## 1 Introduction  
Poisson's deformations are known in the literature as a material expansion or contraction in a direction perpendicular to the external loading. Typical stone materials have Poisson coefficients between 0.1 and 0.3 [1]; for example, for concrete it is 0.15... 0.2 and for steel it is 0.27... 0.3. It should be mentioned that these values are related to the elastic stage of the behavior of materials under pure compression or tension.  
At non-elastic strains, the ratio between the transverse and axial strains becomes inconstant; therefore, it cannot be described by the Poisson coefficient. In such cases, the Poisson effect is a function of real non-linear strain and the Poisson coefficient is replaced by the Poisson function (Poisson ratio) [2]. For metals, the transfer from a Poisson coefficient to a Poisson function is less important as their non-linear behavior is due to the yielding of the material. However, for stone-type materials such as concrete, this transfer is very important as it begins at stress values that are about half of the ultimate one, which is when elastic-plastic deformation development begins [3, 4].  
The many ways of defining the Poisson ratio are known from the literature, but little is known regarding its accuracy [5]. The initial values of the Poisson function at the beginning of the concrete elastic-plastic stage are equal to the Poisson coefficient, but the Poisson function values at higher strains are not constant. This factor is significant and should be considered in designs.  
Following the available data, the Poisson effect before shear cracking is characterized by the well-known Poisson coefficient, but after cracking it is characterized by two ratiosthat were studied in [6]. Experiments were performed to measure these ratios to provide a basis to develop a corresponding model for predicting shear stress vs. shear strain curves, including post-peak descending branches.  
Many researchers in the fields of concrete and polymer materials have extended the definition of the Poisson function to linear viscoelasticity [7]. The Poisson function can increase, decrease or have a non-monotonic function of time. Theoretical and practical examples have been provided to highlight that any evolution of concrete viscoelastic Poisson functions is possible.  
The available methods for an ultimate limit state analysis of reinforced concrete structures or structural elements use the modelling of the material characteristics such as strain, stress and the strain rate [8]. Such modelling is based on theoretical concepts, including plastic fracturing [9], monotonic and cyclic behavior under multiaxial stress conditions [10] and viscoplasticity [11].  
It was observed that the Poisson ratio of concrete lies in the range of 0.15 to 0.22 when determined from strain measurements under a compressive load [12]. A similar range of the Poisson ratio was obtained under a tensile load. Although the Poisson ratio for the pure compression or tension of concrete has been investigated by many researchers, the problem has not been practically studied for bending elements. The authors performed a long-term experimental program on high-performance two-layer beams (TLB). In the frame of this program, we tested small-scale (0.7 m long) single supported TLBs [13], full-scale (3 m long) simple supported beams [14; 15], continuous (4 m long) two-span TLBs (each span was 2 m) [16] and pre-stressed 3 m long TLBs [17; 18].  
In this paper, the available experimental data for real reinforced concrete (RC) single- and two-layer beams were analyzed in order to investigate the transverse deformations, Poisson ratio and corresponding function for elastic-plastic deformations in the compressed zone. The analysis was performed using a recently developed theoretical model of RC tensile behavior for different stress-strain conditions [19] and structural phenomena [4].