Nonlinear Calculations of the Strength of Cross-sections of Bending Reinforced Concrete Elements and Their Practical Realization

Abstract

Calculation methodology of reinforced concrete elements based on the calculated resistance of reinforced concrete is presented. The basic depending which allows setting the strength of bending sections and elements is obtained. The reliability of the dependencies is experimentally confirmed. There are calculation examples of bending elements by the developed methodology. According to the given method, tables have been developed, which depending on the accepted parameters allow determining the resistance of the concrete, the stresses in the reinforced concrete and reinforcement, and the total relative deformation of the cross section. Using the calculated resistances of reinforced concrete allowed to reduce the calculation of reinforced concrete elements according to the nonlinear deformation model to the application of the formulas of the classical resistance of materials and to significantly simplifies the process of their calculation.

1. Introduction

Concrete is a composite material that is made of gravel, sand, cement, water, and various types of additives. Each of the components has its own characteristics, which together determine the physical and mechanical parameters of concrete. The current normative documents regulate to establish these parameters by testing the experimental samples of specified sizes—prisms or cylinders—the quality of the resulting concrete is controlled by cubes. The resistance of the concrete to the compression is determined by dividing the maximum compressive load into the cross-sectional area of the experimental sample of the appropriate sizes [1, 2]. Considering the fact that for different sizes of samples, this ratio will have different meanings, it can be confirming that the term “calculated resistance of concrete” is relative. However, the introduction of this term allowed to have starting points when calculating the cross sections of concrete and reinforced concrete elements under the influence of various force factors.

A similar situation with the tension. Tension is a characteristic of a stress-strain state, which is determined by multiplying the corresponding deformations into a deformation module. Thus, it is not possible to determine the tension directly by experimental way. We determine deformations and then tension by using certain assumptions. Again, without lowering the values of the accepted terms, we have rather relative parameters. Based on these considerations, the introduction of the term calculated resistance of reinforced concrete should also take place. At first sight, this term is perceived quite difficult, especially in conditions of classical reinforced concrete. But at the same time its introduction reduces the calculation of cross sections of reinforced concrete elements to formulas of resistance of materials.

2. The term calculated resistance reinforced concrete

The basic idea of accepting calculated resistance is to separate the geometric parameters from the physical and mechanical ones. When we talk about elements from a single material, this does not cause any contradictions. In the case of composite materials, there are physical, mechanical, and geometric parameters of each material. In many cases geometric parameters can be selected in general from all physical and mechanical ones, but not individually. That is why the calculated resistance of composite materials will depend on the physical and mechanical parameters of all materials of which the cross section of the element is formed. In general, this can be expressed by the following equation:

$$f_i\left(a_1,\dots ,a_n\right)=\frac{F_{\mathit{Ed}}}{f\left(b_1,\dots ,b_n\right)},$$E1

where f_i(а₁, …, a_n) is the calculated resistance of the cross section of the element of a composite material under the condition of destruction on the i material, MPa; F_Ed the external calculated force factor, which corresponds to the limiting state of the element; f(b₁, …, b_n) the corresponding geometric characteristic; а₁, …, a_n the physical and mechanical parameters of material’s cross section of the composite element; and b₁, …, b_n the geometric parameters of the cross section of the composite element.

For a single cross-section of a composite element there may be a large number of calculated resistances due to the fact that the strength of the cross section is determined by the strength characteristics of all materials from which the composite element is formed. Therefore, the total calculated resistance of a composite material is determined by the minimum value of the calculated resistances under conditions of destruction on all materials from which the cross section of the element is formed:

$$f=\text{min}\left(f_1\left(a_1,\dots ,a_n\right),\dots ,f_i(a_1,\dots ,a_n),\dots ,f_n(a_1,\dots ,a_n)\right),$$E2

where f is the calculated (total) resistance of the cross section of composite material.

The calculated resistance of composite materials can be obtained both theoretically and experimentally. To determine it theoretically, the necessary valid hypotheses and statics equations are adopted. The calculated resistance (obtained by this way) does not contain empirical coefficients, but is determined by generally accepted experimental and theoretically grounded hypotheses and prerequisites. In the case of the experimental setting of the calculated resistance, it is more appropriate to determine the calculated resistance separately for each condition of destruction. This allows balancing various experimental studies to the same conditions.

The feature of the use of calculated resistance is that they are determined for specific tabulated values of the corresponding classes or characteristics of the materials. In particular cases, these may be parameters that determine the characteristics of materials of a certain class. For concrete, this characteristic may be K = 1.05(Е_cdε_c1/f_cd) [3].

Introduction of the calculated resistance of composite materials allows the use of no less important term tension in the cross-section of the composite material σ_i. It is also a conditional hypothetical term by which it is possible to determine the parameters of a stress-strain state at different levels of load. These tensions are determined by a formula similar to expression (1):

$${\sigma }_i\left(a_1,\dots ,a_n\right)=\frac{F_i}{f\left(b_1,\dots ,b_n\right)},$$E3

where σ_i(а₁,…,a_n) is the tension in the cross section of an element of a composite material, MPa, and F_i the external force factor, which corresponds to a certain level of load.

The geometric characteristic in expressions (1) and (3) for the same type of deformation has the same meaning. To theoretically obtain these tensions it is necessary to consider systems of equations of equilibrium for a certain type of deformation and to lead them to dimensionless quantities. The parameters that are obtained this way are tabulated depending on the load level, the accepted parameters, the classes of materials, etc. Typically, the tension is determined on the condition that the material does not reach the limit values of the deformations in the operating stages of work of the cross section of the element, and therefore, unlike the calculated resistance, they will have a single value, so there is no need for the introduction of formulas of type (2).

Finally, it is worthwhile to note the features of using the method of calculated resistances:

1. The basics of calculation contain experimentally and theoretically grounded preconditions and hypotheses.

2. Establishment of geometric parameters allows to balance calculated systems of equations to the clear separation of geometrical, physical, and mechanical parameters of cross sections.

3. Diagrams of deformation of materials are established. It should be noted that the adopted diagrams do not play a significant role for this method. The calculated resistance for a certain type of deformation can be established for practically all existing diagrams.

4. By conducting preliminary calculations, the main calculated parameters are tabulated.

The advantages of this method should include:

1. The only methodology for calculating composite materials for nonlinear deformation of materials with classical material resistance.

2. Simplicity and convenience of the calculating process.

3. Ability to use different diagrams of deformation of materials.

4. Setting parameters of a stress-strain state at different load levels.

5. When obtaining new knowledge about the features of deformation of composite materials, it is enough to specify the value of the calculated resistance, and the method of calculation will remain unchanged, which greatly simplify the process of balance of norms [4].

6. Conducting comparative and estimating calculations of cross sections from different materials.

Regarding the disadvantages, then they primarily relate suitability of this method for some classes of materials, and some discomfort associated with using tables.

3. Calculation of bending reinforced concrete elements of a rectangular cross section

Consider the definition of the calculated resistance of reinforced concrete for bending reinforced concrete elements with single reinforcement. In order to show the universality of this method, regardless of the calculation method (force or deformation), first consider the term of the calculated resistance of the reinforced concrete for the force model laid down in the design standards SNiP 2.03.01-84* [5].

For the stress-strain state shown in Figure 1, the equilibrium equation is written when ξ ≤ ξ_R, taking the sum of the moments relative to the neutral line:

$$f_{\mathit{yd}}{A}_s-f_{\mathit{cd}}\mathit{bx}=0.$$E4

$$f_{\mathit{yd}}{A}_s\left(d-x\right)+f_{\mathit{cd}}b\frac{x^2}{2}={М}_{\mathit{Ed}}.$$E5

Figure 1.

Scheme of forces in the cross section of bending reinforced concrete element for single reinforcement.

The value of x is determined from Eq. (4) and substituted by expression (5). As a result of simple transformations received:

$$f_{\mathit{yd}}{\rho }_f-\frac{f_{\mathit{yd}}^2{\rho }_f^2}{2f_c}=\frac{M_{\mathit{Ed}}}{{\mathit{bd}}^2}.$$E6

In formula (6) the left part is denoted by D₁; then D₁ = M_Ed/bd².

For the formula to take a familiar form, which is used in the resistance of materials in the calculations of metal, wooden, and stone structures, the left and right sides are multiplied by 6 and written like

$$6D_1=\frac{6{М}_{\mathit{Ed}}}{{\mathit{bd}}^2}\mathit{\text{or}}6D_1=\frac{{М}_{\mathit{Ed}}}{W_c}.$$E7

In this formula W_c is the elastic moment of the resistance of the working cross section of concrete; 6D₁ is nothing more than the calculated resistance of the reinforced concrete to the bend f_zМ,1, namely:

$$f_{\mathit{zМ},1}=\frac{{М}_{\mathit{Ed}}}{W_c}.$$E8

Similarly, it is obtained with ξ > ξ_R:

$${\alpha }_R{f}_c=\frac{M_{\mathit{Ed}}}{{\mathit{bd}}^2}.$$E9

The left part is denoted by D₂ and then D₂ = M_Ed/bd², finally receiving

$$f_{\mathit{zМ},2}=\frac{{М}_{\mathit{Ed}}}{W_c}.$$E10

The conditions are used (2) for the expression of the total calculated resistance:

$$f_{\mathit{zМ},\mathit{\text{SNiP}}}=\text{min}\left\{\begin{array}{l}6f_{\mathit{yd}}{\rho }_f-\frac{6f_{\mathit{yd}}^2{\rho }_f^2}{2f_c}\\ 6{\alpha }_R{f}_c\end{array}.\right.$$E11

Obtained by this way, calculated resistance of reinforced concrete for reinforcement А-400 and А-500 is shown in Table 1.

Class of concrete	Percentage of reinforcement ρf
	0.05	0.50	1.00	1.25	1.50	1.75	2.00	2.50	3.00
	fyd = 375 MPa (A400С)
С8/10	1.11	9.49	15.47	15.66	15.66	15.66	15.66	15.66	15.66
С12/15	1.11	10.01	17.54	20.37	21.73	21.73	21.73	21.73	21.73
С16/20	1.12	10.33	18.83	22.39	25.50	28.14	28.61	28.61	28.61
С20/25	1.12	10.52	19.59	23.58	27.20	30.46	33.36	35.06	35.06
С25/30	1.12	10.63	20.02	24.25	28.17	31.78	35.07	40.08	40.08
С30/35	1.12	10.71	20.34	24.74	28.88	32.75	36.35	42.73	44.77
С32/40	1.12	10.77	20.58	25.13	29.44	33.50	37.33	44.26	49.13
С35/45	1.12	10.83	20.81	25.49	29.95	34.21	38.25	45.70	52.31
С40/50	1.12	10.87	20.97	25.73	30.30	34.68	38.86	46.66	53.69
С45/55	1.12	10.90	21.09	25.93	30.59	35.07	39.38	47.46	54.84
С50/60	1.12	10.93	21.22	26.13	30.87	35.46	39.89	48.26	55.99
	fyd = 450 MPa (A500С)
С8/10	1.32	10.97	15.33	15.33	15.33	15.33	15.33	15.33	15.33
С12/15	1.33	11.71	19.85	21.22	21.22	21.22	21.22	21.22	21.22
С16/20	1.34	12.18	21.72	25.50	27.87	27.87	27.87	27.87	27.87
С20/25	1.34	12.45	22.81	27.20	31.07	34.06	34.06	34.06	34.06
С25/30	1.34	12.61	23.43	28.17	32.46	36.31	38.85	38.85	38.85
С30/35	1.34	12.72	23.88	28.88	33.49	37.71	41.54	43.31	43.31
С32/40	1.34	12.81	24.24	29.44	34.29	38.79	42.95	47.42	47.42
С35/45	1.34	12.89	24.57	29.95	35.03	39.81	44.28	51.91	51.91
С40/50	1.34	12.95	24.79	30.30	35.53	40.48	45.16	53.69	55.27
С45/55	1.34	12.99	24.98	30.59	35.94	41.05	45.90	54.84	58.28
С50/60	1.35	13.04	25.16	30.87	36.36	41.61	46.64	55.99	61.44

Table 1.

Calculated resistance of reinforced concrete to bend for single reinforcement f_zМ,SNiP, MPa.

Note: Intermediate values are determined by straight-line interpolation.

Similar expressions are obtained for nonlinear calculations. Write the value of the corresponding calculated resistances for different cases of destruction:

$$f_{\mathit{zМ},1\mathit{dm}}=6\left(\frac{\underset{0}{\overset{{\epsilon }_c}{\int }}{\sigma }_c{\epsilon }_{c}d{\epsilon }_c}{{\left(\underset{0}{\overset{{\epsilon }_c}{\int }}{\sigma }_{c}d{\epsilon }_c\right)}^2}-\frac{{\epsilon }_c}{\underset{0}{\overset{{\epsilon }_c}{\int }}{\sigma }_{c}d{\epsilon }_c}\right){{\rho }_f}^2{f}_{\mathit{yd}}^2+6{\rho }_f{f}_{\mathit{yd}},$$E12

$$f_{\mathit{zМ},2\mathit{dm}}=6\left(\frac{\underset{0}{\overset{{\epsilon }_c}{\int }}{\sigma }_c{\epsilon }_{c}d{\epsilon }_c}{{\left(\underset{0}{\overset{{\epsilon }_c}{\int }}{\sigma }_{c}d{\epsilon }_c\right)}^2}-\frac{{\epsilon }_c}{\underset{0}{\overset{{\epsilon }_c}{\int }}{\sigma }_{c}d{\epsilon }_c}\right){\rho }_f^2{E}_s^2{\epsilon }_c^2+6{\rho }_f{E}_s{\epsilon }_c.$$E13

To determine the corresponding calculated resistance, it is necessary for expressions (12) and (13) to apply an extreme criterion in the form:

$$\frac{{\mathit{df}}_{\mathit{zМ},\mathit{idm}}}{d{\epsilon }_c}=0,{\epsilon }_cє\left[{\epsilon }_{\mathit{cl},}{\epsilon }_{\mathit{сu}}\right].$$E14

The total calculated resistance to bend in calculating by the deformation model will be determined by the condition.

$$f_{\mathit{zМ},\mathit{dm}}=\text{min}\left\{\begin{array}{l}{f}_{\mathit{zМ},1\mathit{ДМ}},\frac{{\mathit{df}}_{\mathit{zМ},1\mathit{dm}}}{d{\epsilon }_c}=0,{\epsilon }_cє\left[{\epsilon }_{\mathit{cl},}{\epsilon }_{\mathit{сu}}\right];\\ f_{\mathit{zМ},2\mathit{ДМ}},\frac{{\mathit{df}}_{\mathit{zМ},2\mathit{dm}}}{d{\epsilon }_c}=0,{\epsilon }_cє\left[{\epsilon }_{\mathit{cl},}{\epsilon }_{\mathit{сu}}\right].\end{array}\right.$$E15

For the further use of the expression (15), it is necessary to adopt a concrete deformation diagram. Adopted function of the deformation diagram does not have essential value, but it must satisfy the conditions for deformation of concrete. Accepting for deformation diagram for concrete Eurocode-2 [3], expression (3.14), the tabulation is executed so that the maximum fault in interpolation will not be more than 5%. The value of the calculated resistance to bending for single reinforcement for all classes of concrete and reinforcement classes А-400 and А-500 are shown in Table 2.

Class of concrete	Percentage of reinforcement ρf
	0.05	0.50	1.00	1.25	1.50	1.75	2.00	2.50	3.00
	fyd = 375 MPa (A400С)
С8/10	1.10	9.44	14.68	15.12	15.43	15.67	15.86	16.13	16.32
С12/15	1.11	9.97	17.38	20.09	20.85	21.27	21.60	22.10	22.45
С16/20	1.11	10.30	18.70	22.19	25.20	27.38	27.90	28.71	29.29
С20/25	1.11	10.49	19.48	23.40	26.95	30.11	32.88	34.82	35.65
С25/30	1.11	10.60	19.91	24.08	27.93	31.46	34.66	39.64	40.69
С30/35	1.12	10.68	20.24	24.59	28.66	32.45	35.96	42.12	45.45
С32/40	1.12	10.75	20.49	24.98	29.23	33.22	36.96	43.69	49.26
С35/45	1.12	10.81	20.72	25.35	29.76	33.94	37.90	45.16	51.52
С40/50	1.12	10.84	20.88	25.60	30.11	34.42	38.53	46.14	52.94
С45/55	1.12	10.87	21.01	25.80	30.40	34.82	39.05	46.95	54.10
С50/60	1.12	10.90	21.14	26.00	30.69	35.21	39.56	47.75	55.26
	fyd = 450 MPa (A500С)
С8/10	1.32	10.90	14.57	15.02	15.35	15.60	15.79	16.07	16.27
С12/15	1.33	11.66	19.40	20.15	20.70	21.13	21.48	21.99	22.35
С16/20	1.33	12.13	21.53	25.17	26.50	27.16	27.71	28.53	29.13
С20/25	1.33	12.41	22.65	26.95	30.69	32.67	33.42	34.57	35.43
С25/30	1.34	12.57	23.28	27.93	32.12	35.82	37.89	39.33	40.40
С30/35	1.34	12.69	23.74	28.66	33.18	37.28	40.93	43.80	45.10
С32/40	1.34	12.78	24.11	29.23	33.99	38.39	42.40	48.01	49.53
С35/45	1.34	12.86	24.44	29.76	34.75	39.42	43.78	51.43	54.52
С40/50	1.34	12.92	24.67	30.11	35.26	40.11	44.68	52.94	58.52
С45/55	1.34	12.96	24.86	30.40	35.68	40.69	45.43	54.10	61.40
С50/60	1.34	13.01	25.04	30.69	36.10	41.25	46.17	55.26	63.37

Table 2.

Calculated resistance of reinforced concrete to bend for single reinforcement f_zМ,dm, MPa.

Note: Intermediate values are determined by straight-line interpolation.

Similarly, the calculated resistance for bend for double reinforcement is obtained. For this purpose, the calculated resistance for different conditions of destruction of bending reinforced concrete elements for double reinforcement are determined:

$$f_{\mathit{zМ}2,1\mathit{dm}}=6\frac{\frac{\underset{0}{\overset{{\epsilon }_c}{\int }}{\sigma }_c{\epsilon }_c\mathit{d\epsilon }}{{\epsilon }_c^2}+{\rho }_{\mathit{fc}}{f}_{\mathit{yc}}\left(k-{\mathit{nk}}^2\right)+{\epsilon }_c{\left(k-1\right)}^{2}k{\rho }_f{Е}_s}{k^2}.$$E16

$$f_{\mathit{zМ}2,2\mathit{dm}}=6\frac{\frac{\underset{0}{\overset{{\epsilon }_c}{\int }}{\sigma }_c{\epsilon }_{с}\mathit{d\epsilon }}{{\epsilon }_{с}^2}+{\rho }_{\mathit{fc}}{f}_{\mathit{yc}}\left(k-{\mathit{nk}}^2\right)+{\rho }_f{f}_{\mathit{yd}}\left(k^2-k\right)}{k^2}.$$E17

$$f_{\mathit{zМ}2,3\mathit{dm}}=6{\epsilon }_c\frac{\frac{\underset{0}{\overset{{\epsilon }_c}{\int }}{\sigma }_c{\epsilon }_{с}\mathit{d\epsilon }}{{\epsilon }_{с}^3}+E_s{\rho }_{\mathit{fc}}{\left(1-\mathit{nk}\right)}^{2}k+E_s{\rho }_f{\left(k-1\right)}^{2}k}{k^2}.$$E18

$$f_{\mathit{zМ}2,4\mathit{dm}}=6{\epsilon }_c\frac{\frac{\underset{0}{\overset{{\epsilon }_c}{\int }}{\sigma }_c{\epsilon }_{с}\mathit{d\epsilon }}{{\epsilon }_{с}^3}+E_s{\rho }_{\mathit{fс}}{\left(1-\mathit{nk}\right)}^{2}k+\frac{f_{\mathit{yd}}}{{\epsilon }_c}{\rho }_f\left(k-1\right)k}{k^2}.$$E19

In the given expressions, k is determined from the first equation of equilibrium under the conditions of the destruction of the element.

The total calculated resistance to bend in double reinforcement according to the deformation model will be determined by the condition:

$$f_{\mathit{zМ}2,\mathit{dm}}=\text{min}\left\{\begin{array}{l}{f}_{\mathit{zМ}2,1\mathit{dm}},\frac{{\mathit{df}}_{\mathit{zМ},1\mathit{dm}}}{d{\epsilon }_c}=0,{\epsilon }_cє\left[{\epsilon }_{\mathit{cl},}{\epsilon }_{\mathit{сu}}\right];\\ f_{\mathit{zМ}2,2\mathit{dm}},\frac{{\mathit{df}}_{\mathit{zМ},2\mathit{dm}}}{d{\epsilon }_c}=0,{\epsilon }_cє\left[{\epsilon }_{\mathit{cl},}{\epsilon }_{\mathit{сu}}\right];\\ f_{\mathit{zМ}2,3\mathit{dm}},\frac{{\mathit{df}}_{\mathit{zМ},3\mathit{dm}}}{d{\epsilon }_c}=0,{\epsilon }_cє\left[{\epsilon }_{\mathit{cl},}{\epsilon }_{\mathit{сu}}\right];\\ f_{\mathit{zМ}2,4\mathit{dm}},\frac{{\mathit{df}}_{\mathit{zМ},4\mathit{dm}}}{d{\epsilon }_c}=0,{\epsilon }_cє\left[{\epsilon }_{\mathit{cl},}{\epsilon }_{\mathit{сu}}\right].\end{array}\right.$$E20

In Table 3 the expression of the calculated resistance to bend for double reinforcement is derived taking the diagram of deformation of concrete in the form of the function Eurocode-2 [3].

Class of concrete	ρf = 0.01			ρf = 0.02			ρf = 0.03
	ρfc/ρf
	0.25	0.50	0.75	0.25	0.50	0.75	0.25	0.50	0.75
n = 0.06–0.1	fyd = 375 MPa (А400С)
С8/10	18.09	20.02	21.04	25.81	35.75	41.17	31.54	46.74	60.39
С12/15	19.28	20.55	21.16	31.33	38.53	41.70	37.42	52.38	61.58
С16/20	20.03	20.88	21.19	35.78	39.85	42.03	43.99	56.93	62.33
С20/25	20.46	21.07	21.23	37.52	40.63	42.22	50.11	58.67	62.76
С25/30	20.71	21.17	21.26	38.51	41.06	42.32	53.39	59.66	63.01
С30/35	20.89	21.22	21.29	39.24	41.39	42.35	55.04	60.39	63.19
С32/40	21.02	21.26	21.32	39.80	41.64	42.38	56.31	60.95	63.33
С35/45	21.14	21.31	21.36	40.33	41.87	42.41	57.50	61.48	63.44
С40/50	21.21	21.35	21.39	40.68	42.03	42.44	58.30	61.84	63.50
С45/55	21.28	21.39	21.43	40.98	42.14	42.46	58.95	62.13	63.53
С50/60	21.35	21.43	21.46	41.27	42.25	42.50	59.61	62.42	63.55
n = 0.06–0.1	fyd = 450 MPa (А500С)
С8/10	20.23	23.59	25.13	27.70	39.59	48.97	34.49	52.69	70.91
С12/15	22.45	24.35	25.25	33.11	44.76	49.73	40.27	58.17	73.21
С16/20	23.52	24.82	25.31	39.06	46.91	50.20	46.71	64.31	74.28
С20/25	24.14	25.08	25.36	43.38	48.03	50.45	52.71	68.76	74.90
С25/30	24.49	25.18	25.41	44.81	48.65	50.50	57.47	70.19	75.25
С30/35	24.74	25.26	25.45	45.87	49.11	50.55	61.98	71.25	75.50
С32/40	24.92	25.33	25.49	46.67	49.46	50.60	65.05	72.05	75.68
С35/45	25.08	25.40	25.53	47.42	49.78	50.65	66.84	72.80	75.74
С40/50	25.18	25.46	25.56	47.92	49.98	50.69	67.99	73.29	75.78
С45/55	25.27	25.50	25.60	48.32	50.13	50.73	68.92	73.70	75.82
С50/60	25.35	25.55	25.64	48.70	50.24	50.77	69.87	74.08	75.87

Table 3.

Calculated resistance of reinforced concrete to bend for double reinforcement f_zМ2,dm, MPa.

Note: Intermediate values are determined by straight-line interpolation.

As can be seen from Tables 2 and 3 in some cases, double reinforcement significantly (more than three times) increases the calculated resistance of reinforced concrete of bending elements and accordingly increases their bearing capacity. In this way, the reinforcement can greatly enhance the compressed area of concrete of bending reinforced concrete elements. For comparison, the data of the calculated resistance to bend for double reinforcement are presented by methodology of SNiP 2.03.01–84* [5] (Table 4).

Class of concrete	ρf = 0.01			ρf = 0.02			ρf = 0.03
	ρfc/ρf
	0.25	0.50	0.75	0.25	0.50	0.75	0.25	0.50	0.75
n = 0.06–0.1	fyd = 375 MPa (А400С)
С8/10	18.20	20.06	21.04	26.23	36.61	41.20	31.51	47.37	60.48
С12/15	19.37	20.58	21.17	32.29	38.67	41.71	37.58	53.43	61.64
С16/20	20.10	20.90	21.25	36.06	39.97	42.04	44.47	57.20	62.37
С20/25	20.52	21.09	21.30	37.77	40.73	42.23	50.91	58.91	62.80
С25/30	20.76	21.20	21.32	38.73	41.15	42.33	53.91	59.87	63.04
С30/35	20.94	21.28	21.34	39.45	41.47	42.41	55.52	60.59	63.21
С32/40	21.08	21.34	21.36	40.00	41.72	42.48	56.77	61.14	63.35
С35/45	21.21	21.40	21.37	40.52	41.95	42.53	57.93	61.66	63.48
С40/50	21.30	21.43	21.38	40.87	42.10	42.57	58.71	62.00	63.57
С45/55	21.37	21.47	21.39	41.15	42.23	42.60	59.36	62.29	63.64
С50/60	21.44	21.50	21.40	41.44	42.36	42.63	60.01	62.58	63.71
n = 0.06–0.1	fyd = 450 MPa (А500С)
С8/10	20.51	23.15	24.78	26.60	37.87	47.32	32.24	49.15	66.05
С12/15	22.31	24.07	25.11	32.49	43.72	48.64	38.12	55.03	70.58
С16/20	23.44	24.64	25.32	39.14	46.02	49.47	44.78	61.69	72.44
С20/25	24.10	24.98	25.44	43.13	47.37	49.96	50.97	67.18	73.54
С25/30	24.47	25.17	25.51	44.63	48.13	50.23	55.76	68.89	74.15
С30/35	24.75	25.31	25.56	45.73	48.70	50.43	60.22	70.16	74.61
С32/40	24.97	25.42	25.60	46.59	49.14	50.59	64.33	71.15	74.97
С35/45	25.17	25.52	25.64	47.39	49.55	50.74	66.68	72.07	75.30
С40/50	25.30	25.59	25.66	47.93	49.82	50.84	67.88	72.68	75.52
С45/55	25.41	25.65	25.68	48.37	50.05	50.92	68.88	73.19	75.70
С50/60	25.52	25.70	25.70	48.82	50.27	51.00	69.89	73.70	75.89

Table 4.

Calculated resistance of reinforced concrete to bend for double reinforcement f_zМ2,SNiP, MPa.

Note: Intermediate values are determined by straight-line interpolation.

Compare the calculated resistance of reinforced concrete to bending defined by the force model and deformation method for single and double reinforcement. As can be seen from Tables 5 and 6, the calculated resistance of reinforced concrete to the bend differs within the limits of the calculated fault. This makes it possible to say that for heavy concrete classes С8/10÷С50/60 and ordinary reinforcement classes А-400 and А-500, calculations of the strength of the cross sections of bending reinforced concrete elements with single and double reinforcement can be performed on any of the mentioned methods. By these ways, the maximum difference will be within 8% and only for certain conditions.

Class of concrete	Percentage of reinforcement ρf
	0.05	0.50	1.00	1.25	1.50	1.75	2.00	2.50	3.00
	fyd = 375 MPa (A400С)
С8/10	0.997	0.995	0.949	0.965	0.985	1.001	1.013	1.030	1.042
С12/15	0.997	0.996	0.991	0.986	0.960	0.979	0.994	1.017	1.033
С16/20	0.996	0.997	0.993	0.991	0.988	0.973	0.975	1.003	1.024
С20/25	0.996	0.997	0.994	0.992	0.991	0.989	0.986	0.993	1.017
С25/30	0.996	0.998	0.995	0.993	0.992	0.990	0.988	0.989	1.015
С30/35	0.996	0.998	0.995	0.994	0.992	0.991	0.989	0.986	1.015
С32/40	0.996	0.998	0.996	0.994	0.993	0.992	0.990	0.987	1.003
С35/45	0.996	0.998	0.996	0.995	0.994	0.992	0.991	0.988	0.985
С40/50	0.996	0.998	0.996	0.995	0.994	0.993	0.991	0.989	0.986
С45/55	0.996	0.998	0.996	0.995	0.994	0.993	0.992	0.989	0.986
С50/60	0.996	0.997	0.996	0.995	0.994	0.993	0.992	0.990	0.987
	fyd = 450 MPa (A500С)
С8/10	0.998	0.993	0.951	0.980	1.001	1.018	1.030	1.049	1.061
С12/15	0.997	0.995	0.977	0.950	0.976	0.996	1.012	1.037	1.054
С16/20	0.997	0.996	0.991	0.987	0.951	0.975	0.994	1.024	1.045
С20/25	0.996	0.997	0.993	0.991	0.988	0.959	0.981	1.015	1.040
С25/30	0.996	0.997	0.994	0.992	0.990	0.987	0.975	1.012	1.040
С30/35	0.996	0.997	0.994	0.992	0.991	0.989	0.985	1.012	1.042
С32/40	0.996	0.997	0.995	0.993	0.991	0.990	0.987	1.012	1.044
С35/45	0.996	0.998	0.995	0.993	0.992	0.990	0.989	0.991	1.050
С40/50	0.996	0.998	0.995	0.994	0.992	0.991	0.989	0.986	1.059
С45/55	0.996	0.998	0.995	0.994	0.993	0.991	0.990	0.986	1.053
С50/60	0.996	0.998	0.995	0.994	0.993	0.991	0.990	0.987	1.031

Table 5.

Comparison of the calculated resistance of reinforced concrete to bend for single reinforcement f_zМ,dm/f_zМ,SNiP.

Class of concrete	ρf = 0.01			ρf = 0.02			ρf = 0.03
	ρfc/ρf
	0.25	0.50	0.75	0.25	0.50	0.75	0.25	0.50	0.75
1	2	3	4	5	6	7	8	9	10
n = 0.06–0.1	fyd = 375 MPa (А400С)
С8/10	0.994	0.998	1.000	0.984	0.977	0.999	1.001	0.987	0.999
С12/15	0.996	0.998	1.000	0.970	0.996	1.000	0.996	0.980	0.999
С16/20	0.997	0.999	0.997	0.992	0.997	1.000	0.989	0.995	0.999
С20/25	0.997	0.999	0.997	0.993	0.998	1.000	0.984	0.996	0.999
С25/30	0.997	0.999	0.997	0.994	0.998	1.000	0.990	0.996	1.000
С30/35	0.998	0.997	0.998	0.995	0.998	0.999	0.991	0.997	1.000
С32/40	0.997	0.996	0.998	0.995	0.998	0.998	0.992	0.997	1.000
С35/45	0.997	0.996	0.999	0.995	0.998	0.997	0.993	0.997	0.999
С40/50	0.996	0.996	1.001	0.996	0.998	0.997	0.993	0.997	0.999
С45/55	0.996	0.996	1.002	0.996	0.998	0.997	0.993	0.997	0.998
С50/60	0.996	0.997	1.003	0.996	0.997	0.997	0.993	0.997	0.998
n = 0.06–0.1	fyd = 450 MPa (А500С)
С8/10	0.986	1.019	1.014	1.041	1.045	1.035	1.070	1.072	1.074
С12/15	1.006	1.012	1.005	1.019	1.024	1.022	1.056	1.057	1.037
С16/20	1.003	1.007	0.999	0.998	1.019	1.015	1.043	1.042	1.025
С20/25	1.002	1.004	0.997	1.006	1.014	1.010	1.034	1.024	1.019
С25/30	1.000	1.000	0.996	1.004	1.011	1.005	1.031	1.019	1.015
С30/35	0.999	0.998	0.996	1.003	1.009	1.002	1.029	1.015	1.012
С32/40	0.998	0.996	0.995	1.002	1.007	1.000	1.011	1.013	1.010
С35/45	0.996	0.995	0.996	1.001	1.005	0.998	1.002	1.010	1.006
С40/50	0.995	0.995	0.996	1.000	1.003	0.997	1.002	1.008	1.004
С45/55	0.994	0.994	0.997	0.999	1.002	0.996	1.000	1.007	1.002
С50/60	0.993	0.994	0.997	0.998	0.999	0.995	1.000	1.005	1.000

Table 6.

Comparison of the calculated resistance of reinforced concrete to bend for double reinforcement f_zМ2,dm/f_zМ2,SNiP.

One of the main advantages of the deformation model in comparison with the force one is the possibility of obtaining the parameters of the stress-strain state for the operational load. Let’s show how this can be done using the method of calculated resistance of reinforced concrete. For this purpose, the tensions in the bending reinforced concrete element are determined σ_zМ,ДМ under operational loads, at which cross sections of the element can work without cracks at М < М_W, with cracks in the stretched zone at М ≥ М_W, without cracks at М ≥ М_W (areas in the block between the cracks). In this case, it is proposed to determine the tension in the reinforced concrete for cross sections until formation of cracks.

$${\sigma }_{\mathit{WzМ}}=6{\epsilon }_{c,W}\times \frac{\frac{\underset{0}{\overset{{\epsilon }_{c,W}}{\int }}{\sigma }_c{\epsilon }_{с}\mathit{d\epsilon }}{{\epsilon }_{c,W}^3}+\frac{\underset{0}{\overset{{\epsilon }_{\mathit{сtu}}}{\int }}{\sigma }_{\mathit{сt}}{\epsilon }_{\mathit{ct}}\mathit{d\epsilon }}{{\epsilon }_{c,W}^3}+E_s{\rho }_{\mathit{fc}}{\left(1-{\mathit{nk}}_W\right)}^2{k}_W+E_s{\rho }_f{\left(k_W-1\right)}^2{k}_W}{k_W^2}.$$E21

It is noted that the tension σ_WzМ also allows to determine the moment of formation of cracks, so depending on the tasks, it can also be called the calculated resistance of the reinforced concrete to the bend until formation of cracks.

Tension in a cross section with a crack in the stretched zone at М ≥ М_W is determined by the

$${\sigma }_{2\mathit{zМ}}=6\frac{\underset{0}{\overset{{\epsilon }_c}{\int }}{\sigma }_c{\epsilon }_{с}\mathit{d\epsilon }}{k^2{\epsilon }_c^2}+\frac{\underset{0}{\overset{{\epsilon }_{\mathit{сtu}}}{\int }}{\sigma }_{\mathit{сt}}{\epsilon }_{\mathit{ct}}\mathit{d\epsilon }}{k^2{\epsilon }_c^2}+6{\epsilon }_c{E}_s{\rho }_{\mathit{fc}}\frac{{\left(1-\mathit{nk}\right)}^2}{k}+6{\epsilon }_c{E}_s{\rho }_f\frac{{\left(k-1\right)}^2}{k}+\frac{\left(k-1\right)}{k}\mathrm{\Delta }{\sigma }_{s,х}{\rho }_f.$$E22

The tensions between the cracks are determined by

$${\sigma }_{\mathit{mzМ}}=6\frac{E_s^2{\left(k-1\right)}^2\underset{0}{\overset{\frac{{}_{{\sigma }_{s,m}}}{E_s\left(k-1\right)}}{\int }}{\sigma }_c\mathit{\epsilon d\epsilon }}{k^2{\sigma }_{s,m}^2}+6\frac{{\left(k-1\right)}^2\underset{0}{\overset{{\epsilon }_{\mathit{сtu}}}{\int }}{\sigma }_{\mathit{сt}}{\epsilon }_{\mathit{сt}}\mathit{d\epsilon }}{k^2{\epsilon }_{\mathit{ctu}}^2}+6\frac{k-1}{k}{\sigma }_{s,m}{\rho }_f+6\frac{{\left(1-\mathit{kn}\right)}^2}{k\left(k-1\right)}{\rho }_{\mathit{fc}}{\sigma }_{s,m}.$$E23

All of the above tensions in concrete are determined by expression:

$${\sigma }_{\mathit{izМ}}=\frac{М}{W_c}.$$E24

Parameters of the stress-strain state at the operating load levels are necessary for determining the deflection and width of the crack opening. Therefore, the basic parameters that are necessary for this will be: tension in the reinforcement and curvature.

The tension in the reinforcement until formation of cracks is determined by expression

$${\sigma }_{s,W}=\left(k_W-1\right){\epsilon }_{с,W},$$E25

under certain values k_W, ε_c,W.

The tension in the reinforcement in the cross section with the crack in the stretched zone are calculated at known values k, ε_c, Δσ_s,x. Average tensions in the reinforcement on the section in the block between the cracks σ_s,m are defined as the arithmetic mean of the tensions that are determined by expressions (22) and (23).

The curvature of the cross sections of reinforced concrete elements, taking into account the hypothesis of plane cross sections, is determined by the expression

$$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$r$}\right.=\sum \epsilon /d,$$E26

where Σε is the total deformation of fibrous concrete fibers and stretched reinforcement.

The total deformation of fibrous concrete fibers and stretched reinforcement must be determined by the following formulas:

• For cross sections without cracks at М < М_W:

  $$\sum \epsilon ={\epsilon }_{с,W}+{\epsilon }_{s,W}={\epsilon }_{с,W}+{\sigma }_{s,W}/E_s.$$E27

• For cross sections with a crack in the stretched zone:

  $$\sum \epsilon ={\epsilon }_{c,2}+{\epsilon }_{s,\mathit{fic}}={\epsilon }_{c,2}+\left(k_{W,2}-1\right){\epsilon }_{c,2}.$$E28

• For cross sections without cracks at М ≥ М_W

  $$\sum \epsilon ={\epsilon }_{c,m}+{\epsilon }_{s,m}=\frac{{\sigma }_{s,m}}{E_s\left(k-1\right)}+\frac{{\sigma }_{s,m}}{E_s}.$$E29

Deflections are determined by curvature by using numerical methods.

According to the given method, tables have been developed, which depending on the accepted parameters allow to determining the resistance of the concrete, the stresses in the reinforced concrete and reinforcement, and the total relative deformation of the cross section. For this purpose, the deformation diagram was adopted in the form of Eurocode-2 function [3]. These tables are given in [6].

The calculation of the strength of the cross sections of bending reinforced concrete elements and crack resistance is recommended to be performed according to the formula:

$$\frac{{М}_{\mathit{Ed}}}{W_c}\le f_{\mathit{zМ}}\left(f_{\mathit{WzМ}}\right).$$E30

Calculation of tension limitation in the reinforcement is carried out as follows:

$${\sigma }_{\mathit{zМ}}=\frac{{М}_{е}}{W_c}\to {\epsilon }_{\mathit{sі}}\to {\sigma }_s\le {\sigma }_{s,\mathit{\text{Table}}},$$E31

where σ_s,Table is the tension in the reinforcement, in which there is no need to determine the width of the cracks opening, which are determined by Table 2.5 [7].

The calculation of the width of the cracks opening is carried out in same scheme:

$${\sigma }_{\mathit{zМ}}={М}_{е}/W_c\to {\epsilon }_{\mathit{sі}}\to {\sigma }_{\mathit{sі}}\to S_r\to W_k.$$E32

Calculation of deflections is performed in the following order:

$${\sigma }_{\mathit{zМ}}={М}_{е}/W_c\to \mathrm{\Sigma }\epsilon \to 1/r=\mathrm{\Sigma }\epsilon /d\to f.$$E33

A separate important issue in the theory of reinforced concrete is the consideration of regime loads and influences: long-term, quasi-constant, low cycle, temperature, humidity and others. Thus, taking into account the long-term load can be realized by introducing the creep coefficient to the curvature or by introducing into the calculation of the deformation diagrams with the corresponding parameters. The calculation of regime load under the first condition can be carried out according to the given method by using tables for short-term load. When performing calculations under the second condition, it is necessary to use the tables obtained for the corresponding parameters of the diagrams. Similar tables can be made for virtually all regime loads and influences, which greatly simplify the calculations of strength, crack resistance, stiffness and width of crack opening. This is an issue that needs to be studied in detail, but the use of the calculated resistance of reinforced concrete gives confidence in the successful solution of this problem.

4. Examples of calculation of bending reinforced concrete elements

Example 1. Reinforced concrete beam with working cross section b × d = 20 × 45 sm is made of concrete of class С25/30 and reinforced 4∅25 of steel of class А500С. Determine the carrying capacity of the beam.

Solution. The percentage of beam reinforcement with stretched reinforcement is calculated:

$${\rho }_f=\frac{{А}_s}{\mathit{bd}}\times 100\%=\frac{19.63}{20\times 45}=2.181\%.$$

According to the tables the calculated resistance of the reinforced concrete to the bend is determined:

$$f_{\mathit{zМ}}=37.12\mathrm{MPa}.$$

The carrying capacity of the beam is calculated by the formula:

$${М}_{\mathit{Еd}}=W_c{f}_{\mathit{zМ}}=\frac{{\mathit{bd}}^2}{6}{f}_{\mathit{zМ}}=\frac{20\times {45}^2}{6}37.12\times {10}^{-3}=250.56\mathrm{kNm}.$$

Example 2. Reinforced concrete beam with working cross section b × d = 30 × 45 sm is made of concrete of class С25/30 and steel of class А400С and should take an external moment М_Ed = 266.46 kNm. Determine element reinforcement.

Solution. The moment of resistance of the concrete cross section is determined:

$$W_c=\frac{{\mathit{bd}}^2}{6}=\frac{30\times {45}^2}{6}=10125{\mathrm{sm}}^3.$$

The required calculated resistance of the reinforced concrete to the bending is calculated:

$$f_{\mathit{zМ}}=\frac{M_{\mathit{Ed}}}{W_c}=\frac{266.46\times {10}^3}{10125}=26.32\mathrm{MPa}.$$

According to the tables the required percentage of reinforcement is determined:

$${\rho }_f=1.453\%.$$

The area of the cross section of the working reinforcement is equal:

$$A_s={\rho }_f\times b\times d=0.01453\times 30\times 45=19.62{\mathrm{sm}}^2.$$

By gage 4∅25, А_s = 19.63 sm² is accepted.

Example 3. Determine the cross-sectional dimensions of the beam of concrete class С16/20 and the area of the cross section of the working reinforcement of steel of class А400С, if the beam perceives the bending moment М_Ed = 136 kNm, and the contents of the working armature are ρ_f = 1.25%.

Solution. According to the tables, the calculated resistance of the reinforced concrete to the bend is found: f_М = 21.60 MPa. The moment of resistance is determined:

$$W_c=\frac{{\mathit{bd}}^2}{6}=\frac{M_{\mathit{Ed}}}{f_{\mathit{zМ}}}=\frac{136\times {10}^3}{21.60}=6296{\mathrm{sm}}^3.$$

Accepting the ratio b = 0.5d, calculate

$$d=\sqrt[3]{12\times W_c}=\sqrt[3]{12\times 6296}=42.27\mathrm{sm}.$$

Accepting b × d = 20 × 42 sm, then the area of cross section of the working reinforcement will be A_s = ρ_f × b × d = 0.0125 × 20 × 42 = 10.5 sm². By gage 2∅20 + 2∅18, А_s = 11.37 sm².

Example 4. Reinforced concrete beam with working cross section b × d = 30 × 55 sm has to perceive the bending moment М_Ed = 486 kNm. To define the conditions under which the bearing capacity of the beam will be provided and accept the reinforcement.

Solution. The required calculated resistance of the reinforced concrete to the bend is calculated:

$$f_{\mathit{zМ}}=\frac{M_{\mathit{Ed}}}{W_c}=\frac{6M_{\mathit{Ed}}}{{\mathit{bd}}^2}=\frac{6\times 486\times {10}^3}{30\times {55}^2}=32.12\mathrm{MPa}.$$

From the tables it is clear that such a calculated resistance can be provided for different classes of concrete, reinforcement, and the percentage of reinforcement, starting with the concrete of class С20/25 and percentage of reinforcement 1.5 and more. Following physical, economic, and technological considerations, the designer takes the option that best suits the customer, for example, concrete class С25/30 and reinforcement of class А500С with percentage of reinforcement ρ_f = 1.42%. Then the area of the cross section of the working reinforcement will be A_s = ρ_f × b × d = 0.0142 × 30 × 55 = 23.43 sm². By gage 3∅28 + 2∅20, А_s = 24.75 sm² is accepted.

5. Conclusions

Obtained parameters of the stress-strain state of bending reinforced concrete elements: the calculated resistance of reinforced concrete to bend and tension in the cross section of the bending reinforced concrete element, the above dependencies allow to solve a number of problems, namely:

1. Calculation of the strength of the cross section of the bending reinforced concrete element with known cross-sectional dimensions of the concrete and reinforcement area.

2. Determination of the required cross-sectional area of the reinforcement for a given load with known cross-sectional dimensions of concrete.

3. Foundation the dimensions of the cross section of concrete and reinforcement for a certain percentage of reinforcement and the given load.

4. Checking strength with known cross-sectional area of the reinforcement and given cross-sectional dimensions of the concrete.

5. Verification of the conditions for ensuring the strength of the cross section of reinforced concrete element.

6. Calculation of the moment of formation of cracks.

7. Calculation the width of the crack opening under operating load.

8. Determination of the deflections of the elements under the operational load.

Using the calculated resistances of reinforced concrete allowed to reduce the calculation of reinforced concrete elements according to the nonlinear deformation model to the application of the formulas of the classical resistance of materials and to significantly simplify the process of their calculation.