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

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

f ¼ min f <sup>1</sup>ð Þ a1; …; an ;…; f <sup>i</sup>

this characteristic may be K = 1.05(Еcdεc1/fcd) [3].

formulas of type (2).

and hypotheses.

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,

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

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 σi. 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):

where σi(а1,…,an) is the tension in the cross section of an element of a composite material, MPa,

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

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

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.

Fi f bð Þ <sup>1</sup>;…; bn

, (3)

σið Þ¼ a1; …; an

and Fi the external force factor, which corresponds to a certain level of load.

ð Þ …; an …; <sup>f</sup> <sup>n</sup>ð Þ …; an , (2)

http://dx.doi.org/10.5772/intechopen.75122

15

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:

$$f\_i(a\_1, \ldots, a\_n) = \frac{F\_{Ed}}{f(b\_1, \ldots, b\_n)},\tag{1}$$

where fi(а1, …, an) 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; FEd the external calculated force factor, which corresponds to the limiting state of the element; f(b1, …, bn) the corresponding geometric characteristic; а1, …, an the physical and mechanical parameters of material's cross section of the composite element; and b1, …, bn 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:

Nonlinear Calculations of the Strength of Cross-sections of Bending Reinforced Concrete Elements and Their… http://dx.doi.org/10.5772/intechopen.75122 15

$$f = \min\{f\_1(a\_1, \ldots, a\_n), \ldots, f\_i(\ldots, a\_n), \ldots, f\_n(\ldots, a\_n), \tag{2}$$

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

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

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

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,

where fi(а1, …, an) 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; FEd the external calculated force factor, which corresponds to the limiting state of the element; f(b1, …, bn) the corresponding geometric characteristic; а1, …, an the physical and mechanical parameters of material's cross section of the composite element; and b1, …, bn the geometric parameters of the

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

FEd f bð Þ <sup>1</sup>;…; bn

, (1)

various force factors.

14 Cement Based Materials

elements to formulas of resistance of materials.

this can be expressed by the following equation:

cross section of the composite element.

section of the element is formed:

2. The term calculated resistance reinforced concrete

f i

ð Þ¼ a1; …; an

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(Еcdεc1/fcd) [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 σi. 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):

$$\sigma\_i(a\_1, \ldots, a\_n) = \frac{F\_i}{f(b\_1, \ldots, b\_n)}\,'\,\tag{3}$$

where σi(а1,…,an) is the tension in the cross section of an element of a composite material, MPa, and Fi 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:


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

The advantages of this method should include:


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