1 Introduction

One of the main tasks of designing buildings and structures operated in earthquake zones is to ensure the ability to perceive impacts excluding the infliction of harm to health and lives of people. When assessing the risks they need to monitor a condition of structural systems from background influences and to assume the admissibility of structures injuries, sites, to preclude collapse of the building. Analysis of different cases of damage under seismic loading allows to identify the three most typical case of the destruction:

  • General loss of a building stability;

  • the destruction of the vertical frame elements;

  • the destruction of horizontal and vertical elements junction.

2 General Statements

From the energy point of view, the destruction of nodes or elements takes place due to a concentrated pulse energy impact of the external environment on the structure. At the Reinforced Concrete Structures Department of The Novosibirsk State University of Architecture and Civil Engineering (Sibstrin) the energy theory of reinforced concrete resistance was established (Mitasov and Adishchev 2010a, b). In the framework of this theory the problem of “smearing” of the pulse energy to the whole structural system, as well as the dissipation of energy through special events can be posed and solved.

With increasing loads in the bending a cross section of the reinforced concrete goes through several stages of the stress–strain state, which are qualitatively different from each other. Herein the so-called limit states differ, and calculation methods on two groups of limit states are based on contradictory hypotheses and assumptions. Each stage is described by the equations of equilibrium and the relevant hypotheses, and the change of cross-section parameters in the transition processes from one state to another are calculated by means of empirical coefficients. In fact, during a crack formation there is not only a “catastrophic” change in the stress–strain state in the cross section with a crack, but also an actual change in the structure itself. Calculation of the two groups of limit states virtually ignores the real physical properties of the concrete, and also additionally does not take into account the peculiarities of the stress–strain state and physically adequate conditions for the transition from one state to another. The transition from stage I (before cracking) to stage II (operational) is conditioned by the limit value of deformations of the extreme stretched fiber of the concrete. Conditions of the transition from the operational stage to the fracture stage (hypothetical) in the theory of reinforced concrete resistance are not determined. Thus, the “pass-through” calculation from the beginning of loading to the loss of bearing capacity of the reinforced concrete element is impossible in the frame of this theory.

For the evaluation of the bearing capacity, hardness, and crack resistance of the reinforced concrete element only the operational stage presents an interest, namely, the beginning of this stage (crack initiation) and the end (loss of load-carrying ability). Consequently, the problems of a physically adequate description of the stress–strain state of reinforced concrete structures at the main stages of loading, as well as the conditions of the transition from one stage to another come to the foreground. Stage III cannot be realized, and it is of interest to assess the nature of the fracture—plastic or brittle.

At the heart of the calculation method of limit states there are three stages of the stress state of the cross section, which are qualitatively different from each other. Stage I is characterized with the displacements (deflections) of the construction, stage II—the moment of cracking, stage II—displacements and crack width. Stage IIa is a pre-destruction stage at which the formation of a plastic hinge takes place. The calculation of the bearing capacity of the first group of limit states assumes implementation of a hypothetical stage III. Analyzing the second group of limit states, it should be noted that the applied methods resemble “allowable stresses” but instead of stresses the control values of deflections and crack width are used. The absence of conditions of transition from a “continuity’ state to a “section with a crack” state does not allow us to make a consistent calculation of strength, hardness, and crack resistance.

Taking into account the nonlinear deformation of concrete and discreteness of reinforcement of concrete out of the known integral calculation methods, suitable for the quite rigorous description of the deformation process of reinforced concrete structures, we find the method, based on the use of energy relations and criteria, to be the most acceptable. As the area of a subgraph of a diagram of the deformation of the material corresponds to the energy density (or specific energy) of deformation, using diagrams of deformation of materials in the construction of the computational model organically fits into the scope of the problem under consideration.

Since before cracking the qualitative state of cross sections is presented through stages Ia, and the stabilized condition—through stage II, to preserve the existing gradation, we introduce stage Ib—the stage of formation and stabilization of the cracks.

3 The Basic Hypotheses

Let us formulate the basic stages of changing the stress–strain state of stage Ib:

  1. (1)

    at the time of the cracking the energy of concrete in tension is redistributed to the reinforcement, which acting as a braking element, restrains crack propagation along the cross section of the element:

    $$\mathop \sum \limits_{j} A_{\text{sj}} \mathop \int \limits_{{\varepsilon_{\text{sj}} }}^{{\varepsilon_{\text{sj}} + \Delta \varepsilon_{\text{sj}}^{\text{st}} }} \sigma \left( \varepsilon \right){\text{d}}\varepsilon - \Delta W_{\text{cr}}^{\text{bt}} = 0,$$
    (14.1)

    where \(\Delta \varepsilon_{\text{sj}}^{\text{st}}\)—increment of deformation in the j-layer of reinforcement after the redistribution of energy of the concrete in tension \(\Delta W_{\text{cr}}^{\text{bt}}\) on the reinforcement;

  2. (2)

    the sudden nature of the crack appearance is accompanied by an instantaneous change in the stress state of the cross section, which is dynamic in nature; as there are no changes along the length of the reinforced concrete element (all redistribution occurs in the vicinity of one cross section), the voltage \(\sigma_{\text{s}}^{\text{d}}\) («dynamic» value) in the reinforcement at the maximum disclosure of a newly formed crack is determined from the condition of equality of the increment of the specific energy of deformation and work of stresses \(\sigma_{\text{s}}^{\text{d}}\) («static» value) on the total increment of deformations \(\varepsilon_{\text{s}}^{\text{d}} - \varepsilon_{\text{s}}^{*}\):

    $$\mathop \int \limits_{{\varepsilon_{\text{s}}^{*} }}^{{\varepsilon_{\text{s}}^{\text{d}} }} \sigma \left( \varepsilon \right){\text{d}}\varepsilon = \sigma_{\text{s}}^{\text{st}} \left[ {\varepsilon_{\text{s}}^{\text{d}} - \varepsilon_{\text{s}}^{*} } \right],$$
    (14.2)

    where \(\sigma_{\text{s}}^{*}\), \(\varepsilon_{\text{s}}^{*}\)—stresses and deformations in the reinforcement, respectively before cracking, “fictitious static” stresses and deformations \(\sigma_{\text{s}}^{*}\), \(\varepsilon_{\text{s}}^{\text{st}}\) after redistribution of the energy of the fractured zone of the concrete cross section are determined by the Eq. (14.1); “dynamic” stresses and deformations \(\sigma_{\text{s}}^{\text{d}}\), \(\varepsilon_{\text{s}}^{\text{d}}\) for each layer of the reinforcement are determined by the Eq. (14.2) and correspond to the maximum (unbalanced by external loads) crack opening Fig. 14.2;

  3. (3)

    at the moment of the realization of the “dynamic” stresses \(\sigma_{\text{sj}}^{\text{d}}\) in the reinforcing layers the statistical equilibrium in a section with cracks is absent; then the stresses in the reinforced metal decrease, the process of vibrations is of a fading character, stresses in the reinforced metal are stabilized and become equal to \(\sigma_{\text{sj}}^{\text{cr}} < \sigma_{\text{sj}}^{\text{d}}\). This condition is characterized by a stable equilibrium. In the general case for the two-valued stress distribution in the cross section the equilibrium equation after the crack stabilization the can be written as follows:

    $$\mathop \int \limits_{ - x}^{{h_{\text{cr}} - x}} \sigma \left( y \right)b\left( y \right){\text{d}}y - \mathop \sum \limits_{j} \sigma_{\text{sj}}^{\text{cr}} A_{\text{sj}} = \pm N,$$
    (14.3)

    where hcr—the distance from the most compressed fiber to the tip of the crack; by—a variable width of the cross section; N—the resultant of all the external longitudinal forces (the sign is taken according to the nature of the impact: compression-tension).

The scheme of changing stresses in the reinforcement after the crack formation and its stabilization is shown in Figs. 14.1 and 14.2.

Fig. 14.1
figure 1

The scheme of changes of stresses and deformations in the reinforcement after cracking a—The specific energy of deformation of concrete in tension before cracking; b—The scheme of determination of stresses in the reinforcement in a cross section with a crack

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Fig. 14.2
figure 2

The scheme determining the maximum stress in the reinforcement at the time of the formation of cracks

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4 Deformation of Reinforced Concrete Beam with a Crack

Stage II occurs after the cracks formation and their stabilization in the tension zone. It is characterized by cracks growth along the height of the cross section and the formation of new ones with increasing of the external load.

The study of this stage is required for the determination of displacements, crack width and a possibility of their closure at lower loads.

Stage IIa is characterized by the formation of a “plastic hinge” in a cross section. The development of this stage can occur in two scenarios.

  1. (a)

    The stresses in the reinforcement have reached the yield point (physical or conditional). The determination of stress–strain state is estimated according to the scheme of Stage Ib with the replacement of the unknown quantity, instead of a certain moment of the external forces the limit stress value in the reinforcement is given. For a closed solution in this case one equilibrium equation is enough (14.1). By (14.2) the load-carrying capacity of the cross section is determined;

  2. (b)

    The compressive zone of the concrete has entered the stage of pre-fracture. This state should be evaluated in accordance with the well-known postulate: the specific energy in a pre-fracture state does not depend on the method of the load application and the deformation rate, i.e.,

$$\mathop \int \limits_{0}^{{\varepsilon_{\text{b}}^{*} }} \sigma_{\text{b}} \left( \varepsilon \right){\text{d}}\varepsilon = {\text{const}}.$$
(14.4)

Studying this stage allows us to describe the pre-fracture state from energy positions.

Let us consider the process of cracks formation and their stabilization.

The moment of cracking corresponds to achieving the critical value by the specific energy of deformation of concrete in tension:

$$\Delta W_{\text{cr}}^{\text{bt}} = \mathop \int \limits_{0}^{h - x} b\left( y \right)\left[ {\mathop \int \limits_{0}^{\varepsilon \left( y \right)} \psi \left( \varepsilon \right){\text{d}}\varepsilon } \right]{\text{d}}y,$$
(14.5)

where ε(y)—fiber deformation, located at a distance y from the neutral layer of the element.

After crack formation in the cross section there is a redistribution of forces. The energy of concrete in tension (14.5) is transmitted to the reinforcement, a crack “grows” in width and height, stresses in the reinforcement in the crack increase. The instantaneous nature of the cracking process generates the dynamics of the process of the redistribution forces in the cross section—on the tip of a dynamically grown crack there are the highest deformations of concrete in tension \(\varepsilon_{\text{bt}}^{*}\). Then under the action of elastic forces in the reinforcement a crack is partially closed, its height decreases, the oscillations are faded. This state corresponds to the balancing of the work of the external load on the appropriate movements by the work of the internal forces.

The specific energy of deformation of the concrete in tension before the crack formation at the two-valued stress distribution in the cross section is represented by the expression (14.5). In accordance with the accepted hypotheses the dependence of any fiber deformations in the cross section from the vertical coordinate is as follows:

$$\varepsilon \left( y \right) = \left( {\frac{{\varepsilon_{\text{bt}} }}{h - x}} \right)y.$$
(14.6)

Then the energy of concrete in tension is determined from the formula

$$\Delta W_{\text{cr}}^{\text{bt}} = \mathop \int \limits_{0}^{h - x} b\left( y \right)\mathop \int \limits_{0}^{{\left( {\varepsilon_{\text{bt}} /h - x} \right)y}} \psi \left( \varepsilon \right){\text{d}}\varepsilon {\text{d}}y.$$
(14.7)

After the crack formation there is an unloading of the concrete in tension and energy \(\Delta W_{\text{cr}}^{\text{bt}}\) passes completely into the energy of the tensile reinforcement. When this happens, the stresses in the reinforcement have increased to a value \(\sigma_{\text{s}}^{*}\), a and the corresponding increase in the energy of reinforcement deformation is equal to \(\sigma_{\text{s}}^{*}\):

$$\Delta W_{\text{s}} = \mathop \sum \limits_{j} A_{\text{sj}} \mathop \int \limits_{{\varepsilon_{\text{sj}}^{*} }}^{{\varepsilon_{\text{sj}}^{\text{st}} }} \sigma_{\text{s}} \left( \varepsilon \right){\text{d}}\varepsilon = \mathop \sum \limits_{j} A_{\text{sj}} \mathop \int \limits_{{\sigma_{\text{sj}}^{*} }}^{{\sigma_{\text{sj}}^{\text{st}} }} \sigma \varphi^{{\prime }} \left( \sigma \right){\text{d}}\sigma ,$$
(14.8)

where \(\varepsilon_{\text{s}} = \varphi \left( {\sigma_{\text{s}} } \right)\)—the analytical expression of the diagram of reinforcement deformation.

Then \(\sigma_{\text{sj}}^{\text{st}}\) can be found from the equation

$$\Delta W_{\text{cr}}^{\text{bt}} = \mathop \sum \limits_{j} A_{\text{sj}} \mathop \int \limits_{{\sigma_{\text{s}}^{*} j}}^{{\sigma_{\text{sj}}^{\text{st}} }} \sigma \varphi^{{\prime }} \left( \sigma \right){\text{d}}\sigma .$$
(14.9)

Cracking occurs so quickly that a change of stress in the reinforcement can be considered as instantaneous. Stresses in the reinforcement thus increase to some dynamic value \(\sigma_{\text{s}}^{\text{d}} > \sigma_{\text{s}}^{\text{st}}\). As already noted, this quantity is calculated from the Eq. (14.1), which takes the following form:

$$\mathop \int \limits_{{\sigma_{\text{sj}}^{*} }}^{{\sigma_{\text{sj}}^{\text{d}} }} \sigma \varphi^{{\prime }} \left( \sigma \right){\text{d}}\sigma = \sigma_{\text{sj}}^{\text{st}} \left[ {\varphi \left( {\sigma_{\text{sj}}^{\text{d}} } \right) - \varphi \left( {\sigma_{\text{sj}}^{*} } \right)} \right].$$
(14.10)

The most “dynamic” deformation of the extreme fiber from the compressive zone of the concrete \(\varepsilon_{\text{b}}^{\text{d}}\) is determined from the equation

$$\frac{{x_{\text{cr}} }}{{\varepsilon_{\text{b}}^{\text{d}} }}\mathop \int \limits_{{\varepsilon_{\text{b}}^{\text{d}} }}^{{\varepsilon_{\text{bt}}^{*} }} \psi \left( \varepsilon \right)b\left( {\frac{{x_{\text{cr}} }}{{\varepsilon_{\text{b}}^{\text{d}} }}\varepsilon } \right){\text{d}}\varepsilon = \mathop \sum \limits_{j} \sigma_{\text{sj}}^{\text{d}} A_{\text{sj}}$$
(14.11)

After reaching the highest values of stresses in reinforcement and concrete the unloading take place. The unloading for the concrete can be considered in accordance with the recommendations of the work (Zhunusov and Bespaev 1972), for the reinforcement—with a linear law

$$\varepsilon_{\text{sj}}^{\text{cr}} = \varepsilon_{\text{sj}}^{\text{d}} - \frac{{\left( {\sigma_{\text{sj}}^{\text{d}} - \sigma_{\text{sj}}^{\text{st}} } \right)}}{{E_{\text{s}} }}$$
(14.12)

Thus, the energy relations to solve one of the problems of the theory of resistance of reinforced concrete have been obtained—the problems of determining the stresses in the reinforcement in the cross section with a crack.

Stage III, as noted above, cannot be used in the calculation method and can serve only to distinguish between fracture scenarios—quasi-plastic and brittle.

We obtained fundamental solutions to the problem of transition from continuity state to the “cross-section with a crack” state that will allow us to create a uniform calculation method of strength, hardness, and crack resistance.

The fundamental solution of the problem of transition from the state continuity to the state “section with a crack” was obtained. That will allow creating a uniform method for calculating strength, stiffness, and fracture toughness. Analysis of Eqs. (14.1)–(14.12) allowed us to outline a series of constructive events (Fig. 14.3), which will allow to prevent the collapse of buildings and structures by increasing the stability of vertical elements, improving resistance to the efforts of extension and shear, as well as due to energy dissipation through pre-organized cracks (special plates).

Fig. 14.3
figure 3

The proposed structural measures to improve the seismic resistance

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5 Constructive Measures

If the schemas of increasing stability by pre-stressing (Fig. 14.3a) are well known and used in the construction of high-rise buildings and earthquake-resistant constructions (Zhunusov and Bespaev 1972; Zhou and Jiang 1985; Saatcioglu and Yalcin 2003; Moghaddam and Samadi 2009; Dong et al. 2006), the other two (Fig. 14.3b, c) were developed at our University (RF patent No. 67603, No. 112693).

The authors conducted theoretical and experimental studies, the results of which prove that stochastic cracking (the greatest damage is to the first crack) generate shock waves that spread ahead of them, preparing the ground for the destruction of construction, and low stretch of concrete at the impact on compressed elements leads to fault them on the inclined sections.

  1. (a)

    the increasing resistance of vertical elements;

  2. (b)

    the increasing resistance to the efforts of extension and shear;

  3. (c)

    the increase of hardness and fracture toughness by controlling crack formation

1—plates

Figure 14.4 shows diagrams and photographs of the physical experiments in the studies that were conducted by the authors at the Department of reinforced concrete structures (Logunova and Peshkov 2011; Panteleev et al. 2012; Mitasov and Logunova 2011). There were studied the work of compression elements with the interior forming frame to a vertical impact and shear with bending.

Fig. 14.4
figure 4

Experimental study of constructive solutions to improve seismic resistance

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  1. (a)

    the junction of the columns with the internal forming frame and slab;

  2. (b)

    test compressed elements with various types of internal cages;

  3. (c)

    tests of columns with inner cage on shear and bending;

  4. (d)

    tests of beams with pre-organized cracks and graphs of deflections.

Experiments conducted by the authors (Fig. 14.4) allow to draw the following observations:

  1. (1)

    post-tensioning of compressed elements ensures their stability and together with the internal forming frame will facilitate the work of plate-column junction.

  2. (2)

    The internal forming frame almost eliminates the shear of compressed element on the inclined section.

  3. (3)

    Pre-organized cracks retains the reinforced concrete element not only before but also after a shock, dissipating part of the energy of the seismic impact.

6 Conclusions

The article presents main provisions of energy theory of reinforced concrete resistance and methods to improve the earthquake resistance of structures. The presented information allows to draw the following conclusions:

  • The problem of a reinforced concrete section transition from “”solid” state to state “with a crack” in a stretched zone was solved.

  • The energy theory of reinforced concrete resistance allows to provide a uniform methodology for calculating strength, stiffness and fracture toughness of structures.

  • The use of pre-stressing with internal forming frame in compressed elements and a pre-organized cracks in bending elements allows to increase shear resistance of columns, their stability and to damp part of seismic energy.