Rationalization of the parameters of the cylindrical bridge support (theoretical basis)

. Paper deals with a new constructive solution of a massive concrete pillar intended, mostly, for construction of railway and road bridge supports and with procedure of rationalization of its parameters. Statement and solution of the problem of strength and stability of a concrete support, which has an external mesh cage, reinforcing rings (frames) and an internal cavity is considered. The contact stresses between the concrete core and the metal cage are determined, which, in turn, depend on the attributes of the cage and the frames. The rationalization criterion based on minimizing the potential energy of deformation is introduced. For the indicated task, the variables are the thickness of the mesh cage, the size of its cell, the step and the cross-section of the frames. Support with rational parameters which depend on characteristics of cage and frames gives the lowest material consumption and the highest bearing capacity that is the aim of the approach of investigation.


Introduction
Developing the idea of a concrete support with external reinforcement, presented in [1], this paper proposes the detail approach to the rationalization of its parameters. The effectiveness of external reinforcement made of expanded metal sheet can be extremely increased in case of using the additional transverse steel rings. One can see the interest in the task of finding the rational dependency between parameters of the mesh cage and reinforcing rings. However, together with mentioned above, the task of this work is to establish the bond between the value of the potential energy of deformations (PED) and characteristics of external reinforcement (the cage and the frames at one time) and, using the energy approach [2], to find out the decision which gives minimum value of PED. excluded. In turn, mesh сage is supported by transverse steel rings (frames), set with ac step (Fig. 1).
For high massive supports, the cross-sectional area A(z) should be formulated in accordance with the expression [3]: where d0 -diameter of the internal cavity at the origin of the coordinates of the cylindrical system (Fig. 2); D -external diameter of the support; c -specific weight of concrete; fc -compression tensile strength of concrete.
To the ends of the element (concrete part), axially applied constant stresses of q intensity. Equilibrium of these stresses is equal to F.

Strength Problem
Let us denote by p yet unknown contact voltages emerging between the mesh cage and the concrete core. Here it is assumed that radial deformations of the core and circular mesh cage are joint throughout the whole load range (life-cycle).
The transverse cross sections of the considered element (core + cage) under the designated central compression will have the following stress strain state (Fig. 3): -the core -axially symmetric transverse compression; -mesh cage -axially symmetric radial tension. Let's set the following designations: R -external radius of a concrete core; D -internal radius of a concrete core; r -current radius (coordinate of cylindrical system); Ec, Gc -deformation modules of the 1st and 2nd kind of concrete core, 2(1 ) Fig. 3. Cross section of the core and the cage.

Concrete core
To determine the radial deformations of the concrete core, we use the solution of the Lame problem [4]. Radial displacement of the core under compression: At the point of contact between the core and the cage r = R. Deformation of the concrete in the transverse direction: At the point of contact between the core and the cage r = R and 2 2 Expansіon of the core under longitudinal forces in the case of axial symmetry: In case of simultaneous action of q and p (solid cylinder) [5]: Considering the direction of compression forces In case of an empty cylinder, after transformations, and considering the results [6]: For verification when d=0 we have: And assuming the validity of 2 which gives a complete coincidence with [5].

Steel Mesh Cage
Let's assume that the size of the cells of mesh cage are small in comparison with radius and height of the support. Then, on the basis of [7], the mesh cage can be roughly replaced by a continuous orthotropic shell. Introducing further a three-orthogonal coordinate system, in the spirit of cited work, we will have: In expressions of stiffness (12) for spiral and circular ribs under tension-compression and shear conditions it is indicated (Fig. 4): φ is the angle of the inclination of the spiral ribs to the creature; h -thickness of the mesh structure (height of ribs); δh, δc -the width of the spiral and circular ribs, respectively; ah, aс -the distance between spiral and circular ribs; Es, Gs -elastic and shear modules of ribs (steel); nh -number of pairs of spiral ribs; D is the diameter of the shell.
where a is the distance between the points of intersection of the symmetrical spiral ribs over the circumference of the cross-section of the shell. Equating the radial deformation of the concrete core to the annular deformation of steel cage, we get: Let's set the designation: Futher, considering the relationship between the contact tensions p and the longitudinal (given) stresses q, written for the isotropic shell, we obtain [5]: where c s E n E  -reduction factor; red h -reduced thickness of the mesh cage.
Having a sort of cut sheets, it is possible on the basis of (20) to determine the contact tension between the cage and the core, depending on the main attributes of the sheet and the step of the steel rings (frames).

Stress State of the Core
To estimate the stress state of the core we define the main stresses, considering that the components of the stress tensor, in this case, are the following [

Сonclusions
Listed, in aggregate, predetermines the following conclusion: