Cement-based composite materials have minimum of two components, one of which has higher strength compared to the other. Such materials include concrete, reinforced concrete (RC), and ferrocement, applied in single- or two-layer RC elements. This paper discusses experimental and theoretical results, obtained by the authors in the recent three decades. The authors have payed attention to a structural phenomenon that many design features (parameters, properties, etc.) at ultimate limit state (ULS) of a structure are twice higher (or lower) than at initial loading state. This phenomenon is evident at material properties, structures (or their elements), and static and/or dynamic structural response. The phenomenon is based on two ideas that were developed by first author: quasi-isotropic state of a structure at ULS and minimax principle. This phenomenon is supported by experimental and theoretical results, obtained for various structures, like beams, frames, spatial structures, and structural joints under static or/and dynamic loadings. This study provides valuable indicators for experiments’ planning and estimation of structural state. The phenomenon provides additional equation(s) for calculating parameters that are usually obtained experimentally and can lead to developing design concepts and RC theory, in which the number of empirical design coefficients will be minimal.

Behavior of structures in certain cases is similar to basic principles in human society. One of the main principles in human society is that all people are equal to each other. In other words, if, for example, there are two persons,

It will be demonstrated in this paper that a similar principle is an objective property of a structure, made of elastic-plastic or ideally elastic-plastic material in ultimate limit state (ULS), from the load bearing capacity viewpoint. This property is further called a “structural phenomenon.” The essence of this phenomenon is that values of some parameters of the structure that are measured experimentally under static or/and dynamic loadings increase or decrease by about a factor of two. Therefore, an alternative name of this phenomenon is doubling of structural design parameter at ULS.

A scheme of the principle for a general case is presented in Figure

Dominant technical frequency,

Another example that demonstrates increase in structural parameter is shown in Figure

Deflections in a fill scale RC shell versus damping ratios.

Earthquake type | Damping ratio, % | ||||||
---|---|---|---|---|---|---|---|

0 | 2 | 4 | 6 | 8 | 10 | 12 | |

Deflection, cm | |||||||

Low frequency | 2.50 | 2.25 | 2.21 | 2.15 | 2.09 | 2.05 | 2.00 |

Medium frequency | 3.63 | 2.38 | 2.23 | 2.17 | 2.10 | 2.06 | 2.00 |

High frequency | 4.40 | 3.32 | 2.81 | 2.63 | 2.41 | 2.30 | 2.19 |

24 × 24 m shell deflection versus damping ratio for various ground motions: 1: high-frequency earthquake, 2: medium-frequency earthquake, and 3: low-frequency earthquake.

Comparing Figures

Relation between structural phenomenon and minimax principle:

Figure

Structural phenomenon evidence for a fixed concrete beam: (a) static scheme, (b) bending moments diagram in elastic stage, and (c) bending moments diagram in ultimate limit state.

In the ULS after the moments’ redistribution,

In both cases,

The examples that were presented in this chapter are not the only ones. Other relevant cases will be discussed below. It should be mentioned that, as the structural phenomenon was not investigated previously, there is a very limited number of publications, in which relevant experimental data were obtained, but not analyzed. For example, experimental results [

Four-point static tests in ultimate stages show that steel fibers increase the deflections in the middle-span of a high strength concrete bending element approximately twice [

The ultimate equilibrium method, proposed by Gvozdev in 1938 [

Following modern design codes, methods, based on plastic analysis, are suitable for checking structures at ultimate limit state [

Later the ultimate equilibrium method was enlarged for application in structures that are made of concrete type materials with downloading branch in the stress-strain diagram [

Following Gvozdev [

static theorem, estimating the lower bound of the ultimate loading capacity of the structure,

kinematic theorem, estimating the upper bound of the ultimate loading capacity of the structure,

unity theorem that could directly obtain the ultimate loading capacity but does not allow finding the maximum static or minimum kinematic loads.

It should be mentioned that the required ductility of an RC structure or its element should be sufficient for the suggested failure mechanism.

Series of experiments were performed in the previous century in order to overcome the problem of the unity theorem and to find a more accurate value of the structural load bearing capacity [

Another possible way for exact estimation of RC structures’ load bearing capacity is simultaneous application of the static and kinematic methods at the ULS. However, this way is not practical, because it is impossible to estimate the convergence of results, obtained by these methods [

Development of this idea yielded to minimax principle formulation [

The authors have shown that the minimax term means minimum of maxima or the lowest board of upper bounds for a two-variable function of the structural bearing capacity [

Let us consider a rectangular RC element section that contains reinforcing

A quasi-isotropic state requires minimal total reinforcing of the section: that is,

Let us calculate, for example, an RC element with rectangular section of

Graphical representation of the three above functions is given in Figure

Minimization of total RC section reinforcement (

The present study is based on selected experimental and theoretical data that were reported by many researchers in the last three decades of the previous century and till today. As known, very many researchers have investigated behavior of structures under loads that increase from elastic state up to its failure. In this case if a structure is symmetric and the load is also symmetric, usually structural parameters in elastic state increase or decrease twice at failure. Therefore, we have presented and analyzed indeed those data.

It is logical to analyze the structural phenomenon from the following three different groups of experiments:

investigation of structural concrete at material level,

behavior of RC structures and elements under static loads,

response of RC structures and elements to dynamic loads.

Using high strength concrete in construction became very popular in recent decades. At the same time, the following fact is evident: in spite of the fact that concrete compressive strength increases with concrete class, the mean value of concrete tensile deformations,

As an extension of this idea, let us discuss the relation between the displacements and Poisson deformations under ultimate load and after unloading of two-layer beams consisting of normal and fibered high strength concrete in tensile and compression zones, respectively [

Following modern codes ([

Influence of concrete element deformations on its stiffness parameter.

As it was shown experimentally, increasing the load, acting on cylindrical concrete specimens with different steel fibers contents (from 0 to 60 kg/m^{3}), yields increase in Poisson coefficient, ^{3}. Following the experimental data, in specimens without fibers,

Concrete creep increases with time and correspondingly yields a decrease in the modulus of elasticity ([

Additionally, concrete creep depends on the compression stress in service limit state, surrounding environment humidity, composition and class of concrete, and so forth. Considering the concrete class only, the concrete creep depends on concrete creep coefficient

In the previous section examples of structural phenomenon for cement-based composite nonlinear material was discussed (concrete was considered as a private case of such material). The present section focused on behavior of RC elements from the viewpoint of statics. A rectangular RC bending element section with double reinforcement of

The reinforcement sections depend on

A sum of the bending moment, taken by the compressed reinforcement,

At the same time (

The phenomenon, related to doubling of various structural parameters, is also evident in problems of section design to shear forces [

Following modern codes [

The main parameters, affecting the section shear bearing capacity, are angles

A theoretical value of

Thus, the extreme value of

Structural phenomenon is also evident in dynamic behavior of buildings and proved by experimental data. For example, based on experimental results, obtained for a full-scale three-story beamless precast framed RC building part [

The influence of gravitation stresses on RC section energy dissipation under cyclic forces was examined [

As the section plastic energy dissipation decreases proportionally to the gravitation stresses, the ductility factor correspondingly decreases twice.

A six floor flat slab RC building with braced frame was analyzed [

Seven possible structural static schemes, running from fully braced to unbraced, were analyzed (Figure

Changes in the basic frame scheme under growing horizontal dynamic loading (following [

Following the results, obtained in the same research [

Additionally, a self-variable stiffness RC frame adapts its response to an earthquake by using the basic concrete properties [

As known, it is impossible to test real full-scale structures in ULS. In most cases, structural response in limit elastic state is tested. At the same time, it is very important to know structural dynamic parameters close to the ULS. For this reason, modern numerical techniques are applied. Experimental results for the limit elastic state are effective for verification of numerically obtained initial structural parameters. The authors have shown that experimentally obtained dominant vibration period for a full-scale 9-floor RC building was 0.72 s and the corresponding base shear force (BSF) was 3400 kN [

For adapting the building to a region with PGA = 0.3 g, a base isolation system (BIS) was used. The BIS was designed so that the BSF in the isolated building, subjected to an earthquake with PGA = 0.3 g, would be close to those in a fixed-base structure under an earthquake with PGA = 0.15 g [

Analysis of structural ductility enables us to prove the above-mentioned conclusion. As the input seismic energy that affects the building is independent of structural dynamic parameters, especially of ductility, hence according to Figure

Structural dynamic response,

The present study is focused on analysis and discussion of available experimental and theoretical data from the viewpoint of a structural phenomenon. It was shown that the phenomenon is valid for various design parameters at ultimate limit state (ULS) of a structure or its elements. It was demonstrated that the phenomenon is evident for material properties and structure (or its elements), as well as for structural static and/or dynamic response.

The phenomenon is based on quasi-isotropic state of a structure at ULS and minimax principle. It is supported by many experimental and theoretical results, obtained for different structures (beams, frames, spatial structures, and structural joints) under static or/and dynamic loadings.

The structural phenomenon enables us

to predict the ULS of the building or appropriate safety factor to this state,

to assess the limit changes of strength and deformation parameters in buildings before beginning their real design,

to solve strengthening problems of a building,

to carry out certification of a building (durability problem),

to find the limit values of steel fibers, confining effect, compressed reinforcement section in the element, and so forth,

to find the seismic resistance of a structure, that is, the level of structural load bearing capacity under a strong earthquake,

to reveal the stage, when the structural static scheme is changed.

The structural phenomenon, discussed and analyzed in the frame of the present study, can be also applied for other important issues in structural design. For example, one of logical suggestions for selecting an upper limit for a number of passive damping units in a structure is that maximum reduction in dynamic response of a building with effective supplemental passive devices is two times, compared to the original one (without dampers).

Thus, the results of this study provide valuable indicators for experiments planning, estimation of structural state (elastic, elastic-plastic, plastic, or failure), evaluating possibilities of retrofitting, and so forth. From the mathematical viewpoint, the phenomenon provides additional equation(s) that enable us to calculate parameters, usually obtained experimentally or using some empirical coefficients. Therefore, using this phenomenon can lead to developing proper design concepts and new RC theory, in which the number of empirical design coefficients will be minimal.

The authors declare that they have no competing interests.