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Russian Journal of Building Construction and Architecture

BUILDING STRUCTURES,BUILDINGS AND CONSTRUCTIONS

DOI 10.25987/VSTU.2019.44.4.001

UDC 624.012.45

V. S. Fedorov 1, Vl. I. Kolchunov 2, A. A. Pokusaev 3, N. V. Naumov 4

CALCULATION MODELS OF DEFORMATION OF REINFORCED CONCRETE

CONSTRUCTIONS WITH SPATIAL CRACKS

Russian University of Transport (RUT) (MIIT)

Russia, Moscow 1, 3

Southwest State University (SWSU)

Russia, Kursk 2, 4

1 Academician of the RAACS, D. Sc. in Engineering, Prof.,

Head of the Department of Structures, Buildings and Facilities, e-mail: fvs_skzs@mail.ru

2 D. Sc. in Engineering, Prof. of the Department of Unique Building and Structures, e-mail: vlik52@mail.ru 3 PhD student of the Department of Structures, Buildings and Facilities

4 PhD student of the Department of Unique Building and Structures, e-mail: lich1992@hotmail.com

Statement of the problem. A model of deformation of reinforced concrete structures with spatial cracks is proposed.

Results. In the article working prerequisites are presented and the determining equations of the model of the resistance of reinforced concrete structures to torsion with bending are derived. Conclusions. Methods for calculating the resistance of reinforced concrete structures, as well as calculating the distance between spatial cracks and the width of crack opening under the joint action of the bending moment, torque and lateral force in the second stage of the stress-strain state are considered for two cases (case 1 –– when the spatial cracks of the first type appear on the lower edge of the reinforced concrete structure, case 2 –– when the spatial cracks of the first type appear on the lateral edge of the reinforced concrete structure).

Keywords: calculation method, torsion with bending, stress – strain, reinforced concrete structures, spatial crack.

Introduction. Resistance of spatial sections of ferroconcrete structures is regarded as complex if apart from bending moments and transverse forces, a structure is affected by rolling moments. They cause spiral cracking in ferroconcrete structures that form a spatial section inside three faces of an element along with a compressed area in the fourth face isolating it.

© Fedorov V. S., Kolchunov Vl. I., Pokusaev A. A., Naumov N. V., 2019

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Designing calculation models for complex resistance and bending rolling is gaining momentum [1, 2]. Firstly, there have been few studies on the topic [3––10]. Secondly, there is a growing need for analyzing spatial operation of a vast majority of ferroconcrete structures that are causing major transformations in modern cities [13––15]. Thirdly, it is known for a fact that there is nothing more practical than coming up with a solid theory for calculating them [16––23].

The current guidelines [12] for calculating operation of bending rolling suggest rather complicated and elaborate formulas. Despite the fact that these guidelines do not take into consideration the effect of a complex stress-strain on strain stress in compressed concrete Rb. Therefore it is not only the viability of the solution that is lost but its essential accuracy as well.

In the paper by А. S. Zalesov [8] a major step was taken towards identifying an equilibrium equation for the problem which was written for longitudinal and transverse planes, which makes calculation formulas considerably easier compared to those specified in the guidelines. Axial efforts in the transverse reinforcement in the side faces of an element are given consideration (the guidelines only specify efforts in the reinforcement near the face). However, the deformation equations of a spatial section used here and replacing it with a simplified diagonal one have not been proved in a number of experiments. Assuming longitudinal strains in the compressed zone of concrete to equal Rb has no substantial reason for a complex stressstrain of the zone while determining the projection of a dangerous spatial crack the condition of a minimum function of a lot of variables, etc. is not used.

In this paper the authors propose a calculation method which was not found to have any of the above disadvantages.

1. Solid considerations for a model of resistance of ferroconcrete structures to bending rolling. The following calculation considerations were made for the suggested method:

––a spatial crack on the lower face of a ferroconcrete element occurs perpendicularly to the main deformations of concrete lengthening and direction of the end of the spatial crack front near the compressed face of a ferroconcrete element coincides with that of main deformations of concrete shortening. Therefore a spatial crack has a spiral shape with three possible schemes of the compressed zone –– Fig. 1;

––a calculated scheme is the one consisting of a support block (formed by a spatial crack and a vertical section passing through the end of its front in compressed concrete) and the second block formed by a vertical section perpendicular to the longitudinal axis of the reinforced concrete element along the edge of the spatial crack –– Fig. 1;

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Russian Journal of Building Construction and Architecture

––calculated forces in the spatial are the following: normal and tangential forces in concrete of the compressed zone; components of axial forces in the reinforcement opposite to that where the compressed zone is located; components of axial forces in the transverse reinforcement located at the side faces of the reinforced concrete element;

— for medium fiber strains of compressed concrete and tensile reinforcement in section I––I, the hypothesis of their proportionality to the heights of the compressed and stretched section zones is considered fair;

––relationship between the strain intensity εі and the stress intensity σі of concrete.

For solving a direct engineering problem between external influences, their ratio is always set (Q : M : T). Therefore having determined one of them, for example, a support reactionRsup , it

is easy to identify the remaining influences, e.g, M and T.

Based on the equations conditions in section I––I and in the spatial section, the following calculated parameters are determined (Fig. 1): specific support reaction Rsup ; the height of the

compressed zone x in section I––I; stresses in the longitudinal reinforcement σs at the intersection of its spatial crack; the height of the compressed zone of concrete x in the vertical plane passing through the end of the front of the spatial crack; linear force in the transverse reinforcement located at the side faces of the spatial section qsw, Q generated by the transverse force; linear force in the transverse reinforcement located at the side faces of the spatial section qsw,T generated by the torque; linear force in the transverse reinforcement located at the lower edge of the spatial section qsw,σ generated by the torque.

Fig. 1. The design diagram of the resistance of the reinforced concrete structure under a combination of bending moment, torque and shear force (case 1):

is the compressed area of the spatial section; is the compressed area of the section I––I

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The shear stress Q and shear stress of torsion in compressed concrete Т are identified by projecting the σі – εі diagram onto the – plane (considering the distribution in the Q : T ratio) and onto the I––I plane and projecting the stress components of the k plane onto the plane perpendicular to the longitudinal axis of the reinforced concrete element.

In order to design the calculation equations, two blocks will be separated from the reinforced concrete element by means of the section method (Fig. 1). The first block is separated by a cross section I––I passing at the end of a spatial crack. This block is in equilibrium under the impact of external forces.

2. Determining equations of the model of resistance of reinforced concrete structures to torsion with bending. Based on the equilibrium equation of the moments of internal and external forces in this I––I section in relation to the z axis relative to the point of application of the resultant forces in the tensile reinforcement (∑MO,I = 0), we get:

Here y xb ,x is a static and geometric parameter that takes into account the location of the center of gravity of the compressed concrete zone in the section I––I (in the хb section, the diagram of compressive stresses is rectangular in the х – хb section is triangular); Rsup is the support reaction in the first block (Fig. 1), –– for the second group of limit states this parameter is known; а is the horizontal distance from the support to the section I––I. Based on this equation, the unknown σb,I .is determined.

It should be noted that for the limit states of the second group, the support reaction in the first block Rsup is a known parameter. We will further need a similar parameter Rsup,crc once the first spatial crack occurs which is known from solving the problem of the formation of spatial cracks. Based on the equilibrium equation of the projections of all the forces acting on the x axis in the sectionI––I,we identifythe heightofthecompressedzoneofconcretex inthissection(∑X =0):

into account the projection of the component stresses in the k plane onto the I––I plane perpendicular to the longitudinal axis of the reinforced concrete element; y,1 xb ,x is the parameter equal to a numerical coefficient to the parameter y xb ,x .

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