yield strength fsy, the tensile strength fst and the corresponding strain e(fst). Unloading after exceeding fsy is modeled with a slope equal to Es. The thermal expansion coefficient of both concrete and steel is assumed as aT = 1  105 K1. For the bond between concrete and reinforcement, a bond law formulated as bond stress–slip (s–s) relation is applied. Therefor, supporting points (s, s) are defined, see Fig. 3. Between the supporting points, it is interpolated by means of piecewise cubic Hermite polynomials (PCHIP) according to [17]. The PCHIP

Fig. 2. Segment model and discretization.

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J.U. Hartig, U. Häussler-Combe / Theoretical and Applied Fracture Mechanics 74 (2014) 39–47

assumed that each Y is influenced by its neighborhood. Such an influencing is referred to as correlation in statistics. In the model, a so-called autocorrelation function qY,Y0 (x, x0 ) is applied representing the correlation between two random variables Y and Y0 belonging to one material property at two positions x and x0 . The following bell-shaped form is used

 2 ! jx  x0 j ; 0 6 qY;Y0 ðx; x0 Þ 6 1: qY;Y0 ðx; x Þ ¼ exp  lcorr 0

Fig. 3. Bond law for validation simulations.

approach shows continuity of the first derivatives of the interpolating functions at the transitions between the intervals, which is advantageous for a stable numerical solution of the non-linear system of equations arising from the non-linearities in the model. Details are described in [13,14,18]. The supporting points are chosen such that s is limited by a bond strength smax at smax. After exceeding smax, it is usually distinguished between confined and unconfined concrete, compare e.g. [1]. In case of confined concrete, plastic deformations occur in the interface due to successive shear failure of the concrete between the ribs of the reinforcement bars. Bond degradation is then associated with gradually decreasing s at increasing s. When bond degradation is finished at sf, a residual bond stress sf is assumed, which is explained with remaining frictional load transfer. For unconfined concrete, splitting failure occurs and, thus, plastic deformations after exceeding smax are missing. The bond stress decreases directly after exceeding smax until the residual bond stress level sf is reached. Unloading is modeled based on the concept of plasticity for both cases. Fig. 3 contains an exemplary set of bond parameters, which are used subsequently for validation of the model. In this case, unconfined concrete was assumed. The non-linear bond law, limited tensile strength of the concrete and the elasto–plastic material law for the reinforcement steel lead to nonlinear systems after discretization. Incremental–iterative methods have to be used for the solution, where loading is applied in steps. The BFGS method [19] combined with line search is used for the equilibrium iteration in each step. In the current investigations, loading is given by imposed deformations, which are applied to the bar elements as additional strains ec,imp and es,imp according to Eqs. (11) and (12). In the case of thermal loading, the imposed strain is determined with eimp = aT DT where DT is the incremental temperature change. When the tensile strength in a concrete bar element is exceeded, the stiffness is set to zero and the system is recalculated on the same load level. Concrete cracking is restricted to one element per equilibrium iteration. Furthermore, it is assumed that one element might only represent one crack and, thus, the crack width of the respective element is given directly as the relative displacement of the two nodes. 2.2.3. Spatial scatter in material properties Material properties are subject to spatial scatter, which influences also the load-bearing response of the material. Especially the material properties of the concrete show relatively strong spatial fluctuations due to the heterogeneity. An example is given in Fig. 6 with the spatially scattering concrete tensile strength fct. The random field approach in [20] based on the Karhunen– Loève expansion is applied to model the spatial fluctuations. In this approach, the material properties at certain positions x corresponding to the integration points of the bar elements representing the concrete are modeled as random variables Y. Furthermore, it is

ð13Þ

depending on the distance |x–x0 | and a so-called correlation length lcorr. The correlation length is assumed to be a material property related to the structure of the material. For concrete, lcorr is assumed to depend primarily on the aggregate size. The correlation between two positions x and x0 increases with increasing lcorr. For each combination of Y and Y0 at the integration points of the bar elements, a value qY,Y0 (x, x0 ) is determined and assembled in a socalled autocorrelation matrix P of order nrv  nrv where nrv is the number of random variables corresponding to the number of integration points of the concrete bar element chain. Autocorrelation matrix P can be decomposed into eigenvalues k and eigenvectors w by means of solving the standard eigenvalue problem Pw = kw. A discretized sample of the random field can then be established as

S¼

nrv pffiffiffiffi X ki ni wi

ð14Þ

i¼1

where n are uncorrelated standard Gaussian random numbers. The vector S contains a realization for each random variable Y corresponding to the material property at the integration point. The random numbers are generated during a so-called Monte-Carlo simulation, which is described in the next section. It shall be noted that this procedure is only appropriate for standard Gaussian distribution functions. In the case that the standard Gaussian distribution is not appropriate for the modeling of the material property, which is rather rule than exception, the autocorrelation coefficients qY,Y0 (x, x0 ) have to be transformed to the Gaussian domain. Therefore, the so-called Nataf transformation can be applied, see, e.g., [21]. Then the sample is established according to Eq. (14). The values of the sample are then transformed back to the non-Gaussian domain, see, e.g., [21]. For a more detailed description of the entire procedure, see, e.g., [14,20]. 2.2.4. Monte-Carlo simulation Due to the introduced scatter of concrete material properties a single simulation of the load-bearing response of the RC structural member is not sufficient anymore as it would be the case in a deterministic model. Thus, a series of simulations has to be carried out in a so-called Monte-Carlo type simulation (MCS), see e.g. [22]. In such a simulation, a number nset of sets, which each consists of nrv random numbers n, are sampled by a random number generator. For each set, the realization of the random field representing the spatial fluctuation of a material property is carried out according to Eq. (14). The respective values are then applied to the Finite Element model and the simulation is performed. This procedure is carried out for all nset sets. The results of the simulations are then analyzed to gain result distributions and their parameters, e.g., mean values, standard deviations and quantile values. In the following, the procedure is specified. The requirements for random number sampling and evaluation of the results are focused on crack widths in this contribution. The appropriate sampling of sets of random numbers is a topic on its own, see, e.g., [22,23]. As statistical parameters derived from a limited number of samples are subject to scatter itself, the concept of confidence intervals has to be applied. With some prior knowledge, statistical parameters of the crack width distribution may be

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J.U. Hartig, U. Häussler-Combe / Theoretical and Applied Fracture Mechanics 74 (2014) 39–47

estimated. This leads to relations between number of samples, level of significance and confidence interval. A number of nset = 1000 sets gives a high level of significance with narrow confidence interval in the current context. In each loading step of each simulation, the crack width w, the reinforcement stress rsr of each crack and the actual value of imposed deformation eimp are recorded for subsequent evaluation. 3. Results and discussion 3.1. Validation of numerical model Experimental data concerning restrained thermal loading is rare. For the validation of the numerical model, results of experiments performed in [8] are used. In these experiments, seven RC concrete bars of a length of 6 m, a width of 0.17 m and a height of 0.35 m were exposed to thermal loading. Essential properties of the specimens are summarized in Table 1. The specimens were heated at first in a water bath up to a temperature of 80 °C. When the temperature was reached the specimens were fixed at both ends and a moderate pre-stressing corresponding to about one third of the concrete tensile strength was applied. Afterwards, the specimens were cooled down to ambient temperature of about 20 °C, which imposed further tensile stresses in the concrete exceeding the concrete tensile strength and led to cracks in the concrete. The ends of each specimen were equipped with additional reinforcement on lengths of 0.5 up to 0.8 m to ensure load transfer from the bearing. This shortened the lengths were concrete cracks could occur in the specimens. The specimens were reinforced with different reinforcement ratios, reinforcement diameters and reinforcement bar types. For specimen 1a, ribbed reinforcement bars with a diameter of 4 mm were applied while for the other specimens twisted reinforcement bars (Torstahl) with different diameters were used, see Table 1. The Young’s modulus of the steel is Es  200,000 N/mm2. For the steel bars used in specimen 1a, the yield strength is fsy  600 N/mm2 and for the other specimen fsy  400 N/mm2, which led in the experiments 1b and 1c to plastic deformation of the reinforcement. This is taken into account in the model with Eq. (12) considering a strength limit fst = 480 N/mm2 at a strain e(fst) = 0.01. For the concrete, the mean values of Young’s modulus and tensile strength were determined with Ec = 31774 N/mm2 and fct = 2.1 N/mm2. While mean Young’s modulus was used for all simulations, the mean concrete tensile strength for each specimen determined with prisms of 10  10  53 cm3 was used for the corresponding simulation, see Table 1. The thermal expansion coefficients for both concrete and steel are aT  1  105 K1.

Both reinforcement bar types are not used anymore. While the material laws for the used steels are given in [8], information concerning the bond behavior in the concrete is missing. Some information regarding the bond behavior of Torstahl is given in [24], which indicate lower transferable bond stresses compared to deformed steel bars as used today. For the model, the bond law parameters were calibrated such that a good agreement of the cracking characteristics between experiments and simulations was achieved. The supporting points for the bond law were defined with a bond strength of smax = 8 N/mm2 at smax = 0.05 mm and a constant frictional bond stress of sf = 1 N/mm2 starting at sf = 0.15 mm, see Fig. 3. The same values were used for all simulations. A bar element length of 0.01 m is applied in the model, which results in 425 up to 500 elements per bar element chain depending on the not strengthened length of the specimens (cracking length) according to Table 1. The nodal displacements of the first node of the concrete and the reinforcement bar element chain at x = 0 were prescribed equal to zero. In the first step the pre-load was applied corresponding to the experiments as prescribed displacements of the last nodes of the bar element chains. The respective displacements are given in Table 1 for each simulation. Afterwards, the end displacement was fixed and thermal loading DT was applied incrementally decreasing from 60 K to 0 K as imposed strains eimp = aT DT to all bar elements. Spatial scatter of material parameters was neglected in the simulations for model validation and constant values were applied as defined previously. Fig. 4 shows experimental and numerical imposed force versus temperature change (Fimp–DT) relations for three of the seven parameter combinations. It can be seen that the general tendencies of the load-bearing response are appropriately reproduced by the model. The successively increasing forces during multiple concrete cracking are not reflected by the model due to the assumption of spatially constant concrete tensile strength. Moreover, the imposed force level for the crack development is overestimated by the model. One of the main reasons is that mean values of concrete tensile strength given in [8] were applied, which were determined in direct tension tests on small-sized concrete prisms. It is known and referred to as size effect that local strength values in large concrete specimens might be considerably lower than in small specimens. It will be shown subsequently in the example of application that the model has the ability to reflect this behavior when spatial scatter of fct is taken into account. On the other hand, this increases numerical effort considerably due to the necessity of carrying out a series of simulations for each parameter combination in a MCS as already pointed out. Table 1 provides mean crack width wm, crack numbers and reinforcement stresses at the cracks rsr as determined in the

Table 1 Results of experiments in [8] and comparison to results of the model. Specimen no.

1a

1b

1c

2a

2c

3a

3c

Specimen properties ds (mm) Number of bars qeff (%) Cracking length (m) Pre-deformation (mm) fct (MN/m2)

4 24 0.50 4.35 0.081 2.06

6 10 0.48 4.28 0.069 1.83

10 4 0.51 4.25 0.000 1.86

6 16 0.77 5.00 0.086 2.11

12 4 0.75 4.60 0.079 2.02

8 12 1.00 4.50 0.051 2.40

14 4 1.02 4.75 0.071 2.29

Experimental results (DT  60 K) Number of cracks wm (mm) rsr (MN/m2)

12 0.19 420

14 0.17 399

5 0.32 365

27 0.10 291

10 0.23 278

24 0.08 252

14 0.16 236

Model results (DT = 60 K) Number of cracks wm (mm) rsr (MN/m2)

18 0.14 420

13 0.18 397

6 0.41 364

25 0.11 301

15 0.18 280

23 0.11 262

15 0.17 265

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J.U. Hartig, U. Häussler-Combe / Theoretical and Applied Fracture Mechanics 74 (2014) 39–47 Table 2 Parameters for the calculations with the numerical model.

experiments and the simulations at a thermal loading of DT  60 K. It can be seen that the reinforcement stresses at the cracks rsr are appropriately reproduced by the model. Moreover, it can be stated that the values of rsr are relatively insensitive to changes in the bond law parameters. This implies also that concrete tension softening has only small influence on the cracking behavior of these RC bars because otherwise rsr would be significantly overestimated by the model. The cracking characteristics are also well reproduced by the model although the deviations between experiments and simulations are somewhat larger compared to rsr. It should be noted that crack width results are in general subject to relatively large scatter and only one experiment was performed for each parameter combination. Corresponding to the experimental results, the model shows decreasing crack widths and increasing numbers of cracks with increasing reinforcement ratio and decreasing reinforcement diameter due to increasing bond area. It is also worth to note that the cracking characteristics in the simulations are quite sensitive to the bond law parameters. In summary, the validity of the model can be confirmed with the limits of its deterministic character. The quality of the model is better than the analytical model in Section 2.1 because a more realistic description of the interaction between concrete and reinforcement is applied.

3.2. Example of application In the following, an example of application of the model is presented, which aims for analyzing the statistics of crack widths. Therefor, the tensile strength of the concrete is modeled spatially scattering and a Monte-Carlo simulation is performed. For the example, a RC tension bar of 2 m length with one reinforcement bar is considered. The RC tension bar is modeled with bar elements of 0.01 m length and bond elements as previously described. Boundary conditions are given with fixed nodes at both ends of the bar elements chains. Thermal loading of DT = 100 K is applied as imposed strains incrementally decreasing in steps of 0.5 K. The cross-sectional and bond areas are given in Table 2. The material properties of the concrete are chosen corresponding to concrete grade C30 according to [1]. The material parameters of reinforcement steel and concrete are also given in Table 2. Both materials are assumed to have a thermal expansion coefficient of aT = 1  105 K1. The bond parameters are chosen corresponding to the recommendations of [1] for local bond stress–slip relations. The interpolation between the supporting points is performed again with the PCHIP approach. For establishing the random field of the concrete tensile strength, a correlation length has to be defined. It is estimated with 32 mm corresponding to twice the assumed maximum aggregate size of 16 mm. For the concrete

34,000 MN/m2 30 MN/m2 2.9 MN/m2 0.547 MN/m2 0.05 m 0.01 m2

Reinforcement Young’s modulus Yield strength Number and diameter Cross-sectional area Effective reinforcement ratio

200,000 MN/m2 500 MN/m2 1 Ø 14 mm 154 mm2 1.54%

Bond Maximum bond stress smax Frictional bond stress sf Slip at maximum bond stress smax Slip at frictional bond stress sf Bond area

11.0 MN/m2 1.6 MN/m2 0.6 mm 1.0 mm 4398 mm2/m

tensile strength, a Gaussian distribution is assumed in a first approach. Furthermore, the local bond stress values sloc of the bond laws are directly coupled with the local concrete tensile strength values fct,loc according to the relation sloc = smeanfct,loc/fct,mean. As the integration point positions of the bar elements and the bond elements do not coincide, the values sloc are determined by interpolation from fct,loc values. The analytical model of [4], cf. Section 2.1, is also applied to analyze this example of application and to have a basis for comparison regarding crack width estimation in current design codes. In this model, the interaction between concrete and reinforcement is taken into account with a constant bond stress sm, which is defined corresponding to [1] with sm = 1.8fct = 5.22 N/mm2, and the shape parameter related to the bond stress distribution bt = 0.4. In the Monte-Carlo simulation, 1000 simulations with different realizations of the random field for the concrete tensile strength were carried out. In the following, the results of these simulations are analyzed. At first, the reaction forces depending on imposed deformations are shown in Fig. 5 as rsr–eimp relations. For comparison also the results of the analytical model according to [4] and a deterministic simulation with spatially constant concrete tensile strength are shown in Fig. 5. The load levels where the first concrete cracks develop are in all simulations of the MCS smaller compared to the deterministic models (both analytical and numerical). The reason is that due to the spatial scatter of the concrete tensile strength always regions with lower strength values compared to the mean

300 numerical model (deterministic)

σ sr [MN/m²]

Fig. 4. Experimental results on imposed forces with decreasing temperature in [8] and comparison to results of the model (notations ‘‘1c, 2c, 3c’’ refer to specimen numbers in [8]).

Concrete Young’s modulus Compressive strength Mean value of tensile strength Standard deviation of tensile strength Correlation length of tensile strength Cross-sectional area

200

analytical model 100

0

numerical model (stochastic)

0

20

40

60

80

100

|ΔT| [K] Fig. 5. Reinforcement stresses at the crack due to temperature reduction (grey lines represent the results of the Monte-Carlo simulation).

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J.U. Hartig, U. Häussler-Combe / Theoretical and Applied Fracture Mechanics 74 (2014) 39–47

5.0

σc and fct [MN/m²]

crack fct

4.0 3.0 2.0

σc

1.0 0

0

0.5

1.0

1.5

2.0

x [m] Fig. 6. Stochastic sample: tensile strength and normal stress of concrete along x at DT = 100 K.

Fig. 7. Stochastic sample: normal stress in reinforcement along x at DT = 100 K.

Fig. 8. Stochastic sample: bond stress along x at DT = 100 K.

0.4

numerical model (stochastic)

0.3 w [mm]

strength exist, which are preferential positions for concrete cracking. In contrast to the analytical model, the stabilized cracking state is not reached in the numerical simulations. Moreover, it can be seen that the results of the numerical simulations show relatively strong scatter due to scattering concrete tensile strengths and bond strengths. The comprehension of the load-bearing behavior can be improved when the stress distributions are taken into account. Fig. 6 shows the calculated concrete stresses along the bar axis x for a thermal loading of DT = 100 K for one of the stochastic simulations. At the cracks, the concrete stresses are zero, which also visualizes scattering crack spacing. Crack spacing shows scatter due to scattering concrete tensile strength. The cracks develop in the vicinity of strength minima but depend also on the maxima in concrete normal stress. The course of the reinforcement stresses along x is reversed compared to the concrete with stress maxima at the cracks where the concrete has stress minima, see Fig. 7. The course of the bond stresses along x, which is observable in Fig. 8, shows sign changes at the cracks and zero-crossing between neighboring cracks as well as maximum values. It can be seen that the bond stresses do not reach the bond strength values smax, which are also subject to scatter corresponding to the assumed bond law. Hence, it can be assumed that with further temperature reduction more cracks will develop. The scatter in crack spacing also results in scatter in crack widths. Fig. 9 shows the mean crack widths of each sample in the Monte-Carlo set depending on imposed deformations or temperature change, respectively. Also the results of the analytical and a deterministic simulation with spatially constant concrete tensile strength are shown. The mean crack widths show strong scatter in the MCS. At maximum loading, the mean crack width

numerical model (deterministic)

0.2 analytical model

0.1 0

0

20

40 60 |ΔT | [K]

80

100

Fig. 9. Mean crack width due to temperature reduction (grey lines represent the results of the Monte-Carlo simulation).

is in a range of approximately 0.2–0.35 mm. The prediction of the analytical model is close to the lower limit of the MCS. Mean values of crack width might be, however, not very significant for characterizing the potential detriment the cracks may cause. Thus, a more detailed statistical evaluation of the crack widths is carried out. Frequency distributions of discrete crack width intervals related to discrete rsr intervals are determined and plotted in histograms, see Fig. 10. Therefore, the crack width values were evaluated for every crack at each load step in all simulations associated with the corresponding values of rsr. Fig. 10 shows crack width histograms for three different rsr intervals. The shapes of the histograms reveal that crack widths increase with increasing rsr or loading, respectively. This is expressed by the mean value E(w), which increases almost linearly. Also the 5% and 95% quantile values (w0.05 and w0.95), which characterize the percentage of cracks having a crack width smaller than the quantile value, increase with increasing loading. Furthermore, the variability of the crack widths increases with increasing loading, which is associated with increasing standard deviation D(w). The histograms are asymmetric exhibiting a left-skewness. This might be caused by a multimodality of the underlying distribution. Multimodality is explained by preferred patterns of crack spacing basically associated with Eq. (9), which indicates an upper limit of crack spacing. Another preferred crack spacing value for higher load levels is given by the half of the value of Eq. (9). This leads also to preferred crack width values because crack spacing is strongly related to crack widths. In Fig. 11, the histograms over the entire loading range are summarized. It can be seen again that mean value, standard deviation and quantile values of the crack width increase with increasing loading. Due to the increasing standard deviation, the maximum crack widths, as indicated with the 95%-quantile, increase stronger

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J.U. Hartig, U. Häussler-Combe / Theoretical and Applied Fracture Mechanics 74 (2014) 39–47

0.5 0.4

σsr = 80-90 MN/m² E(w) = 0.109 mm D(w) = 0.013 mm w0.05 = 0.078 mm w0.95 = 0.122 mm σsr = 180-190 MN/m² E(w) = 0.207 mm D(w) = 0.032 mm w0.05 = 0.131 mm w0.95 = 0.246 mm

0.3 0.2 0.1 0

σsr = 280-290 MN/m² E(w) = 0.301 mm D(w) = 0.053 mm w0.05 = 0.180 mm w0.95 = 0.365 mm

0.00-0.01 0.02-0.03 0.04-0.05 0.06-0.07 0.08-0.09 0.10-0.11 0.12-0.13 0.14-0.15 0.16-0.17 0.18-0.19 0.20-0.21 0.22-0.23 0.24-0.25 0.26-0.27 0.28-0.29 0.30-0.31 0.32-0.33 0.34-0.35 0.36-0.37 0.38-0.39 0.40-0.41 0.42-0.43

relative frequency

0.6

w intervals [mm]

<30 40-50 60-70 80-90 100-110 120-130 140-150 160-170 180-190 200-210 220-230 240-250 260-270 280-290

E(w) E(w) - D(w) E(w) + D(w) w0.05 w0.95 wanalytical

0.00-0.01 0.02-0.03 0.04-0.05 0.06-0.07 0.08-0.09 0.10-0.11 0.12-0.13 0.14-0.15 0.16-0.17 0.18-0.19 0.20-0.21 0.22-0.23 0.24-0.25 0.26-0.27 0.28-0.29 0.30-0.31 0.32-0.33 0.34-0.35 0.36-0.37 0.38-0.39 0.40-0.41 0.42-0.43

σsr intervals [MN/m²]

Fig. 10. Crack width histograms for various rsr intervals.

w intervals [mm] Fig. 11. Evaluation of crack width properties.

than the mean values. This illustrates that it might be important to design rather for maximum crack width than mean crack width. In this context, it is important to point out again that current recommendations by design codes are based on models only providing mean crack widths values, which considerably underestimate maximum crack widths. In fact, even the mean crack width is considerably underestimated by the analytical model in the parameter combination under consideration as it can be seen in Fig. 11 for wanalytical. Moreover, only a value of bt = 0.4 was applied in the analytical model, while [1] recommends, e.g., a value of 0.6 for short-term loading. Applying the larger value results in more cracks and, thus, even smaller crack widths as well as larger differences to the mean and maximum crack width values predicted by the MCS.

4. Summary and conclusions In this contribution, a numerical model within the framework of the stochastic Finite Element Method was presented and applied to RC bars under uniaxial imposed deformations due to thermal loading. The concrete tensile strength and the bond strength between concrete and reinforcement were modeled spatially scattering applying a random field approach. Due to the scatter in material parameters also scatter in the response of the material to loading occurs. To evaluate these dependencies a Monte-Carlo simulation was carried out. The statistical evaluation was focused on crack widths. It was shown that mean crack width limits as usually recommended by design codes to ensure serviceability underestimate occurring maximum crack widths significantly. Even worse is the

fact is that also mean crack width is considerably underestimated by the analytical model, at least for the model under consideration. This reduces the reliability of the structural design. The presented model is an efficient means for analyzing the statistical properties of crack widths. The quality of the model can be evaluated as better than the analytical approach provided by current design codes because the geometric resolution is the same (one-dimensional) but the description of the material is more sophisticated in the numerical model. The available results are more extensive in the numerical model providing, e.g., statistics of crack width. Nevertheless, these investigations have to be regarded as a first example of application. Scatter in material parameters was only considered for concrete tensile and bond strength. Other material properties and also the geometry were assumed constant in a first approach. Scatter of a number of parameters may in principle be modeled with coupled stochastic fields in the same way as has been demonstrated in this contribution. Furthermore, all loading types and material properties like creep and shrinkage may be taken into account. Some uncertainty remains in the selection of distribution types and statistical properties of scattering parameters. Nevertheless, the discussed models and methods may give a better comprehension of the stochastic cracking of reinforced concrete based on further comprehensive studies. This might be the basis for improved recommendations for determining cracking properties of RC structures in future design codes, e.g. by means of improved construction rules as already available in [2] without the necessity of performing extensive Monte-Carlo simulations.

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