Strength and deformability models of cellular structure shells at static and short-term dynamic loading

. The paper presents the results of a comprehensive experimental and numerical study of the destruction of reinforced beams made of concrete and fibroconcrete under short-term dynamic action. Experimental studies on were carried out on a copra rig. A short-term dynamic load on the beam was created with the help of a falling load weighing 450 kg from a height of 700 mm. In experiments, the magnitude of the dynamic load was determined using a force gauge sensor, linear displacement sensors were used to calculate linear displacements. Numerical modeling is carried out in a full three-dimensional dynamic formulation within the framework of the phenomenological approach of continuum mechanics with the explicit allocation of reinforcing elements. For the numerical solution, the finite element method is used, modified to solve dynamic problems. In the calculations, the impact of the load on the beam was replaced by an impulse. The dependence of the pulse on time was determined from the experiment. The effect of reinforcement on the deformation and destruction of the beam is investigated. The comparison of experimental and numerical results is carried out.


Introduction
Practical application of effective methods of reinforcement and restoration of reinforced concrete structures with the use of steel-fiber concrete allows to reduce the material consumption and increase the reliability and survivability of reinforced structures of reconstructed buildings and structures.
It is known from the available literature sources that fiber concrete has increased strength and deformation characteristics, both under static and short-term dynamic loads [1...21].An increase in crack resistance when reinforced concrete structures are reinforced is achieved by using zone reinforcement of steel fiber in the stretched zone of the element, and an increase in the strength of bent and compressed-curved elements is achieved by using zone fiber reinforcement in the compressed zone of the reinforced concrete element.
The type of dispersed reinforcement closest to the deformative parameters of concrete is basalt and carbon fiber, which has high resistance to aggressive environment, environmental friendliness, fire resistance, and long service life.However, until now, the strength and performance characteristics of carbon fiber remain poorly understood, there is a lot of contradictory information in the literature, the technology of introducing basalt and carbon fiber into concrete has not been debugged: the technologies recommended by the manufacturer for the production of such fiber concrete have significant differences with the technologies used in the works of domestic and foreign researchers.To date, there are no approved regulatory and guidance documents in Russia for the creation and calculation of products and structures made of fiber concrete with carbon fiber.
In this paper, the strength, deformability and crack resistance of bent laminated reinforced concrete beams reinforced with layers of fibro concrete with carbon fiber with destruction in normal cross-section under static, cyclic and short-term dynamic loads are investigated.

The calculation model
The purpose of the research of wide layered beams is the experimental and numerical determination of the features of their deformation, the identification of schemes of cracking and destruction of reinforced concrete elements of the layered structure under conditions of variable magnitude static and single short-term dynamic loading.
The experimental beams consist of three different reinforced layers in cross-section height: fibroconcrete -concrete -fibroconcrete.The samples are a fragment of a multilayer protective shell of the cellular structure of the coating of a nuclear power plant (Fig. 1, 2).In numerical simulation of the loading conditions, the geometric dimensions of the beam and its reinforcement scheme correspond to the parameters of the experiment described above.Numerical modeling is carried out in a full three-dimensional dynamic formulation within the framework of the phenomenological approach of continuum mechanics with the explicit allocation of reinforcing elements.For the numerical solution, the finite element method is used, modified to solve dynamic problems.For calculations, the author's software package and algorithm are used, which allows for parallel calculations with high performance.
The behavior of concrete and fiber concrete is described in the framework of the model using the Hoffman strength criterion, taking into account the different tensile and compressive strength of concrete and fiber concrete: where coefficients С 1-9 are found from the following equations: (2) where X t , X c are the limits of tensile and compressive strength along the axis X, respectively Y t , Y c are the limits of tensile and compressive strength along the axis Y, Z t , Z c are the limits of tensile and compressive strength along the axis Z, S xy , S yz , S zx are the limits of shear strength along the corresponding axes.
The behavior of steel reinforcement was described in the framework of an elasticplastic model.
Figure 3 shows a general view of the simulated beam.Fiber concrete layers are indicated in blue, concrete -in gray, reinforcement -in red.The areas of load application on the front surface of the beam are indicated in green, the places of attachment on the back surface are indicated in black.In the places where the beam was fixed, movements in the negative direction along the Z axis were limited.The beam was divided into tetrahedral finite elements.The total number of finite elements in the calculations was 1.8•10 7 .
Figure 4 shows the calculated distributions of fracture zones in the beam at various points in time.In the beam, normal cracks are distributed almost along the entire length of the beam.
The highest levels of compressive stresses σ xx and σ zz are observed in the upper zone of the beam.The highest level of tensile stresses is achieved in the lower zone of the beam section.The greatest values of shear stresses σ zx are concentrated in the areas located in the zone of application of the load.
Figure 5 shows the curves of changes in time of movement of the reinforced beam in the middle of its span, obtained from the results of the experiment (blue line) and calculation (red line).The discrepancy in the values of movements in the middle of the span was 22%.Taking into account the dimension of the problem being solved, the convergence of the calculation data with the experiment is quite satisfactory.To calculate the strength and test the bearing capacity of compressed-curved steelreinforced concrete elements, a method and program for calculating the strength of normal sections based on a nonlinear deformation model taking into account real nonlinear diagrams of concrete, reinforcement and steel-reinforced concrete under static and short-term dynamic loads have been developed.
Using this method, the load-bearing capacity of experimental wide layered beams was evaluated and the relative resistance region for them was constructed (Fig. 6).
It can be seen from the calculation that the experimental data obtained are in good agreement with the results of analytical calculations of the bearing capacity according to the developed method.Fig. 6.The area of relative resistance by a wide layered beam reinforced with layers of carbon fiber concrete in the lower and upper cross-section zones span

The experimental research
Tests of wide layered beams were carried out under different types of loading: static once before destruction, alternating until the destruction of the beam, repeated before destruction and a single short-term dynamic impact on the drilling rig with the destruction of the experimental beam.
The length of the experimental beams was 2200 mm with a design span of 2000 mm.The cross-sectional dimensions of all wide beams were adopted 220*150 mm.Each beam consisted of three layers: the lower and upper layers, each 20 mm thick, are made of reinforced carbon fiber concrete, and the middle layer, 110 mm thick, is made of reinforced concrete.The coefficients of reinforcement of fiber concrete layers of beams with carbon fiber were assumed to be equal to 0.2% of the binder weight.
Reinforcement of beams with rod reinforcement is made symmetrical in the form of a frame and grids, which are accepted knitted.The spatial reinforcement frame consists of a longitudinal working armature 8 Ø 10 A400, connected by means of clamps made of wire Ø4B500.
In the middle of the thickness of the upper and lower fiber-concrete layers of the experimental beams is a knitted mesh consisting of wire Ø4 B500 with a pitch of 66mm in both directions.
To obtain information about the operation of experimental layered beams during their loading, a complex of primary measuring information converters was used.
During the static test, the load value was created using hydraulic jacks DG-50, oil was supplied to the piston from the pumping station.The magnitude of the applied load was recorded by a pressure gauge.
During a short-term dynamic test, the dynamic load value was determined using a force gauge sensor.Linear displacement sensors were also used to calculate linear displacements.Accelerometers were used to measure accelerations during a short-term dynamic test.
To measure the deformations of the stretched and compressed reinforcement of experimental beams, tensoresistors of the PKB-10 type with a resistance R = 201.6 ohms were installed on the middle of the rods of the reinforcement frames.Strain gauges of the PKB-20 and PKB -50 types were also installed on the surface of experimental beams to determine the deformations of concrete and fibroconcrete.
The BS-1 wide layered beam was tested for single static loading, BS-2 beams for multiple static loading, BS-3 beams for alternating static loading.
The test of the DB-4 wide layered beam was carried out for short-term dynamic loading on a coping rig.The beam was mounted on dynamometric supports, allowing to fix the magnitude of the support reactions during dynamic testing.A short-term dynamic load on the beam was created with the help of a falling load weighing 450 kg from a height of 700 mm.

The experimental results
As a result of the experimental studies carried out, all the beams were destroyed along the normal cross-section with the formation of cracks in the zone of pure bending by the destruction of the compressed zone of concrete.The destructive dynamic load for the beam was Fu=105...120 kN.
Analysis of the crack formation and destruction scheme (Fig. 7...10) of wide layered beams showed that the layered fiber-concrete beams collapsed due to the formation and further opening of normal cracks in height with further destruction of the compressed concrete zone.
In a beam tested under alternating load, destruction is observed in the lower and upper parts of the beam with the formation of more cracks along its length.At the same time, it was revealed that layered wide beams under load behave as a single element, there was no peeling of carbon fiber concrete and concrete layers.During the experimental studies, the beam displacements were measured at five points equidistant from each other.After processing the data, characteristic dependences of displacement changes for all beams were constructed.The graph of comparison of changes in the maximum displacements of beams is shown in Fig. 11.
It can be seen from the graph that the values of maximum displacements for carbon fiber reinforced concrete beams tested under different types of loading on average coincide with each other.In the BD-4 beam tested under short-term dynamic loading and the BS-2 beam tested under repeated static loading, the maximum deflections in the middle of the span were on average 5-10% less than the deflections of the BS-1 beam tested under single static loading.The maximum deflections in the middle of the span were reached by the BS-3 beam tested under alternating loading.1.New experimental data characterizing the process of resistance of fibro-reinforced concrete structures have been obtained: changes in deformations of concrete, reinforcement and steel-fiber concrete, deflections, loads at various stages of deformation of the sample.
2. For the first time, the influence of carbon fiber zone reinforcement on the strength and deformability of bent reinforced concrete elements under different types of loading was revealed and analyzed.
3. It was experimentally revealed that layered wide beams under load behave as a single element, there was no peeling of carbon fiber concrete and concrete layers.These structures can be calculated as reinforced concrete with zone (in compressed and stretched zones) carbon fiber reinforcement.
4. The proposed model of concrete and fiber concrete behavior adequately describes the dynamics of the stress-strain state and the process of destruction of materials.
5. The implemented algorithm and calculation methodology allow us to investigate the behavior of the structure as a whole in a full three-dimensional dynamic formulation.The obtained results of numerical and analytical calculations are in satisfactory agreement with experimental data.

Fig. 1 .Fig. 2 .
Fig. 1.General view of the spatial design scheme of the outer protective shell of the coating (17550 nodes, 31000 finite elements)

Fig. 7 .
Fig. 7. Schemes of cracking and destruction of the BS-1 beam

Fig. 11 .
Fig. 11.Comparison graph of maximum displacements for beams tested under different types of loading 5 Conclusions This work was financially supported by the Ministry of Science and Higher Education of Russian Federation (grant # 075-15-2021-686).Tests were carried out using research equipment of The Head Regional Shared Research Facilities of the Moscow State University of Civil Engineering E3S Web of Conferences 410, 02035 (2023) https://doi.org/10.1051/e3sconf/202341002035FORM-2023