Perspective Chapter: New Theoretical Basics of Calculation of Reinforced Concrete Elements

Abstract

In this chapter, on the basis of the nonlinear deformation model of the mechanics of deformable solids, new theoretical foundations for the design of reinforced concrete elements have been studied. This is the first formulation of the strength conditions of reinforced concrete elements relative to the height of the compressed concrete zone ξ in the following form: for eccentrically tensioned elements $0\le \xi <{\xi }_{\mathit{\text{OR}}}$, where ${\xi }_{\mathit{\text{OR}}}=1/\left(1+\frac{{\epsilon }_{\mathit{\text{sult}}}}{{\epsilon }_{\mathit{\text{bult}}}}\right)$, for bending elements ${\xi }_{\mathit{\text{OR}}}\le \xi \le {\xi }_R$, where ${\xi }_R=1/\left(1+\frac{{\epsilon }_{\mathit{sl}}}{{\epsilon }_{\mathit{\text{bult}}}}\right)$ for eccentrically compressed elements with three cases of calculation: (1) ${\xi }_{\mathit{\text{OR}}}\le \xi \le {\xi }_R$; (2) ${\xi }_R<\xi <1$; and (3) $1\le \xi \le \infty$. The height of the compressed zone of concrete ξ is determined from the condition $M_X\le M_{X,\mathit{ult}}$. The representation of strength conditions in the above form makes it possible to develop new and accurate solutions to the elastic-creep theory of mechanics of reinforced concrete, taking into account the real properties of reinforced concrete materials. The research concluded several important points; for example, it has been established that the developed method for designing reinforced concrete elements is based on an accurate and rigorous application of the nonlinear deformation model of the mechanics of deformed solids to the problems of elastoplastic bending of the theory of reinforced concrete. The developed method for designing reinforced concrete elements makes it possible to take into account the main characteristics of materials: the long-term strength of concrete and the increase in the strength of reinforcement (rebar) after the yield point in accordance with the state diagrams of concrete and reinforcement compared to other important matters worth publishing to shed light on.

1. Introduction

The rules and principles established by the Eurocodes are a strict set of design provisions on the basis of which the design of reinforced concrete structures should be carried out. The European reinforced concrete standards have incorporated numerous scientific research and experience of prominent scientists from various countries, including Russia, the best sections of the national standards of these countries, a reasoned formulation of the main hypotheses, and generally accepted methods.

After the approval of this set as the standards of the European Community, the theory and practice of reinforced concrete begin to develop within the framework of strict compliance with all the requirements, design provisions of the Eurocode deformation model of the section.

It should be noted that the inconsistency between the method of limit states of the Eurocode and the method of limit loads of the Russian standard was repeatedly pointed out by the creators of the standard K.E. Tal and A.A. Gvozdev [1]. Comparing the national standard, now the basis of the updated SNiP 52.01-2003, and European standards, A.A. Gvozdev pointed out their significant difference in the principles and methods of calculation regarding the design of normal sections and the action of duration of the load on reinforced concrete structures.

In the methods and limit states of the national standard, the hypothesis of flat sections, one of the main hypotheses in the Eurocode and the classical theory of mechanics, is categorically rejected.

To determine the creep deformation of concrete, there are two theories based on the basic properties of concrete deformation: the theory of aging and the theory of heredity.

As experiments show, the theory of aging exaggerates the influence of aging of materials and underestimates the calculated values of creep deformations.

Studies to test the theory of heredity led to the conclusion that it not only excludes aging, but also exaggerates the calculated values of creep deformations. As noted by V.M. Bondarenko [2], both of these theories give some limit values of creep deformations: the smallest and the largest. In addition, as the analysis showed, in some cases their use can lead to qualitatively incorrect results. The main reason for their existence and application is the relative mathematical simplicity, which consists in an elementary transition from integral to differential equations of state [2]. To eliminate the shortcomings of mentioned theories of creep, the Soviet scientist N.G. Maslov [2] proposed the theory of elastic-plastic (creeping) body of mechanics of solid deformed bodies, the essence of which is that this theory takes into account both the properties of concrete aging and the properties of heredity.

Considering the above, in this chapter, the state diagrams of concrete and reinforcement, taking into account elastic-plastic deformations, are summarized and presented in the form of a two-line diagram (Figure 1).

Figure 1.

Design scheme of bending elements, (a) diagram of normal stresses taking into account creep and long-term strength of the compressed zone of concrete, (b) deformation diagrams of a flat section at various limit states: I: at eccentric tension; II: at bending; III: at eccentric compression, (c) state diagram of concrete taking into account creep and long-term strength of concrete, and (d) diagram of the state of rebar, taking into account the increase in the strength of rebar after the yield point.

Thus, the study of elastic-creep bending of reinforced concrete elements is reduced to solving the generally accepted method of the theory of plasticity (creep) of deformed solid bodies of mechanics.

For a compressed concrete prism in the mode of proportional development of longitudinal deformations in time, a gradual decrease in the resistance of concrete is detected, the so-called descending branch of the stress-strain diagram (Figure 1c).

The greatest intensity of growth of inelastic deformations is observed in the first 3-4 months and may continue for several years.

In the diagram, section O-A characterizes the deformations that occur during loading, and section A-B characterizes the rising of inelastic deformations (creep) at a constant value of stresses.

According to experimental data, under a long-term load under the influence of developing significant inelastic deformations and structural changes, concrete is destroyed at stresses less than the temporary resistance to axial compression $R_b;R_{\mathit{bl}}=0,9R_b$ and less than [3].

In 2014, the Russian newspaper “Stroitelnaya Gazeta” published an article “Inconsistency between the updated standard for reinforced concrete and the Eurocode - an obstacle in construction” [1]. Because the national standard for reinforced concrete AzDTN 2.16-1 was developed on the basis of the updated standard for reinforced concrete SNiP 52.01-2003 of Russia, this issue was studied at AzNIISA and a decision was made to bring it to conformity with Eurocodes. These differences cannot be eliminated by any correction or conversion coefficient.

When comparing national and European standards, their significant differences in the concepts of limit states and methods for designing reinforced concrete elements were revealed.

Therefore, instead of the limit state, which is proceeded from the stage of destruction of the stressed state of bending elements, a new form of limit states is proposed in the form of strain diagrams obtained on the basis of the hypothesis of flat sections. The essence and form of these limit states are described and given in Refs. [2, 3, 4, 5, 6, 7].

Limit states of bending elements in accordance with the nonlinear deformation model are accepted in the form of strain diagrams of flat sections, which pass through one of these three characteristic points of the section, corresponding to deformations of tensed reinforcement ${\epsilon }_{s,\mathit{max}}={\epsilon }_{s,2}$ under eccentric tension, ultimate deformation of the compressed zone of concrete ${\epsilon }_{b,\mathit{max}}={\epsilon }_{b,2}$ under simple bending, as well as ${\epsilon }_b={\epsilon }_{b0}$ under eccentric compression [4, 8, 9].

As is known, the only correct solution to the problem of mechanics of deformable solids will be when all three aspects of the problem: static, geometric, and physical are combined into a single one. In this formulation, on the basis of the nonlinear deformation model, the problem of elastoplastic bending of reinforced concrete elements was solved and an analytical solution of this problem is given in Refs. [3, 7].

The limit states for the strength of bendable reinforced concrete elements, adopted in Refs. [3, 4], based on a nonlinear deformation model of the mechanics of deformable solids, make it possible to compile strength conditions in a general form, taking into account the main deformation characteristics: both creep and long-term strength of concrete and the increase in the strength of reinforcement after the yield point, the state diagrams of which are shown in Figure 1c and d.

Two-line diagram of concrete:

at $0>{\epsilon }_b\le {\epsilon }_{b1}$${\sigma }_b=E_b\cdot {\epsilon }_b$;

at ${\epsilon }_{b1}\le {\epsilon }_b\le {\epsilon }_{b2}$${\sigma }_b=R_b\left[\left(1-\frac{{\epsilon }_b-{\epsilon }_{b1}}{{\epsilon }_{b2}-{\epsilon }_{b1}}\right)+\frac{R_{\mathit{bl}}}{R_b}\frac{{\epsilon }_b-{\epsilon }_{b1}}{{\epsilon }_{b2}-{\epsilon }_{b1}}\right]\le R_b$.

Two-line diagram of rebar:

at $0>{\epsilon }_s\le {\epsilon }_{\mathit{sl}}$${\sigma }_s={\epsilon }_s\cdot E_s;$.

at ${\epsilon }_{\mathit{sl}}\le {\epsilon }_s\le {\epsilon }_{s2}$${\sigma }_s=R_s\left[\left(1-\frac{{\epsilon }_s-{\epsilon }_{\mathit{sl}}}{{\epsilon }_{s2}-{\epsilon }_{\mathit{sl}}}\right)+\frac{R_{\mathit{su}}}{R_s}\frac{{\epsilon }_s-{\epsilon }_{\mathit{sl}}}{{\epsilon }_{s2}-{\epsilon }_{\mathit{sl}}}\right]\ge R_s$.

The strength conditions of reinforced concrete elements are as follows:

$${\epsilon }_{b,max}\le {\epsilon }_{b,\mathit{ult}};{\epsilon }_{s,max}\le {\epsilon }_{s,\mathit{ult}}$$E1

where ${\epsilon }_{b,\mathit{ult}}={\epsilon }_{b2}$ is the ultimate relative deformation of concrete in compression:

${\epsilon }_{s,\mathit{ult}}={\epsilon }_{s2}$ ultimate relative deformation of reinforcement at breaking.

Depending on the new forms of limit states, a method for designing reinforced concrete bending elements has been developed that meets the main requirements of Eurocodes.

2. The novelty of this work

The novelty of this work lies in the fact that the developed method for calculating reinforced concrete elements for the first time takes into account the long-term strength of concrete and the increase in the strength of reinforcement after the yield point, the account of which leads to savings in reinforcement.

The purpose of developing a methodology for calculating reinforced concrete elements is that its application in existing regulatory documents will ensure their complete compatibility with Eurocodes.

Then, the solution of the problem of bending elements based on a nonlinear deformation model with piecewise linear state diagrams of concrete is reduced to solving the following static equations, subject to the hypothesis of flat sections and analytical expressions of a two-line concrete state diagram (Figure 1c):

1. $M={\int }_{A_b}{\sigma }_b\mathit{\text{bydy}}+{\int }_{A_s}{\sigma }_s\left(h_0-y\right)d{A}_s+{\int }_{A_s^{\prime }}{\sigma }_s^{\prime }\left(y-a^{\prime }\right){\mathit{dA}}_s^{\prime };$

   $${\int }_{A_b}{\sigma }_b\mathit{bdy}-{\int }_{A_s}{\sigma }_{s}d{A}_s+{\int }_{A_s^{\prime }}{\sigma }_s^{\prime }d{A}_s^{\prime }=0;$$

   where

   $${\int }_{A_b}{\sigma }_b\mathit{\text{bydy}}={\int }_0^{y_0}{\epsilon }_b\bullet E_b\mathit{\text{bydy}}+{\int }_{y_0}^y{R}_b\left({\epsilon }_b\right)\mathit{\text{bydy}}$$
   $${\int }_{A_b}{\sigma }_b\mathit{bdy}={\int }_0^{y_0}{\epsilon }_b\bullet E_b\mathit{bdy}+{\int }_{y_0}^y{R}_b\left({\epsilon }_b\right)\mathit{bdy}$$

2. Linear deformation diagram obtained on the basis of the hypothesis of flat sections, where, at a known value of the ultimate deformation of concrete ${\epsilon }_{b2}$, the deformations of the reinforcement ${\epsilon }_s$ and ${\epsilon }_s^{\prime }$ are determined, and then, the stresses at the characteristic points of the concrete’s compressed zone section;

3. Piecewise linear or curvilinear state diagrams of concrete characterizing the deformations and stresses of the compressed zone of concrete.

   As a result, all aspects of the problem are combined: statistical, geometrical (deformational), and physical.

   The application of a mathematically rigorous nonlinear deformation model of the mechanics of deformable solids to the problems of the theory of reinforced concrete made it possible to propose a new version of the limit states of bending elements and, on their basis, develop calculation methods for designing reinforced concrete structures, which is one of the pressing problems of structural mechanics.

3. The content of the work

Due to the fact that the strength conditions (1) are general, depending on the purpose and problem setting, they can be presented in a more convenient form, for example, within the limits of elasticity of concrete and reinforcement, based on Hooke’s law, they are easily expressed in the form of the corresponding well-known theory of allowable stresses:

$${\sigma }_{b,max}\le R_b;{\sigma }_{s,max}\le R_s$$E2

When applying the theory of the nonlinear deformation model of the mechanics of deformable solid bodies to reinforced concrete structures, based on the following formulas of resistance of materials [10, 11, 12, 13]:

$\frac{1}{r}=\frac{{\epsilon }_{b,max}}{y}=\frac{M_x}{D}$ we find:

${\epsilon }_{b,max}=\frac{M_{x}y}{D}$ and ${\epsilon }_{b,\mathit{ult}}=\frac{M_{x,\mathit{ult}}y}{D}$, instead of (1) we get:

$$M_x\le M_{x,\mathit{ult}}$$

On the other hand, depending on the limit states for different types of deformations, the strength conditions (1) are simplified and take a simpler form:

According to the limit state for eccentrically tensed elements, the limit state is accepted in the form where the relative elongation of the reinforcement is taken to be equal ${\epsilon }_{s,\mathit{ult}}$, the strength condition is satisfied ${\epsilon }_{s,max}\le {\epsilon }_{\mathit{\text{sult}}}$, based on this condition, and the concrete deformation is found as follows:

$${\epsilon }_b={\epsilon }_{s,\mathit{ult}}\frac{y}{h_0-y}\le {\epsilon }_{b,\mathit{ult}}$$

we find from here: $y\le y_{\mathit{\text{OR}}}$ or $\xi \le {\xi }_{\mathit{\text{OR}}}$

where ${\xi }_{\mathit{\text{OR}}}=\frac{1}{1+\frac{{\epsilon }_{s,\mathit{ult}}}{{\epsilon }_{b,\mathit{ult}}}}$

Thus, for eccentrically tensed elements, strength conditions (1) are satisfied if:

$$0\le \xi \le {\xi }_{\mathit{\text{OR}}}$$E3

Analogously, according to the limit state for simple bending, where the largest relative deformation of the most compressed concrete fibers is taken to be equal ${\epsilon }_{b,\mathit{ult}}$, then the condition ${\epsilon }_{b,max}\le {\epsilon }_{b,\mathit{ult}}$ is satisfied identically, and based on this condition, the relative deformation of the stretched reinforcement is found as:

${\epsilon }_s={\epsilon }_{b,\mathit{ult}}\frac{h_0-y}{y}\le {\epsilon }_{s,\mathit{ult}},$ and from here we find:

$$y\ge y_{\mathit{\text{OR}}}\text{or}\xi \ge {\xi }_{\mathit{\text{OR}}}$$

On the other hand, at bending, another condition is fulfilled $\xi \le {\xi }_R$, where ${\xi }_R=\frac{1}{1+\frac{{\epsilon }_{\mathit{sl}}}{{\epsilon }_{b,\mathit{ult}}}},$ then we have:

$${\xi }_{\mathit{\text{OR}}}\le \xi \le {\xi }_R$$E4

There are three cases for eccentrically compressed elements, and the strength conditions for each case take the following form:

1. ${\xi }_{\mathit{\text{OR}}}\le \xi \le {\xi }_R$, as for bending elements;

2. $${\xi }_R\le \xi \le 1;$$E5

3. $1\le \xi \le \infty$, for which the calculation method will be developed separately in a special form.

In all considered cases, the strength check is reduced to compliance with the condition (3)–(5) where it is required to determine the height of the compressed zone of concrete, which is determined from the following condition:

$$M_x\le M_{x,\mathit{ult}}$$E6

When comparing the strength conditions compiled on the base of the nonlinear deformation model in the form (3)–(5) with the method based on the plastic hinge model, we see that we have only one strength condition:

$$M_x\le M_{x,\mathit{ult}}$$

And the second strength condition (1) for tensed reinforcement ${\epsilon }_{s,max}\le {\epsilon }_{\mathit{\text{sult}}}$ cannot be established because the law of flat sections is rejected, which leads to inconsistencies with Eurocodes.

Verification of the fulfillment of the strength conditions (3)–(5) is related to the height of the compressed zone of the concrete of bending elements, which is determined from the condition (6), where $M_{x,\mathit{ult}}$ has the form [3]:

$$\begin{array}{l}{M}_x\le M_{x,\mathit{ult}}=\\ =R_b{{\mathit{bh}}_0}^2\left\{\begin{array}{l}\left(1-k_0\right)\xi \left[1-\frac{1}{2}\left(1-k_0\right)\xi \right]+\frac{1}{2}{k}_0\xi \left[1-\left(1-\frac{2}{3}{k}_0\right)\xi \right]-\\ -\frac{1}{2}\left(1-{\gamma }_{\mathit{bl}}\right)\left(1-k_0\right)\xi \left[1-\frac{1}{3}\left(1-k_0\right)\xi \right]\end{array}\right\}+\end{array}$$

$$+{\sigma }_s^{\prime }{A}_s^{\prime }\left(h_0-a^{\prime }\right);$$E7

$$N={\sigma }_s{A}_s-{\sigma }_s^{\prime }{A}^{\prime }-R_b{\mathit{bh}}_0\xi \left[\left(1-\frac{k_0}{2}\right)-\frac{1}{2}\left(1-k_0\right)\left(1-{\gamma }_{\mathit{bl}}\right)\right]$$

here $M_{x,\mathit{ult}}$—limiting moment of internal forces relative to the center of gravity of tensioned reinforcement, the value of which is determined in Ref. [3];

$\xi =\frac{y}{h_0}$—height of the compressed zone of concrete;

${\gamma }_{\mathit{bl}}=\frac{R_{\mathit{bl}}}{R_b},$$R_{\mathit{bl}}$—long-term strength of concrete;

$k_0=\frac{{\epsilon }_{b,1}}{{\epsilon }_{b,max}}$—coefficient of elastoplastic deformation of concrete, which varies within: $k_0=0,15-1,0;$ and the long-term strength of concrete $R_{\mathit{bl}}={\gamma }_{\mathit{bl}}{R}_b$ и ${\gamma }_{\mathit{bl}}=0,8÷1,0$. By varying these coefficients, we find the values of the limiting moment $M_{x,\mathit{ult}}$ and the shapes of the stress diagram in concrete in accordance with $M_{x,\mathit{ult}}$ (Figure 1a).

$k_0=1,$ in this case, concrete and reinforcement work in the elastic stage, therefore, ${\gamma }_{\mathit{bl}}=1$ and concrete creep does not occur.

$A_R=0,5{\xi }_R\left(1-\frac{1}{3}{\xi }_R\right)=0,208$ and therefore $M_{\mathit{\text{xult}}}=0,208R_b{{\mathit{bh}}_0}^2,$which differs from the solution in the limit state by 46.8%.

$k_0=0,15-1,0,$ limiting moment reaches its maximum value at $k_0=0,2$ and minimum at ${\gamma }_{\mathit{bl}}=0,85$:

$$M_{x,\mathit{ult}}=R_b{{\mathit{bh}}_0}^2{A}_R$$

where

$$A_R=0,84{\xi }_R-0,3907{\xi }_R^2=0,3865$$

If we neglect the elastic deformation of concrete near the neutral layer of the section, then we have:

$A_R=0,74{\xi }_R-0,304{\xi }_R^2=0,3583$ and the difference is about 6.7%; therefore, under conditions (7), we neglect the influence of elastic deformations of concrete in the compressed zone of the element and have:

$A_R=0,8{\xi }_R-0,32{\xi }_R^2=0,3912$ and the difference, taking into account long-term strength, is 9.2%.

Thus, the design scheme of bending elements is accepted as shown in Figure 1a, where the influence of the long-term strength of concrete is taken into account, which is fully consistent with the principles and requirements of the Eurocode.

Based on the accepted design scheme, the strength conditions (6) take the following form:

$$M\le R_b{{\mathit{bh}}_0}^2{A}_0+{\sigma }_s^{\prime }{A}_s^{\prime }\left(h_0-a^{\prime }\right)$$E8

$$N={\sigma }_s{A}_s-{\sigma }_s^{\prime }{A}_s^{\prime }-0,8R_b{\mathit{bh}}_0\xi \left[1-0,4\left(1-{\gamma }_{\mathit{bl}}\right)\right]$$

where

$$A_0=0,8\xi \left(1-0,4\xi \right)-0,4\left(1-{\gamma }_{\mathit{bl}}\right)\xi \left(1-0,267\xi \right)$$

Strength conditions (8) are valid if conditions (3)–(5) are observed, written for various types of bending of reinforced concrete elements.

A new view at the limit states and strength conditions (1) made it possible to develop methods for calculating reinforced concrete elements for various forms of bending.

Having determined $A_0=\frac{M}{R_b{{\mathit{bh}}_0}^2},$ from the condition (8), for determining the height of the compressed zone we obtain the following equation:

$$\begin{array}{l}{\xi }^2-\frac{0,8-0,4\left(1-{\gamma }_{\mathit{bl}}\right)}{0,32-0,1068\left(1-{\gamma }_{\mathit{bl}}\right)}\xi +\\ +\frac{M}{R_b{{\mathit{bh}}_0}^2\left[0,32-0,1068\left(1-{\gamma }_{\mathit{bl}}\right)\right]}=0\end{array}$$E9

After finding $\xi$, the conditions (3)–(5) written for each type of bending of reinforced concrete elements are checked. The following technique is proposed for finding the stresses in the reinforcements, taking into account the increase in stresses after the yield point.

4. Design according to the first group of limit states

4.1 Eccentric tension

In accordance with the limit state, for this type of deformation in tensed reinforcement, the relative deformation is equal to ${\epsilon }_{s,\mathit{ult}}$, therefore, the stress ${\sigma }_s=R_{\mathit{su}}$, and in the compressed zone ${\epsilon }_s^{\prime }={\epsilon }_{s,\mathit{ult}}\frac{y-a^{\prime }}{h_0-y},$ and the stress in accordance with ${\epsilon }_s^{\prime }$ is determined as follows (Figure 2):

Figure 2.

Design scheme of eccentrically tensioned elements (a) diagram of stresses taking into account creep and long-term strength of the compressed zone of concrete, and (b) deformation diagram of a flat section at eccentric tension.

at ${\epsilon }_s^{\prime }\le {\epsilon }_{\mathit{sl}},{\sigma }_s^{\prime }=E_s{\epsilon }_s$ and at ${\epsilon }_s^{\prime }>{\epsilon }_{\mathit{sl}}$

$${\sigma }_s^{\prime }=R_s\left[\left(1-\frac{{\epsilon }_s^{\prime }-{\epsilon }_{\mathit{sl}}}{{\epsilon }_{s2}-{\epsilon }_{\mathit{sl}}}\right)+\frac{R_{\mathit{su}}}{R_s}\frac{{\epsilon }_s^{\prime }-{\epsilon }_{\mathit{sl}}}{{\epsilon }_{s2}-{\epsilon }_{\mathit{sl}}}\right]$$

The condition $\xi \le {\xi }_{\mathit{\text{OR}}}$is checked. If this condition is satisfied, then $A_s^{\prime }=0$ and the reinforcement area in the tension zone is determined as follows.

$$A_s=\frac{1}{R_{\mathit{su}}}\left\{\frac{N}{100}+0,8\xi R_b{\mathit{bh}}_0\left[1-0,4\left(1-{\gamma }_{\mathit{bl}}\right)\right]\right\}$$

If $\xi >{\xi }_{\mathit{\text{OR}}},$ then the destruction of the element will be along the compressed zone of concrete. In order to prevent destruction, it is necessary to ensure $\xi \le {\xi }_{\mathit{\text{OR}}}$.

Assuming $\xi ={\xi }_{\mathit{\text{OR}}}$ we find the minimum value of$A_s^{\prime }$:

$A_s^{\prime }=\frac{N\cdot e-A_{\mathit{\text{OR}}}{R}_b{{\mathit{bh}}_0}^2}{{\sigma }_s^{\prime }\left(h_0-a^{\prime }\right)}\mathsf{\text{and}}\begin{array}{l}{A}_s=\frac{1}{R_{\mathit{su}}}\left\{\frac{N}{100}+0,8{\xi }_{\mathit{\text{OR}}}{R}_b{\mathit{bh}}_0\left[1-0,4\left(1-{\gamma }_{\mathit{bl}}\right)\right]\right\}+\\ +A_s^{\prime }\frac{{\sigma }_s^{\prime }}{{\sigma }_s^{\prime }}\end{array}$

4.2 Simple bending of reinforced concrete elements

At simple bending, the conditions ${\xi }_{\mathit{\text{OR}}}\le \xi \le {\xi }_R$must be met. The value of $\xi$ is determined by solving the quadratic Eq. (7).

In contrast to eccentrically tensed elements, here, according to the limit state, the relative deformation of the most compressed extreme fibers of concrete is taken equal to ${\epsilon }_{b,\mathit{ult}}$ and the condition ${\epsilon }_{b,max}\le {\epsilon }_{b,\mathit{ult}}$ is satisfied identically.

Then, the deformations of the reinforcement are determined as follows:

$${\epsilon }_s={\epsilon }_{b,\mathit{ult}}\frac{1-\xi }{\xi };{\epsilon }_s^{\prime }={\epsilon }_{b,\mathit{ult}}\frac{\xi -{\delta }^{\prime }}{\xi };{\delta }^{\prime }=\frac{a^{\prime }}{h_0}$$

Further, depending on the value of ${\epsilon }_s$ and ${\epsilon }_s^{\prime }$, the stresses in the reinforcements $A_s$ and $A_s^{\prime }$are determined:

If ${\epsilon }_s\ge {\epsilon }_{\mathit{sl}}$ and ${\epsilon }_s^{\prime }>{\epsilon }_{\mathit{sl}},$ then the stresses in the reinforcements will be ${\sigma }_s=R_s\left[\left(1-\frac{{\epsilon }_s-{\epsilon }_{\mathit{sl}}}{{\epsilon }_{s2}-{\epsilon }_{\mathit{sl}}}\right)+\frac{R_{\mathit{su}}}{R_s}\frac{{\epsilon }_s-{\epsilon }_{\mathit{sl}}}{{\epsilon }_{s2}-{\epsilon }_{\mathit{sl}}}\right]$ and by the same formula ${\sigma }_s^{\prime }$ at ${\epsilon }_s^{\prime }$.

When determining areas $A_s$and $A_s^{\prime }$ there may be the following cases:

1. the conditions $\xi \ge {\xi }_{\mathit{\text{OR}}}$ and $\xi \le {\xi }_R,$are met and then reinforcement is not required in the compressed zone of concrete $A_s^{\prime }=0;$.

   In the stretched zone $A_s$ is determined by the following formula:

   $$A_s=\frac{1}{{\overline{\sigma }}_s}0,8R_b{\mathit{bh}}_0\xi \left[1-0,4\left(1-{\gamma }_{\mathit{bl}}\right)\right]$$

   If the condition $\xi \ge {\xi }_{\mathit{\text{OR}}}$ is not met, then choosing $A_s^{\prime }$ structurally or assuming $\xi ={\xi }_{\mathit{\text{OR}}}$ by design it is ensured the equation $\xi ={\xi }_{\mathit{\text{OR}}}$ and the areas of reinforcements are determined as follows:

   $$A_s^{\prime }=\frac{M-A_{\mathit{\text{OR}}}{R}_b{{\mathit{bh}}_0}^2}{{\sigma }_s^{\prime }\left(h_0-a^{\prime }\right)};$$
   $$A_s=\frac{1}{{\sigma }_s}0,8R_b{\mathit{bh}}_0\xi \left[1-0,4\left(1-{\gamma }_{\mathit{bl}}\right)\right]+A_{\mathit{so}}^{\prime }\frac{{\sigma }_s^{\prime }}{{\sigma }_s}$$

2. When $\xi >{\xi }_R,$is observed, the condition $\xi >{\xi }_{\mathit{\text{OR}}}$is identically satisfied. The reinforcement area is found as a double reinforcement:

$$A_s^{\prime }=\frac{M-A_R{R}_b{{\mathit{bh}}_0}^2}{{\sigma }_s^{\prime }\left(h_0-a^{\prime }\right)}$$

$$A_s=\frac{1}{{\sigma }_s}0,8R_b{\mathit{bh}}_0\xi \left[1-0,4\left(1-{\gamma }_{\mathit{bl}}\right)\right]+A_s^{\prime }\frac{{\sigma }_s^{\prime }}{{\sigma }_s}$$

4.3 Eccentric compression

As noted above, there are three cases in the design of eccentric compression (5).

The strength conditions for the first two cases of changing the height of the compressed zone have the form (8). Under these strength conditions, there are four unknown parameters $\xi ,{\sigma }_s,A_s^{\prime }$ и $A_s,$ а ${\sigma }_s^{\prime }$and are determined as follows:

${\epsilon }_s^{\prime }={\epsilon }_{b,\mathit{ult}}\frac{\xi -{\delta }^{\prime }}{\xi },$ where ${\epsilon }_s^{\prime }>{\epsilon }_{\mathit{sl}}$. And stresses:

$${\sigma }_s^{\prime }=R_s\left[\left(1-\frac{{\epsilon }_s^{\prime }-{\epsilon }_{\mathit{sl}}}{{\epsilon }_{s2}-{\epsilon }_{\mathit{sl}}}\right)+\frac{{\epsilon }_s^{\prime }-{\epsilon }_{\mathit{sl}}}{{\epsilon }_{s2}-{\epsilon }_{\mathit{sl}}}\cdot \frac{R_{\mathit{su}}}{R_s}\right]$$

$e$—the eccentricity of the normal force and for a rectangular section is equal to:

$$e=e_0\eta +\frac{h_0-a^{\prime }}{2},$$

$\eta$—coefficient taking into account the longitudinal bending:

$$\eta =\frac{1}{1-\frac{N}{N_{\mathit{cr}}}}$$

$N_{\mathit{cr}}$—Euler’s critical force and it is equal to:

$$N_{\mathit{cr}}=\frac{{\pi }^2}{l_0^2}D$$E10

where $D$—bending stiffness is defined as follows [11]:

$$D=\frac{M_{x}y}{{\epsilon }_{b,max}}$$

Here, $M_x$ is determined by the formula (7).

$D$—the stiffness of the element in bending, taking into account the long-term strength of concrete and the stressed state, the value of which was determined in Ref. [3]:

$$D=k_b{E}_b{J}_{x,b}+k_s{E}_s{J}_{x,s};$$E11

here: $k_b=1,5k_0\left(1-\frac{1}{3}{k_0}^2\right)-\left(1-{\gamma }_{\mathit{bl}}\right)\left(1-k_0\right)\left(1+\frac{k_0}{2}\right)k_0,$

$$k_s=\frac{{\epsilon }_{\mathit{sl}}}{{\epsilon }_s{k}_0};J_{x,b}=\frac{{\mathit{by}}^3}{3};J_{x,s}={\mathit{nA}}_s{\left(h_0-y\right)}^2+n{A}_s^{\prime }\frac{{\epsilon }_s^{\prime }}{{\epsilon }_s}{\left(y-a^{\prime }\right)}^2$$

${\epsilon }_s={\epsilon }_{b2}\frac{1-\xi }{\xi },{\epsilon }_s^{\prime }={\epsilon }_{b2}\frac{\xi -{\delta }^{\prime }}{\xi }$

In view of the fact that two strength conditions (8) contain four unknowns, a special calculation technique is required to solve the problem.

In order to reduce the number of unknowns, the following form of replacement of unknowns is proposed [4].

$$N+{\sigma }_s{A}_s=R_{s}A;A_s^{\prime }=A^{\prime };M=N\cdot e$$E12

After replacement, the strength conditions (8) take the following form:

$$M=R_b{{\mathit{bh}}_0}^2{A}_0+{\sigma }_s^{\prime }{A}_s^{\prime }\left(h_0-a^{\prime }\right);$$E13

$$R_{s}A=R_b{\mathit{bh}}_0\xi \left[0,8-0,4\left(1-{\gamma }_{\mathit{bl}}\right)\right]+{\sigma }_s^{\prime }{A}^{\prime }$$

This system completely coincides with the strength equations of elements in pure bending with double reinforcement. Therefore, assuming in them $\xi ={\xi }_R$, we find the cross-sectional area $A$ and $A^{\prime }$.

$$A^{\prime }=\frac{M-R_b{{\mathit{bh}}_0}^2{A}_R}{{\overline{{\sigma }^{\prime }}}_s\left(h_0-a^{\prime }\right)};$$

$$A=\frac{1}{R_s}{R}_b{\mathit{bh}}_0{\xi }_R\left[0,8-0,4\left(1-{\gamma }_{\mathit{bl}}\right)\right]+\frac{{\sigma }_s^{\prime }}{R_s}{A}^{\prime };$$

Using conditions (12), we find $A$ and $A_s^{\prime }$.

$$A_s^{\prime }=A^{\prime }=\frac{M-R_b{{\mathit{bh}}_0}^2{A}_R}{{\overline{{\sigma }^{\prime }}}_s\left(h_0-a^{\prime }\right)};A_s=\frac{R_s}{{\sigma }_s}\left(A-\frac{N}{100R_s}\right)$$

This technique is also valid for the first case of calculation, when ${\xi }_{\mathit{\text{OR}}}\le \xi \le {\xi }_R$ here at $A-\frac{N}{100R_s}\le 0,A_s$ has a negative value and $A_s=0$. Then we have $A_s=0$; $A_s^{\prime }\ne 0$.

Theoretically, it could be $A_s=0$; $A_s^{\prime }=0$. This is possible with $N\cdot e\le R_b{{\mathit{bh}}_0}^2{A}_R$ and $A\le \frac{N}{100R_s}$. The section should be reinforced structurally.

When $A_s=0$, it is required to determine the height of the compressed concrete zone. To this end, from the second equation we find:

$${\sigma }_s^{\prime }{A}^{\prime }=R_{s}A-R_b{\mathit{bh}}_0\xi \left[0,8-0,4\left(1-{\gamma }_{\mathit{bl}}\right)\right]$$

and then substituting it into the first equation, we get.

$$\begin{array}{l}M=R_b{\mathit{bh}}_2^0{A}_0-\\ -\left\{R_{s}A-R_b{\mathit{bh}}_0\xi \left[0,8-0,4\left(1-{\gamma }_{\mathit{bl}}\right)\right]\right\}\left(h_0-a^{\prime }\right)\end{array}$$

From this equation for finding $\xi$ we have:

$$\begin{array}{l}{\xi }^2-\frac{a^{\prime }}{h_0}\frac{0,8-0,4\left(1-{\gamma }_{\mathit{bl}}\right)}{0,32-0,1068\left(1-{\gamma }_{\mathit{bl}}\right)}\xi -\\ -\frac{N}{R_b{{\mathit{bh}}_0}^2\left[0,32-0,1068\left(1-{\gamma }_{\mathit{bl}}\right)\right]}=0\end{array}$$E14

This solution is valid for the second case of calculation (5), when ${\xi }_R\le \xi \le 1$.

Having solved the equations, we find $\xi$ and obtain that $\xi >{\xi }_R,$at which the reinforcement area $A_s=0$, we find:

$$A_s=0,A_s^{\prime }=\frac{N\cdot e-R_b{{\mathit{bh}}_0}^2{A}_0}{{\sigma }_s^{\prime }\left(h_0-a^{\prime }\right)}$$

${\sigma }_s^{\prime }$ is determined by the method described above. Comparing the case ${\xi }_{\mathit{\text{OR}}}\le \xi \le {\xi }_R$, which is also valid for simple bending, we can conclude that in this interval at simple bending, a single reinforcement is possible, i.e., $A_s^{\prime }=0$ while at eccentric compression $A_s^{\prime }>A_s$ and we always have $A_s^{\prime }\ne 0$, $A_s\ne 0$, for which $\xi ={\xi }_R$.

At eccentric compression, the third case of calculation is possible, when $1\le \xi <\infty$. In this case, another calculation method is required that corresponds to the strain diagram for new limit states.

When the section is fully compressed, in accordance with the limit state, the deformation diagram passes through the point “C” where the concrete deformation ${\epsilon }_b$ is equal to ${\epsilon }_{b0}=0,002,$ and the stress diagram in concrete has the form of a trapezoid-parabola (Figure 3) [4].

Figure 3.

Design scheme of eccentrically compressed elements (a) deformation diagram of a flat section, (b) diagram of stresses taking into account long-term strength of the compressed zone of concrete, and (c) state diagram of concrete taking into account creep and long-term strength of concrete.

It should be noted that we solved this problem without taking into account the long-term strength of concrete ${\gamma }_{\mathit{bl}}=1,$ in work [4], only the final results of the calculation are given here.

$$S=\psi R_b\mathit{bh};\psi ={\overline{\gamma }}_{\mathit{bl}}-\frac{3,0416}{{\left(\frac{7y}{h}-3\right)}^2};\delta =\frac{6}{7}-\frac{1}{49\psi }\left(10+7,5{\gamma }_{\mathit{bl}}\right);{\overline{\gamma }}_{\mathit{bl}}=\frac{1}{7}\left[4+1,5\left(1+{\gamma }_{\mathit{bl}}\right)\right]$$

When “у” varies within $h\le y<\infty$, value of coefficient $\psi$ varies within $\left({\overline{\gamma }}_{\mathit{bl}}-0,19\right)\le \psi \le {\overline{\gamma }}_{\mathit{bl}},$ and at ${\gamma }_{\mathit{bl}}=0,85;$$0,778\le \psi \le 0,968$

The strength conditions drawn up relative to the neutral axis of the section take the form:

$$M\le \psi R_b\mathit{bh}\left(y-\mathit{\delta h}\right)+{\sigma }_s^{\prime }{A}_s^{\prime }\left(y-a^{\prime }\right)$$E15

$$N=\psi R_b\mathit{bh}+{\sigma }_s{A}_s+{\sigma }_s^{\prime }{A}_s^{\prime }$$

The stress values in the reinforcements are determined by the values of ${\epsilon }_s$и ${\epsilon }_s^{\prime }$ which were found from the strain diagram depending on ${\epsilon }_{b0}$.

When $y=\infty$, all the fibers of the section experience the same deformations ${\epsilon }_{b0}=0,002$ and the stresses will be equal to:

$${\sigma }_s={\sigma }_s^{\prime }={\sigma }_s=R_s\left[\left(1-\frac{{\epsilon }_{b0}-{\epsilon }_{\mathit{sl}}}{{\epsilon }_{s2}-{\epsilon }_{\mathit{sl}}}\right)+\frac{R_{\mathit{su}}}{R_s}\frac{{\epsilon }_{b0}-{\epsilon }_{\mathit{sl}}}{{\epsilon }_{s2}-{\epsilon }_{\mathit{sl}}}\right]$$

In this case, the strength conditions will take the form:

$$M-{\overline{\gamma }}_{\mathit{bl}}\left(h_0-0,5h\right)R_b\mathit{bh}-A_s^{\prime }{\sigma }_s^{\prime }\left(h_0-a^{\prime }\right)=0$$E16

$$N={\overline{\gamma }}_{\mathit{bl}}{R}_b\mathit{bh}+\left(A_s+A_s^{\prime }\right){\sigma }_s^{\prime }$$

At $y=\infty$;

$$A_s^{\prime }=\frac{M-\left(h_0-0,5h\right){\overline{\gamma }}_{\mathit{bl}}{R}_b\mathit{bh}}{{\overline{{\sigma }^{\prime }}}_s\left(h_0-a^{\prime }\right)};A_s=\frac{N-{\overline{\gamma }}_{\mathit{bl}}{R}_b\mathit{bh}}{{\overline{\sigma }}_s}-A_s^{\prime }$$

From the condition of minimum of the sum $A_s^{\prime }+A_s$is found that at $A_s^{\prime }>A_s$$A_s=0,$and the value of $\psi$ has the form:

$$\psi =\frac{1}{0,8571-\frac{a^{\prime }}{h}}\left[0,204\left(1+0,75{\gamma }_{\mathit{bl}}\right)+\frac{N\left(h_0-a^{\prime }\right)-M}{R_b{\mathit{bh}}^2}\right]$$E17

and the rest of the parameters:

$$y=\frac{h}{7}\left(3+\frac{1,7457}{\sqrt{{\overline{\gamma }}_{\mathit{bl}}-\psi }}\right);$$E18

$$1000{\epsilon }_s^{\prime }=2+\left(3,437-8,019\frac{a^{\prime }}{h}\right)\sqrt{{\overline{\gamma }}_{\mathit{bl}}-\psi };$$

$${\epsilon }_s^{\prime }=\frac{y-a^{\prime }}{y-\frac{3}{7}h}\cdot {\epsilon }_{b0};$$E19

Depending on the ${\epsilon }_s^{\prime }$ is found ${\sigma }_s^{\prime },$ and then $A_s=0$;

$$A_s^{\prime }=\frac{N-\psi R_b\mathit{bh}}{{\overline{{\sigma }^{\prime }}}_s}$$

5. Calculation according to the second group of limit states

Reinforced concrete structures must also meet the requirements of the calculation for the second group of limit states, that is, suitability for normal use.

The second group of limit states according to BAEL-85 is presented in the form of the following phenomena, for which calculations are carried out:

crack formation design in the compressed zone of concrete, caused in the direction of normal stresses ${\sigma }_{b,\mathit{max}}$;

• design for the opening of cracks in the tensed zone of concrete caused by tensile stress in concrete;

• deformation design.

The design for the second group of limit states is based on the well-known method of allowable stresses, which makes it possible to determine the stresses in reinforced concrete elements using the formulas of resistance of materials:

1. The design for the formation of cracks in the compressed zone of concrete in the direction of normal stress ${\sigma }_{b,\mathit{max}}$ is carried out according to the following formula:

   $${\sigma }_{b,\mathit{max}}\le 0,6R$$

   Where R-cubic strength of concrete at axial compression is equal to the concrete class B at compression in MPa.

   ${\sigma }_{b,\mathit{max}}$—the maximum stress in the compressed zone of concrete and is determined by the following formula of resistance of materials:

   $${\sigma }_{b,\mathit{max}}=\frac{M_x\bullet y_1}{J_{x,\mathit{red}}}$$

2. Design for the opening of cracks in the tensed zone of concrete, depending on the category of crack resistance and $\eta$ coefficient of adhesion of reinforcement with concrete in reinforced concrete elements:

   • opening of cracks are not harmful;

     In this case, the value of the ultimate stress in tensed reinforcement is not limited and is taken equal to $\overline{{\sigma }_s}=R_s$.

   • opening of cracks are dangerous.

     In this case, the values of ultimate stresses in tensed reinforcement are limited by taking them equal to one of the lower values of the following condition:

     $$\overline{{\sigma }_s}=\left(\frac{2}{3}{R}_s\mathit{\text{or}}150\eta \right)\mathit{MPa}$$

   • crack opening is very dangerous.

     In this case, the values of ultimate stresses in tensed reinforcement are limited by taking them equal to one of the lower values of the following condition:

     $$\overline{{\sigma }_s}=\left(\frac{1}{2}{R}_s\mathit{\text{or}}150\eta \right)\mathit{MPa}$$

     The conditions of crack resistance of reinforced concrete elements for each of the above categories are ensured by choosing the ultimate stress $\overline{{\sigma }_s}$ in the design formulas of the second group of limit states when determining the area of tensed reinforcement, for example, for tensed reinforced concrete elements:

     $$A_s=\frac{N_x}{\overline{{\sigma }_s}}$$

     for bending elements with single reinforcement:

     $$A_s=\frac{M_x}{\mathit{\beta b}\overline{{\sigma }_s}}$$

     etc. depending on the category of crack resistance of reinforced concrete elements.

     Based on the aforesaid, the values of ultimate stresses in tensed reinforcement depending on the categories of crack resistance and the type of reinforcement surface η are given in Table 1.

3. Deformation design:

Rebar (reinforcement)	Classes	Ultimate stresses$\overline{{\sigma }_s},$MPa
		Cracks are not harmful, $R_s$	Cracks are dangerous	Cracks are very dangerous
Round with a smooth surface η=1	A 240; A 300	210; 260	140; 150	105; 110
Ribbed with a periodic profile η = 1,6	A 400; A 500	350; 435	240; 240	175; 175
Welded nets: smooth wire surface η = 1	B 500	435	150	110
Ribbed with a periodic surface of the wire η = 1,6	$B_p$500	435	240	180

Table 1.

The values of ultimate stresses in tensed reinforcement depending on the categories of crack resistance and the type of reinforcement surface η.

The condition for checking is:

$$f\le f_{\mathit{ult}}$$

here $f—$is defined as follows:

$$f=f_{\mathit{cd}}+f_{\mathit{ld}}$$

$f_{\mathit{cd}}$—deflection (initial) from a short-term load and is determined at $k_0=1$ and ${\gamma }_{\mathit{bl}}=1,$ i.e.

$$D_0=E_b{J}_{x,\mathit{red}}$$

$f_{\mathit{ld}}$—deflection (retarded) from long-acting loads, $k_0=0,2$ and ${\gamma }_{\mathit{bl}}=0,85,$that is,

$$D=K_b{E}_b{J}_{x,b}+k_s{E}_s{J}_{x,s}$$

The coefficients $K_b$and $K_s$ are determined by formula (11).

A detailed presentation of the design according to the second group of limit states is given in Ref. [8].

As a result of the development of a new method for designing reinforced concrete elements, the following conclusions can be drawn:

6. Conclusions

1. It has been established that the developed method for designing reinforced concrete elements, which is based on an accurate and rigorous application of the nonlinear deformation model of the mechanics of deformed solids to the problems of elastoplastic bending of the theory of reinforced concrete, on the basis of which the design of reinforced concrete structures will be carried out, fully meets the rules, principles, and requirements of the Eurocode.

   The essence of this technique lies in the fact that, under the strength conditions of reinforced concrete elements, expressed in terms of ultimate deformations of concrete of the compressed zone ${\epsilon }_{b,\mathit{ult}}$ and tensed reinforcement ${\epsilon }_{s,\mathit{ult}}$, and based on new concepts of limit states and the hypothesis of flat sections, they were reduced to compliance with new conditions written relative to the height of the compressed zone of concrete of bending elements, for example, at eccentric tension to compliance with the condition $0\le \xi \le {\xi }_{\mathit{\text{OR}}}$, at simple bending to ${\xi }_{\mathit{\text{OR}}}\le \xi \le {\xi }_R$, and at eccentric compression to the following three conditions:

   ${\xi }_{\mathit{\text{OR}}}\le \xi \le {\xi }_R$, ${\xi }_R\le \xi \le 1$, and $1\le \xi \le \infty$, where the height of the compressed zone $\xi$ is determined from the general condition $M_x\le M_{x,\mathit{ult}}$. Ultimately, the calculation (design) method has a practical and accessible form.

2. The developed method for designing reinforced concrete elements makes it possible to take into account the main characteristics of materials: the long-term strength of concrete and the increase in the strength of reinforcement (rebar) after the yield point in accordance with the state diagrams of concrete and reinforcement.

3. Comparison of the limit states for the strength of reinforced concrete elements based on a nonlinear deformation model and a plastic hinge model shows that when choosing the limit states, it is necessary to proceed from the deformation diagram obtained on the basis of the hypothesis of flat sections and state diagrams of concrete and reinforcement, which leads to an expansion of the range of tasks that can be solved established by Eurocodes.

4. Given the above, it is proposed to bring the current national standards SNiP 52.01-2003 and AzDTN 2.16-1 for reinforced concrete in accordance with the Eurocodes, by reworking them on the basis of the developed methodology for calculating (designing) reinforced concrete structures.