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The stochastic modelling of the microcracking and the force-displacement behaviour of the tensile steel reinforced tie using the lattice model is presented in the current article. The three-dimension problem of the modelling of the tie is reduced to the two-dimensional so as the main stiffness parameters of the concrete and the reinforcement of the two-dimensional model would be the same as for the three-dimensional. The concrete and steel obey the Hook law. All elastic constants, as well as dimensions of the tie, were assumed as the deterministic quantities except for the critical concrete tensile strains which were treated as a two-dimensional stationary uncorrelated truncated Gaussian random field. The discrete element approach and the explicit integration scheme have been used for the modelling. The estimations of the main parameters of the force-displacement behaviour stochastic process and other statistical indexes were obtained using 72 realization of the force-displacement behaviour of a chosen model. Extra two stochastic realizations of the two different models, as well as three deterministic models, were modelled to compare stochastic and deterministic behaviour of the force-displacement behaviour. The analysis showed that the force-displacement behaviour of the tie under tensile force cannot be treated as a Gaussian stochastic process when the p value is 0.05 at the small displacements and within the interval when the cracking of the concrete is very intensive. However, at the bigger displacements, when the cracking becomes less intensive, the tensile force can be treated as a Gaussian random variable.

The cracking and the force-displacement behaviour of the reinforced concrete structures is still under intensive consideration for several decades, see Hegemier, Murakami and Hageman [

The bond between concrete and reinforcement as well as cracking of the concrete has a crucial influence on the bearing capacity and the deformation behaviour of the reinforced concrete structures. It is well known that such phenomena as deflections and cracking of the reinforced concrete structures have a big variability that has to be taken into account in practical usage of the structures to provide the required reliability and safety. Despite the fact that there is a huge amount of the experimental and theoretical investigations of the force-displacement and tension stiffening behaviour of the reinforced concrete ties, there is very few works of literature in which the force-displacement behaviour of the reinforcement tie is analysed as a stochastic process. Therefore, the identification of the force-displacement stochastic process and the estimation of the parameters of the process are actual issues.

In the present article, the stochastic modelling of the two-dimensional computational model of the steel reinforced tie is presented when the critical concrete tensile strain is a truncated normal random variable while other properties of the tie, concrete and steel, that is moduli of elasticity, Poison’s ratios as well as the dimensions of the two-dimensional computational model are deterministic quantities. The behaviour of the tie tensile force dependent on the displacement of the reinforcement is the principal object of the investigation. The investigation has been conducted by modelling six identical models of the tie except for the critical concrete tensile strains: three deterministic models were modelled with different deterministic (fixed) critical concrete tensile strains, and three stochastic models were modelled with spatially distributed critical concrete tensile strains. Stochastic properties of one model were estimated using 72 realizations. Number of realizations of remaining stochastic models is one. The novelty of the article is the identification and characterisation of the force-displacement stochastic process of the reinforced tie, also the estimation of the parameters of the stochastic process such as the range, coefficient of correlations and so on.

The geometry of the tie is shown in Figure _{0} = 56.80 mm. The height and the width of the model are of the same size:

Geometry of the steel: general view of the tie (a), a cross-section of the tie (b).

The left face of the end of the tie is fixed. The displacement

To apply the planar equilateral triangular elastic spring lattice method for the tie, the 3D tie model is reduced to the 2D model consisting of the concrete and the steel reinforcement. The geometry of the 2D model is shown in Figure

Geometry, boundary conditions, and action of the 2D model of the tie: a scheme of the 2D model (a), a section of the 2D model (b), a geometry of the computational model of the tie (c), and a model of the steel part (d).

For the simulation of the tie, the half of the tie is modelled by using the lattice methodology. The geometry of the model used for the simulation is shown in Figure

It is assumed that the concrete of the 2D model of the tie consists of the spherical aggregate particles of diameter 0.35 mm embedded in the hardened cement paste, Figure

Structure of the concrete part of the 2D model.

The cross-section areas and moduli of elasticity of the 2D model are obtained from the following equations:

The effective cross-section area

where ^{2}. Then, the effective height of the 2D reinforcement ^{2} and these values were adopted for the modelling of the 2D model of the tie. On the bases of the equations ^{2}, and

The concrete and steel parts of the 2D tie model are discretised by the equilateral triangular elastic spring lattices (ETESL). A spring connects the centres of two vicinal aggregate particles. The springs are pin-connected. The length of the leg of a triangular cell is 0.4 mm; that is, the same as the distance between two vicinal aggregate particles. The principal scheme of the discretisation of the 2D model of the ties is shown in Figure

The discrete model of the computational 2D model of the tie: a plane view with the boundary conditions (a); a fragment of the model in the vicinity of the rib (b); the one-dimensional spring element combining two particles acting via the interface (c) and (d), and fragment of the lattice (e).

The final discrete model consists of 42264 spring elements of equal length, 8734 of them belonging to the 2D reinforcement, and 33530 elements belonging to the concrete.

The lattice method is an approximation of the continuum, therefore, has disadvantages. Most important of them are the following: the parameters of the lattice such as the lattice geometry and size, the axial or bending stiffness, the critical tensile strain or stress of the failure criterion, the constitutive relation of the beam or rod have to be determined depending on the continuum, also depending on the chosen criterion of proximity between modelled and the required results of the target continuum. The criterion can be energy stored in the volume of the continuum, displacements or deformations, strength and so on. The next disadvantage is that the Poisson ratio of the continuum approximated by the triangular lattice always equals 1/3. To overcome this problem, the special type of lattices, for example with angular springs, is used; see Ostoja-Starzewski [

According to Hrennikoff [

The fracture of the concrete is modelled by breaking the spring when the strain of the spring exceeds the critical strain, i.e., when

Six kinds of the tie of the same properties except for the critical concrete strains

For the stochastic modelling of the cracking of the concrete and the tie behaviour the critical concrete tensile strains

The random variable

For the sake of the convenience, the elastic constants of the reinforcement, concrete, the effective elastic constants of the 2D model, and the stiffness of the springs of the 2D model are summarised in Table

Summary of properties of the deterministic and random models of the computational models of the ties.

Designation of the specimen | Moduli of elasticity for the 3D model in GPa | Effective moduli of elasticity for the 2D model in GPa | Stiffness of the spring _{s} for the lattice model in MN/m | Critical tensile strains _{crit} of the concrete | ||||
---|---|---|---|---|---|---|---|---|

_{ s } for the steel reinforcement | _{ c } for of the concrete | | | | | Mean values | Coefficient of variation | |

3D tie model | 2D model and computational model | |||||||

| ||||||||

Deterministic models | ||||||||

| ||||||||

DM1 | 210 | 9.954 | 7143.423 | 2333.284 | 2165.235 | 707.239 | 2.260·10^{−3} | 0 |

DM2 | 1.298·10^{−3} | |||||||

DM3 | 0.322·10^{−3} | |||||||

| ||||||||

Spatially variable model | ||||||||

| ||||||||

RM1 | 210 | 9.954 | 7143.423 | 2333.284 | 2165.235 | 707.239 | 2.260·10^{−3} | 0.18 |

RM2 | 1.298·10^{−3} | |||||||

RM3 | 0.322·10^{−3} |

The discrete element methodology, Cundall [

The simulation program has been written in C++11. The geometry of the model and nodes coordinates for the computation program have been created by the finite element program ANSYS 12. One realization of the stochastic process of the force-displacement behaviour of a tie took at least 7 hours using the one core of the computer whose parameters were as follows: RAM 4GB, CPU AMD Phenom II X4 955 3.2 GHz 4 cores.

In this section, the results of the stochastic analysis of the tensile force and the number of the broken springs of the concrete of the tie dependent on the displacement of the reinforcement are presented. The statistical analysis was performed with the program “RStudio” [

The influence of the random critical concrete tensile strain

The dependence of the tensile axial forces

Tensile axial force-displacement relationships of the tie: deterministic

As we can see from Figure

The dependencies of the axial tensile force

Dependencies of the tensile force

Figure

Coefficient of variation

Estimated coefficient of variation

The differences

The differences

The dependencies of the estimated coefficients of variation

As we can see, from Figures

The dependencies of the differences

The histograms and the kernel density estimations of the realizations

Histograms of the tensile force

The histograms and kernel density estimations of the realizations

Histograms of the tensile force

The analysis of the normality of the realizations of the r.v.

The correlation matrices of the correlation coefficients

The correlation matrices of r.v.

Dependencies of the axial force

The weak correlations

The dependencies of the tensile forces

Figure _{crit} the tensile force

The evaluation of the cracks pattern of the specimens DM2 and DM3, when the critical concrete tensile strain

Cracks patterns of the specimen DM2, when

Cracks patterns of the specimen DM3, when

Cracking patterns of the deterministic, DM1, DM2 and DM3, and random, RM1, RM2 and RM3, models at the displacement

Cracking patterns of the specimens DM

From Figure

The force-displacement behaviour of the reinforced tie with spatially variable concrete tensile strains was modelled and investigated as a stochastic process using the regular triangular 2D lattice. The analysis showed that the 2D lattice approach, in general, can be applied to model the force-displacement behaviour of the reinforced concrete ties. The stochastic modelling shows that the distribution of the tensile force with respect to the displacement of the reinforcement is not a Gaussian random variable at the small displacements as well as within the intervals of the displacement within which the formation of the new concrete cracks is very intensive,

Cross-section areas of the concrete and the reinforcement of the 3D model of the tie

Effective cross-section areas of the concrete and the reinforcement of the 2D computational model of the tie respectively

Deterministic specimens

Moduli of elasticity, an effective moduli of elasticity of the concrete, and reinforcement of the 3D and 2D tie models, respectively

An average axial force acting at the free end of the reinforcement

An axial force acting at the free end of the reinforcement

Supremum and infimum of the axial force

The fracture toughness of the concrete

Spring stiffnesses representing the concrete and the steel of the computational 2D model

The initial lengths of 3D and 2D models of the tie and reinforcement

Mean, variance, and coefficient of variation of the random variable

Number of broken springs

Probability of an event

Random variable

Random specimens whose critical concrete strains are random fields

Radius of a particle of the concrete

Estimated standard deviation and mean respectively

Standard deviation

The displacements of the free end of the reinforcement in direction

Poison’s ratios of the concrete and the steel of the 3D and 2D models of the tie

The thickness of the concrete and the reinforcement of the 2D model of the tie

Elongation of a spring

The concrete strain and the critical concrete tensile strain respectively

Mean

Coefficients of correlation

Truncated and untruncated normal random variables of the critical tensile strain of the concrete.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.