1.     Introduction

Reinforced concrete slabs are one of the main structural elements of many industrial and civil construction projects. Reinforced concrete structures have nonlinear properties due to both the material and the structure. The concrete stress-strain diagram is fundamentally different from the steel stress-strain diagram. Steel withstands compressive and tensile stress almost equally well. Concrete withstands compressive stress well and almost does not withstand tensile stress [1, 2]. Significant differences in the deformation of concrete and steel lie in the redistribution of internal forces between structural elements. The formation and development of cracks require special attention.

While the nonlinearity of reinforcing steel is measured only after its stress limits are limited, concrete exhibits nonlinear mechanical properties under all conditions [3–5].

Taking into account physical nonlinearity is associated with large expenditures of computing power. The initial approximation of the parameters of the stress-strain state of reinforced concrete slab structures can be obtained with a linear calculation. However, this approach is only acceptable when the stresses in the reinforcement are less than the proportionality limit and there are no cracks in the concrete [6, 7].

The nature of the stress-strain state of a reinforced concrete slab is determined by the acting loads and the conditions of fastening. Rectangular slabs supported on two opposite sides are correctly described as beams. Their stress-strain state is well studied [8–10]. However, more complex fixation has been little studied.

When studying the stress state of reinforced concrete structures, numerical modeling using the finite element method is most widely used. This is due to the ability of software packages (Ansys, SAP, Nastran, T-Flex Cad, Abaqus, Stark ES, etc.) to take into account the nonlinear properties of the material. As a rule, the stress-strain state was assessed using the Drucker–Prager and Rankine strength criteria uniformly throughout the entire thickness of the material. Calculations in various software packages give a similar picture of the redistribution of stresses in reinforced concrete slabs. For example, in the Ansys software package, the stress-strain state is assessed using the Drucker–Prager and Rankine strength criteria [11–14].

The nonlinear nature of deformation of reinforced concrete slabs is further complicated by the process of crack formation [15, 16]. After cracks form, stress redistribution occurs. The tensile force in the crack is perceived only by the reinforcement. The tensile stress in the concrete near the crack decreases and is redistributed to more distant areas. The crack trajectories pass through points with the greatest principal tensile stresses [17, 18].

Previous studies [19–23] have determined that the process of crack formation is affected by: loading history, nonlinear calculation scheme, and crack formation model. In this regard, it became necessary to describe the nonlinearity of concrete not in the form of a monolithic material, but taking into account its layer-by-layer loading.

The aim of the study is to develop and test the adequacy of an improved mathematical model of the stress-strain state of reinforced concrete slabs through a layer-by-layer description of the physical nonlinearity of concrete and the possibility of crack formation in concrete.

To achieve this goal, the following research objectives were formulated:

1.    Develop an improved mathematical model of the stress-strain state of a reinforced concrete slab with a layer-by-layer description of concrete nonlinearity.

2.    Develop a methodology and conduct experimental studies to assess the stress-strain state of a reinforced concrete slab loaded in the middle and simply supported on three sides.

3.    Analyze and compare the results of the numerical and field experiments to assess the adequacy of the mathematical model.

                                                                                    2.     Materials and Methods

2.1.               Mathematical Model

To calculate reinforced concrete slabs using the finite element method, the proprietary DIVLOC software package was developed jointly with the Bryansk State Engineering and Technological University. The peculiarity of the package was that concrete with discrete reinforcement bars was described as a multilayer material using the finite element method, taking into account the nonlinear behavior of materials (Fig. 1). Concrete deformation occurs in accordance with the theory of N.N. Karpenko [24] when cracks form in concrete based on the mechanics of V.I. Murashev.

Figure 1. Model of the slab: 1 are concrete layers; 2 are reinforcement elements.

The iterative calculation procedure is implemented using the variable elasticity method. In the initial iteration, the slab is modeled linearly with the initial elastic moduli of the materials. In subsequent iterations, the elastic properties are refined in the stiffness matrices based on the previously obtained stresses. Kirchhoff's hypothesis of the absence of interlayer pressure during deformation is adopted as an assumption. The system of linear algebraic equations includes: a stiffness matrix, taking into account the secant elasticity models of the materials, and the external load reduced to the nodes, as well as the nodal displacement vectors.

A 50 mm thick rectangular slab with regular reinforcement of 14 mm diameter longitudinal reinforcement at 200 mm intervals was chosen for the calculation. The calculation model consists of 1,554 multilayer triangular concrete finite elements and 3,254 bar finite elements. The specified materials are B25 concrete and A400 working reinforcement. The plate was freely supported on three sides. The loading was carried out in the middle of the free side (Fig. 2).

Figure 2. Finite element model of a reinforced concrete slab.

During loading, the tearing off of the support faces was allowed with one-sided support of the slab. The finite element model of the slab included 176 multilayer triangular finite elements of concrete and 173 rod finite elements of reinforcement. Also included in the scheme were 20 supporting finite elements, allowing for the modeling of tearing off.

Limitations on strength, rigidity and crack resistance of the slab were taken into account in accordance with the requirements [25]:

-         strength conditions

                                                                          (1)

where  is the third principal stress in concrete;  is the permissible design compressive strength of concrete for the considered loading mode;  is the longitudinal deformation in the reinforcement;  is the permissible deformation of the reinforcement under tension and compression;

-         rigidity condition

                                                                         (2)

where  is the displacement in the vertical direction;  is the maximum allowable deflection;

-         crack resistance conditions

                                                                         (3)

where  is the crack opening width;  is the maximum permissible crack opening width.

According to the theory of N.I. Karpenko [24], the strength of concrete under a complex stress state is found using the limit surface in the space of principal stresses. For heavy concrete, the equation of this surface has the form

                       (4)

where  is the coefficient;    are the strength of concrete under biaxial compression and tension.

According to the static loading condition, the relationships between the calculated resistances  in the principal directions remain the same as between the principal stresses. In the absence of cracks, the secant elastic moduli and secant coefficients of transverse deformations are determined based on the level of the principal stresses

                                                                          (5)

The level of loading allows us to estimate the nature of crack formation when failures occur in stretched concrete and to check it for the condition of strength under compression. If at some step of iterative calculations, the conditions  are fixed, then this is already the basis for stating the formation of a crack in the plane of the main platform [26]. Preliminary elasticity matrices of finite elements are found from the condition of zero elastic modulus in the direction perpendicular to each crack elastic modulus of concrete.

2.2.               Experimental Method

To check the accuracy of the calculation, experimental studies were additionally conducted. Three experimental samples of reinforced concrete slabs were manufactured (B25 class concrete and VR-I reinforcement d4 mm). Working reinforcement with a pitch of 50 × 50 mm was located at a distance of 8 mm from the lower surface of the slab. The experiment was carried out using a stand (Fig. 3).

Figure 3. Scheme (a) and photo (b) of the experimental setup: 1 is experimental plate; 2 is support frame; 3 is beam-lever; 4 is loading stamp; 5 is pallet for loads; 6 are loads; 7 is lever support.

The support frame is a spatial welded structure made of rolled steel sections (I-beams No. 10, channels No. 12 and round rolled steel d16). The experimental plate was loaded using a beam-lever (two I-beams No. 10) through a stamp with a support surface of 250 × 250 mm. The stamp consisted of a centering angle and two rigid plates. The angle ensured the accuracy of the load placement. The plates ensured the load distribution over the entire surface of the stamp. The loading was carried out for 10 minutes. After that, the sensor readings were measured.

Free support of the slab on three sides was achieved using round rolled steel and metal plates. The round profile rested on the frame, and the plates protected the slab from crushing.

The deflections of the plate were measured by clock-type indicators. The displacement indicators were located in the middle of the free edge of the plate and near the supports. The registration of the plate deformations was carried out by constantan foil strain gauges with a measuring base of 50 mm and a resistance of 120±2 Ohm (Fig. 4).

Figure 4. Layout diagram (a) and photo (b) of strain gauges on the test plate.

The strain gauges on the experimental plate were fixed with cold-setting ethyl cyanoacrylate adhesive. The sensors were connected using an eight-channel external half-bridge circuit. Each channel connected a compensation and measuring sensor and two passive resistances.

                                                                                   3.     Results and Discussion

The algorithm implemented in the DIVLOC software package was validated by comparing the obtained results with those of calculations based on the STARK_ES software package. This software package is designed for the strength analysis of large-scale reinforced concrete buildings and structures using the finite element method. It is a domestic development and is certified according to Russian construction standards [27].

According to the calculation, the maximum pressure was 120 kPa. Then the calculations were interrupted, which corresponded to the destruction of the reinforced concrete slab.

At the maximum load (120 kPa), the iteration process converged only after 300–500 iterations. With the growth of bending deformations of the slab, its corners rise above the supports. This is explained by the excess of vertical deformations of corner points from bending (up) over their deformations from the weight of the slab (down). The recorded positive vertical displacement of the nodes (marked with bold dots in Fig. 2) corresponds to their separation from the supports. In the experiment, a separation of the support faces of the slab was also observed. The calculated separation length was 310 mm in the calculation and 340 mm in the experiment (the discrepancy is 11 %). This percentage of discrepancy can be considered sufficient for finite element method models. A more accurate match to experimental values can be achieved by using a finer finite element mesh. However, this will increase the calculation time [28]. The calculation time using DIVLOC was 2.5 hours, while using the STARK_ES software package, it took over 8 hours.

Using the STARK_ES software package, the calculated separation length of the slab corner from the support was 315 mm, which is in good agreement with our calculation results.

The greatest deflection of the plate was observed in the middle of the free edge of the plate, where strain gauges T5, T7, and T8 are located (Fig. 4a). A comparison of the calculated and experimental values of relative linear deformations in the load range of 21.6…124 kPa is shown in Fig. 5.

Figure 5. Calculated and experimental values of relative linear deformation e for sensors: a) T5; b) T7; c) T8.

Comparison of the results of finite element analysis and the experimental results showed fairly close values [29]. The discrepancies between the calculated and experimental values for sensor T5 were no more than 9 %; for sensor T7 – no more than 10 %, and for sensor T8 – 12 %. A significant increase in the difference between the experimental and calculated values with increasing load for sensor T8 is explained by the effect of punching in the area of its location. Before cracks appeared in the slab, the discrepancies between the calculated and experimental values of relative deformation were no more than 7 %, which confirms the adequacy of the proposed mathematical model with a layer-by-layer description of concrete. Similar results were obtained using the STARK_ES computer system, confirming the validity of the algorithm used.

The crack formation patterns obtained through finite element modeling largely replicate the experimental crack formation patterns. According to the finite element calculation, the main tensile stresses were observed on the lower edge of the concrete slab. Already at a load of more than 33 kN, crack initiation was observed. At the first stage of loading, cracks formed in the middle of the free edge, under the loading platform. With increasing load, the crack formation zone expands (Fig. 6). The greatest stresses are concentrated along the lines from the corner of the loading platform to the points of the boundaries of the slab lift areas above the supports. Redistribution of stresses between the loaded platform and the side edges leads to the formation of cracks in these areas [30, 31].

Figure 6. Scheme of crack formation on the lower surface of the plate under different loading.

In the experiment, the sensor readings also changed significantly in the stretched zones of the plate after the cracks formed. This is explained by the redistribution of stresses in the immediate vicinity of the crack.

                                                                                                    4.     Conclusion

An improved mathematical model of the stress-strain state of reinforced concrete slabs with a layer-by-layer description of the physical nonlinearity of concrete has been developed. The model takes into account the redistribution of stresses due to cracking. The adequacy of the model has been verified by a full-scale experiment. The discrepancy between the calculated and experimental results before cracks appeared in the slab did not exceed 23 %.

The use of one-way ties during bending revealed the separation of the slab corners from the supports. This effect was confirmed by the experiment. The difference in the separation length was 16 %.

Step-by-step loading made it possible to identify the onset of cracking in the reinforced concrete slab and obtain a complete picture of their further development. The resulting picture of cracking allows one to identify the most loaded sections of the structure, where additional reinforcement is required to increase their strength. The model can be used in the procedure of optimal design.