Deformation of multilayered physically nonlinear concrete slabs by quasi-static loads

. Three-layer concrete slabs are considered where each layer is formed of its concrete grade. These slabs are widely used in the construction industry, particularly in the construction of agricultural buildings and structures; thus, calculating these slabs under the external loads and forces caused by their weight is an important and relevant task. It is assumed that the concretes are non-linearly formed, and the ratios between stresses and strains are taken as the third-order polynomials with different coefficients for the different grades of concrete. The slabs are supposed to be sufficiently thin, and the Kirchhoff-Lyav hypotheses are valid. The equations of the Genka-Ilyushin deformation theory are used as the equations for the state of phase materials of the considered hybrid slabs. The complete systems of resolving equations are obtained, the equations are solved by the Bubnov-Galerkin method, and the resulting systems of algebraic equations are solved numerically in the Maple mathematical package. The graphs of the slabs' deflection and the deformation values at each point of the slab are obtained. The significant difference is shown in the maximum deflections and deformations with and without considering the slab's weight.


Introduction
Due to the increasing standards and requirements for designed buildings and structures in the context of a growing cost of building materials, the urgent problem is to create building products with improved performance indicators at a comparable cost of provided goods.Concrete products are one of the most common, inexpensive, and easy to manufacture.Modern technologies for producing concrete products make it possible to create multilayered structures, where each layer can be made of the concretes of different physical properties.The multilayered concrete structures are currently used in the construction of agricultural buildings, particularly the large livestock farms and workshops for the production and processing of various agricultural and industrial products.The use of similar hybrid slabs, layered in their cross sections, where each layer is made of concrete-based materials of different properties and compositions, as a frame base makes it possible to provide high thermal insulation properties, chemical resistance, and relatively low production cost [1].Particularly, the paper [1] proposes the implementation of three-layer slabs in the construction of the floors of livestock buildings, where each layer fully corresponds to its functional purpose; for example, the lower layer, made of inexpensive waterproof materials, passes into the second layer of coarse-grained concrete, characterized by low thermal conductivity, which then in the third layer passes into the polymer concrete, characterized by good resistance to the aggressive environments, durability, and increased solidity.The paper [2] considers many prospects for using self-repairing concrete in constructing agricultural buildings.It presents a list of the advantages and disadvantages of different concrete grades.New materials are also released annually to improve the quality of buildings and structures in the agricultural and industrial sectors.They are more resistant to external influence and reduce the labor and resource cost.
There are quite a lot of scientific studies evaluating the action of loads on the slabs.Still, they are limited in calculations, that the composing materials are deformed linearly elastically [3][4][5][6], and do not consider the significant nonlinearity of the deformation diagrams of concretes at tension and compression [7,8].In the papers [9][10][11], the physically nonlinear problems for the reinforced concrete slabs are reduced to solving several elastic problems.The papers [12][13][14][15][16][17][18][19] consider the physical nonlinearity of concretes at the rods and rod systems deformation by quasi-static loads.This paper presents a method for calculating the multilayered slabs based on the nonlinear diagrams of uniaxial tension and compression of concretes of the composing phase materials, where each of the layers can be formed by the concretes with different physical properties.

Materials and Methods
To calculate the stress and strain structures of the agricultural and industrial sector, we consider a structure in the form of a closed rectangular prism with the side dimensions of , , a a a in the Cartesian rectangular coordinate system Oxyz with the origin of coordinates in one of the corners of the prism (Fig. 1)., , a a a in the Cartesian rectangular coordinate system Oxyz According to the real construction practice, all forming slabs are considered concrete three-layer slabs, made by using a single technology (in the general case, from different grades of concrete and with different sizes of phase layer boundaries) and with constant thicknesses of the phase layers along the surface of the given slab.In practice, the external loads, acting on the differently located slabs, are different but not significant for calculations.The vertical slabs are under the pressures, distributed along the external normal and created by the streamlined flows of the air atmosphere, and under the distributed vertical pressures by the total specific gravity of all phase materials.The ceiling slab is loaded by the distributed transverse pressure, consisting of the specific gravity of the phase materials and the weight of various loads from the external reserves to the production process.The floor slab is loaded by the specific gravity of the phase materials and loads of elastic resistance of the foundation.
Let us consider the thin hybrid slabs, which total thickness of the phase layers is significantly less than 1 2 3 , , a a a , and assume that the hybrid slab is deformed from the beginning of loading to its destruction without mutual separation and slippage of the phase materials.The considered assumptions make it possible to use between the components of the displacement vector and the strain tensor for the following dependencies The stress intensity u V in the slab can be represented as a polynomial [19][20] where the coefficients 1 3 , F F in the equation ( 5) can be found by the ratios proposed in [13,17].The calculated values of the coefficients 1 3 , F F , as well as the values of the ultimate deformations * * , i i H H at compression and tension, respectively, were obtained from ( 5), according to the certain limits of concrete solidity at compression and tension of the i-th layer * * , i i V V , are placed in Table 1.The deformations * * , i i H H are considered ultimate permissible, and if they are exceeded, we suppose that the slab loses its bearing capacity.In this table, all values with the stress dimension are related to the modulus of ultimate solidity of B10 concrete at compression.In further calculations, all values with the stress dimension will be related to the modulus of ultimate solidity of B10 concrete at compression.All values with the length dimension will be related to the length 1 a .

Table 1. The physical parameters of concretes used in calculations
No.

Concrete grade
1i H H for the layers . Evaluating the stresses by the ratios (6), we consider the magnitude of stress intensity as invariable along the layer's height and equal to the stress intensity at the lower boundary of the layer.In this case, the function ( ) determines the deformation rule for material in the i-th layer of the slab.This rule is established for a specific grade of concrete by testing its samples according to the standard experimental methods.The experimental results for various grades of heavy and light concrete prove that the functions i \ can be substituted for the dependence , where 1 3 , i i E E are the experimentally determined parameters in the tension-compression diagram of concretes of the i-th phase in the considered slab.
The stresses arising in the layers determine in hybrid slabs the field of forces presented by the expressions , and the fields of bending moments, presented by the formulas , where k is the resistance coefficient of the elastic base, p is the external load acting on the slab.
For the integration of these equations, the necessary boundary conditions are written, depending on the conditions of fixing the slab on the contour in a certain form.

Results
Because the described system of resolving equations is mathematically complex and multivariate, it is not possible to obtain an exact analytical solution.It is necessary to develop sufficiently simple and reliable schemes of numerical calculating procedures.We consider the Bubnov-Galerkin calculating procedure of its modern version suitable for solving the problems, as it is well-known and frequently used in practice [21].
Assume that a rectangular slab is pinched at the edges; then, taking into account only the first term in the Bubnov-Galerkin method, we have where 1 1 1 , , A B C are the required coefficients determining the solution of the problem.Substituting the values of displacements in the equations (1), we obtain The expressions for deformations are obtained from the ratios (1), ( 12) The expression for strain intensity can be obtained from the ratios (2) and ( 13)

S S S S S
Let us assume that the thickness of each layer of the slab is so small that the strain intensity is constant on the thickness in each layer of the slab, so we take the value of intensity at the upper boundary of the layer as an approximate value of strain intensity in the layer.Then, taking into account the expression (5), we rewrite the ratios for forces (7) and moments (8) as ( ) , ( ) The expressions for stresses in the slab, forces, and bending moments can be obtained from the ratios ( 6), ( 8), ( 9), ( 10), ( 11), ( 12), ( 13), (14).
If we substitute the obtained expressions for the forces into the equilibrium equations (10), transferring the expressions from the right side to the left one, and denote the left side by 1 2 3 ( , ), ( , ), ( , ) L x y L x y L x y , then we get

¦
Then, the required coefficients 1 1 1 , , A B C can be determined from the algebraic system of equations where 1 ( , )

Discussion
As an example, we consider a slab with parameters of 1 where each layer is made of the B10 concrete grade.The load value p is taken as p=0.01, and it is assumed that mass forces and support reaction forces are absent ( 0, 0) g k .Then, as a result of numerical calculations in the Maple mathematical package, the solution for slab deflection 3 ( , ) u x y is obtained and shown in Fig. 2(а).Fig. 2(b) presents a solution for the value of strain intensity ( , , ) u e x y z on the lower side of the slab (z=0).
According to these calculations, the maximum value of strain intensity ( , , ) u e x y z in the slab does not exceed the value of 0.00012178, which is less than the ultimate deformation of hardening for the B10 concrete grade.Therefore, the calculations prove that the slab can withstand the specified load.
If we consider a problem similar to the previous case but consider gravity, then the solution has the form shown in Fig. 3.
The obtained solutions show that the strain intensity in the slab reaches the value of 0.000167, which is more than the ultimate deformation of B10 concrete at tension.Therefore, the slab cannot withstand the specified load, considering gravity.Further, the considered homogeneous slab, made from the B10 concrete with the parameters (17), is taken as a reference slab, and the multilayered slabs of the same value as the reference one are considered.
In this case, when the considered multilayered slabs have equal value, then all of them must satisfy the ratios are the width and height of the i-th slab, 1 2 3 1 2 3 , , , , , h h h C C C are the heights and relative costs of materials for the 1, 2, 3 layers, respectively.In further calculations, the relative costs of the B10, B30, and B50 concrete grades are respectively taken equal to where the first layer of the material is made of B10 concrete, the second is B30, and the third is B50; then, for the considered slab in the case of gravity, we get the solutions shown in Fig. 4. In the considered slab, the maximum strain intensity is equal to 0.00005785, which is significantly lower than the ultimate deformation of solidity for the concrete materials used in the slab (2.8 times less than in the reference slab).Thus, the given slab can withstand significantly higher loads.
Let us increase the applied load p by one and a half times.Accordingly, for the strain intensity at the upper edge of the 1, 2, and 3 layers, we get the solutions shown in Fig. 5.
The maximum value of strain intensity in the first layer is equal to 0.000029, in the second layer, it is 0.000058, and in the third layer, it is 0.00008.Thus, the slab can withstand the applied load, as the strain intensity does not exceed the limit values in each layer of the slab.
Suppose we turn over the considered slab without changing the parameters, materials, and loads.In that case, the maximum value of strain intensity becomes equal to 0.0000566, 0.000119, and 0.0001828 in the first, second, and third layers, respectively.The indicated values of strain intensity exceed the limits in the second and third layers, which are higher than their ultimate values; therefore, the slab cannot withstand the applied load.

Conclusion
In this paper, the following results are obtained: 1. a method for calculating the physically nonlinear rods is developed based on the nonlinear diagrams of uniaxial tension and compression, extended over the multilayered thin slabs; complete systems of resolving equations for the deformation of thin multilayered slabs under the quasi-static loads and the examples of calculating the physical coefficients, used in the calculation, are obtained for various and most common grades of concrete; 2. the resulting system of resolving equations is complex and multivariant and does not have an exact analytical solution; therefore, quite simple and reliable schemes of the numerical, analytical procedures are developed based on modern modifications by Bubnov-Galerkin to solve them; 3. the distribution of deflections and the strain intensity at each point of the slab are obtained, the maximum deflections and the maximum strain intensity in the slab are determined; 4. taking into account that the strain intensity should not exceed the limit values in each phase material, it is concluded if the plate can withstand the applied loads; 5. it is proved that the redistributing of phase materials in the slab at the same mass makes it possible to increase the bearing capacity of the slab; 6. it is shown that the calculations without taking into account the own weight of the slab lead to significant calculation errors.

Fig. 1 .
Fig. 1.A prism with the side dimensions of 1 2 3, , a a a in the Cartesian rectangular coordinate system

Fig. 4 .Fig. 5 .
Fig. 4. (а) The value of slab deflection 3 ( , ) u x y at p=0.01, (b) The value of strain intensity ( , , ) u e x y z The components of forces and bending moments for the considered slabs must satisfy the differential equilibrium equations in the form of (18)e consider a slab with materials of the same value(18)as the reference one and the parameters of E3S Web of Conferences 365, 02011 (2023) https://doi.org/10.1051/e3sconf/202336502011CONMECHYDRO -2022