Reinforced Concrete Columns with Local Prestressing Rebars: A Calculation Theory and an Experimental Study

Abstract

Local prestressing of reinforcement can be effective for slender reinforced concrete columns with large longitudinal force eccentricities. This article deals with columns with prestressed reinforcement on the side opposite to the eccentricity of the longitudinal force. Prestressing is created with the help of turnbuckles. The aim of the work is to develop a model for determining the stress–strain state of columns with local prestress and its experimental verification. The article presents the derivation of a resolving equation for the increment of deflection, which considers the non-linearity of the concrete and reinforcement work, the presence of creep and shrinkage of concrete. The solution of the resulting equation was performed numerically by the finite difference method in a MATLAB environment. Experimental studies were carried out according to the hinged support scheme for eight eccentrically compressed samples, four of which had been prestressed. Experiments and numerical modeling of columns with local prestressing showed a significant increase in crack resistance (by 1.3–2.5 times) and bearing capacity (by 12.5–30%) compared to similar structures without prestressing.

1. Introduction

Prestressed concrete columns are not as widely used as prestressed beams and bending slabs. However, with a sufficiently high slenderness of the elements, the prestressing of the reinforcement becomes effective due to an increase in the resistance of the columns to buckling [1]. Moreover, prestressed columns are highly efficient under the combined action of longitudinal forces and large bending moments, which can occur with wind, seismic loads, etc.

Standard methods of manufacturing prestressed reinforced concrete structures involve pulling prestressing reinforcement over the entire length of the element. This method causes negative effects at the end sections—non-closing cracks appear on the upper faces of the elements; local fragmentation of concrete occurs because of the action of concentrated forces at the ends of the structure [2]. In addition, with a bending moment variable along the length of the element, the reinforcement and the level of prestress determined by the maximum force for the end lightly loaded sections are redundant.

To eliminate these shortcomings and to improve the cost indicators of columns made of prestressed reinforced concrete, it is advisable to apply prestressing only in those areas where it will lead to an increase in the characteristics of structures. In lightly loaded areas, it makes no sense to create a prestress and use high-class reinforcement.

For short columns subjected to eccentric compression, the bending moment is constant along the length, and the creation of a local prestress is impractical for them [3,4]. For slender columns, due to the additional eccentricity caused by the deflection of the element, the bending moment along the length changes. Therefore, with an increase in slenderness, the efficiency of creating local prestress in areas with maximum bending moments increases.

One of the first publications on reinforced concrete elements with local prestressing of reinforcement is the patent of A.L. Shagin and M. Rifai [5]. The essence of the proposed method is the formation of local areas around the prestressing reinforcement during concreting, which are not filled with concrete mix. The reinforcement is tensioned by applying a transverse force (Figure 1). After tensioning the reinforcement, the hollows are concreted. To avoid chipping of the concrete cover, this method requires the installation of anchor brackets along the length of the anchoring zone.

In [6], a theoretical and experimental study of reinforced concrete beams with local prestressing is carried out according to the described technology. Four reinforced concrete beams were tested, and a new theoretical approach was proposed to determine the cracking moment of the tested beams, since existing design standards are not applicable to this method of creating prestress. The method mentioned above of creating prestress is also effective for local reinforcement of concrete and steel structures [7,8,9,10].

The disadvantage of this method of creating local prestress is the need to increase the protective layer outside the prestressed zone of the reinforcement and its forced reduction within the prestressed zone. In addition, the described method is not applicable in prestressed columns since it excludes the installation of transverse reinforcement.

In [11], the technology for creating local prestress in beams was modified. As in the patent of A.L. Shagin, sections of the reinforcement, which are subsequently subjected to prestressing, remain open. However, their tension is carried out not with a transverse load, but by means of tension couplings. This method allows for creating not only tensile but also compressive stresses in the reinforcement. Some ideas on the use of this method in columns are presented in [12]; but, so far, they have not received practical implementation.

Current research on the creation of local prestress in reinforced concrete structures is also aimed at introducing new intelligent materials. Article [13] is devoted to the introduction and research of a new prestressing system for railway reinforced concrete sleepers using shape memory alloys. Reinforcing bars made of shape memory alloys are also used in [14,15] to create local prestress in reinforced concrete slabs. In articles [16,17], an experimental study and finite element modeling in the ABAQUS and ANSYS software of reinforced concrete beams reinforced with prestressed iron-based shape memory alloy r is carried out. There are also a significant number of publications on the creation of reinforced concrete elements with local prestressing using composite materials, including carbon fiber reinforced plastics as well as hybrid composites (hybrid fiber reinforced polymer) [18,19,20,21,22,23].

However, the mass use of shape memory alloys and composite materials in prestressed concrete structures is still constrained due to the high cost compared to steel reinforcement.

In this paper, eccentrically compressed columns with pre-tensioned reinforcement are considered. The manufacturing technology of these structures is as follows. During the concreting process, the grooves are leaved in the areas that will be further subjected to pretension (Figure 2). With a constant reinforcement class along the length of the column, the reinforcing bars of the tension zone comprise two sections connected by couplings. In couplings, a right-hand thread is cut in half of the length and a left-hand thread in the other half. The corresponding thread is cut at the ends of the joined reinforcing bars. Depending on the direction of rotation of the coupling, the ends of the rods approach or move away and thus pre-tension or compression stresses are created.

After creating pre-stresses, the grooves are concreted with fine-grained concrete. It is desirable to use concrete with increased tensile strength compared to the base concrete [24,25].

This technology can be applied to slender columns if the direction of the bending moment does not change and the position of the face stretched from the external load is known in advance. If necessary, high-strength reinforcement can be replaced with conventional reinforcement at the end sections by adding one more coupling for each rod.

The main aim of this work is to develop methods for calculating the stress–strain state of columns with local prestress, manufactured according to the described technology, at the stage of manufacture and operation under load, as well as their experimental verification.

2. Materials and Methods

2.1. Deformation Model of Columns with Local Prestress at the Manufacturing Stage

When obtaining the resolving equations, the following hypotheses are accepted:

• The plane section hypothesis;
• The absence in the pre-stressing stage of the influence of pre-stressing reinforcement on the stress–strain state of the support zones ($0\le x\le \frac{l-l_0}{2}$ and $\frac{\left(l+l_0\right)}{2}\le x\le l$, here l₀ is the length of the local prestress zone, l is the length of the column);
• The stress concentration at the junction of the stressed and the non-stressed zone of the column, associated with a change in the dimensions of the cross section, is neglected;
• The joint of the hollows concrete with the main concrete in the stage of work under load is considered ideal.

Hypothesis 1 is a classical hypothesis that is used to calculate Euler–Bernoulli bars, without taking into account transverse shear deformations [26,27]. Hypotheses 2 and 3 are related to the fact that it is possible to consider the neglected factors only by considering the problem in a three-dimensional formulation, using the FEM complexes. The establishing of these hypotheses makes it possible to solve the problem in a one-dimensional formulation (the function that completely determines the stress–strain state is the deflection v, which depends only on x). Hypothesis 2 also assumes that at the stage of creating prestress outside the bare zone, the stresses in the prestressed reinforcement immediately drop from ${\sigma }_{sp}$ to zero (in fact, they drop to zero along the length of the anchoring zone).

Considering the hypotheses introduced in the first stage, the bending moment in the stressed zone is constant, and there are no stresses in concrete and reinforcement in the support sections. The design scheme for the first stage is shown in Figure 3.

When deriving equations, to consider the nonlinear work of concrete, we assume that its modulus of elasticity changes along the height of the cross-section. We consider tensile stresses to be positive for concrete and reinforcement, which is shown in Figure 4.

The increment of concrete deformation at all stages, considering the hypothesis of plane sections, is written as:

$$\mathsf{\Delta }{\epsilon }_b=\mathsf{\Delta }{\epsilon }_0-y\mathsf{\Delta }\chi ,$$

(1)

here $\mathsf{\Delta }{\epsilon }_0$ is the axial strain increment, and Δχ is the element curvature increment.

The stress increments in concrete at the manufacturing stage are determined as follows:

$$\mathsf{\Delta }{\sigma }_b\left(y\right)=E_b\left(y\right)\mathsf{\Delta }{\epsilon }_b=E_b\left(y\right)\left(\mathsf{\Delta }{\epsilon }_0-y\mathsf{\Delta }\chi \right),$$

(2)

here $E_b\left(y\right)$is the tangential modulus of elasticity of concrete.

The stress increments in reinforcement $A_s^{\prime }$from the condition of its joint work with concrete are written as:

$$\mathsf{\Delta }{\sigma }_s^{\prime }=E_s^{\prime }\left(\mathsf{\Delta }{\epsilon }_0-y_s^{\prime }\mathsf{\Delta }\chi \right),$$

(3)

here $E_s^{\prime }$ is the tangential modulus of elasticity of the reinforcement $A_s^{\prime }$.

We compose the sum of the moments of the point C (according to Figure 4), dividing the height of the cross-section in half, in increments of stresses:

$$\mathsf{\Delta }{\sigma }_{sp}{A}_s{y}_s=\underset{A}{{\displaystyle \int }}\mathsf{\Delta }{\sigma }_b\left(y\right)ydA+\mathsf{\Delta }{\sigma }_s^{\prime }{A}_s^{\prime }{y}_s^{\prime },$$

(4)

here A is the area of the concrete cross-section; ${\sigma }_s^{\prime }$ are the stresses in the compressed (at the stage of loading with the compressive force F) reinforcement; ${\sigma }_{sp}$ are the stresses in the pretensioned reinforcement; $A_s$ and $A_s^{\prime }$ are, respectively, the area of the pretensioned and compressed (at the stage of loading with the compressive force F) reinforcement; and $y_s$ and $y_s^{\prime }$ are, respectively, the distances (in absolute value) from the geometric center of gravity of the cross-section to the centers of gravity of the reinforcement $A_s$ and $A_s^{\prime }$.

The integral over the concrete area in (4) is written as:

$$\underset{A}{{\displaystyle \int }}\mathsf{\Delta }{\sigma }_b\left(y\right)ydA=b_0\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{{\displaystyle \int }}}\mathsf{\Delta }{\sigma }_b\left(y\right)ydy+\left(b-b_0\right)\cdot \underset{-\frac{h}{2}+a_0}{\overset{\frac{h}{2}}{{\displaystyle \int }}}\mathsf{\Delta }{\sigma }_b\left(y\right)ydy.$$

(5)

Next, we compose the sum of the projections on the longitudinal axis of the column:

$$\mathsf{\Delta }{\sigma }_{sp}{A}_s+\mathsf{\Delta }{\sigma }_s^{\prime }{A}_s^{\prime }+\underset{A}{{\displaystyle \int }}\mathsf{\Delta }{\sigma }_b\left(y\right)dA=0.$$

(6)

The integral in (6) is calculated by analogy with (5).

Substituting (2) and (3) into (4) and (6), we obtain the following system of equations:

$$\left[\begin{array}{cc}ES& -EI\\ EA& -ES\end{array}\right]\left\{\begin{array}{c}\mathsf{\Delta }{\epsilon }_0\\ \mathsf{\Delta }\chi \end{array}\right\}=\left\{\begin{array}{c}\mathsf{\Delta }{\sigma }_{sp}{A}_s{y}_s\\ -\mathsf{\Delta }{\sigma }_{sp}{A}_s\end{array}\right\},$$

(7)

here

$$EI=b_0\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{{\displaystyle \int }}}{E}_b\left(y\right)y^{2}dy+\left(b-b_0\right)\underset{-\frac{h}{2}+a_0}{\overset{\frac{h}{2}}{{\displaystyle \int }}}{E}_b\left(y\right)y^{2}dy+E_s^{\prime }{A}_s^{\prime }{\left(y_s^{\prime }\right)}^2;$$

$$ES=b_0\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{{\displaystyle \int }}}{E}_b\left(y\right)ydy+\left(b-b_0\right)\underset{-\frac{h}{2}+a_0}{\overset{\frac{h}{2}}{{\displaystyle \int }}}{E}_b\left(y\right)ydy+E_s^{\prime }{A}_s^{\prime }{y}_s^{\prime };$$

$$EA=b_0\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{{\displaystyle \int }}}{E}_b\left(y\right)dy+\left(b-b_0\right)\underset{-\frac{h}{2}+a_0}{\overset{\frac{h}{2}}{{\displaystyle \int }}}{E}_b\left(y\right)dy+E_s^{\prime }{A}_s^{\prime }.$$

The integrals in the stiffness formulas $EI$, $EA$, and $ES$ are calculated numerically using the trapezoidal method.

The calculation at the manufacturing stage is performed according to the scheme of stepwise increase in prestress ${\sigma }_{sp}$. The cross-section height is divided into $n_y$ layers Δy. At the first step, $E_b=E_{b0}=const$. At each step after determining the strain increments, Formula (7) calculates the stress increments in each layer, the total stresses and strains, from which the tangential modulus of elasticity of the layer is determined.

Sargin’s formula is used as a dependence that determines the “stress–strain” curve of concrete:

$$\frac{\sigma }{R}=\frac{k\eta -{\eta }^2}{1+\left(k-2\right)\eta },$$

(8)

here $\eta =\frac{\epsilon }{{\epsilon }_R}$; ${\epsilon }_R$ is the strain value at the top of the diagram; R is the compressive strength of concrete; coefficient k determines the curvature of the diagram $\sigma -\epsilon$; and $k=1/{\lambda }_R$, ${\lambda }_R=R/\left(E_0{\epsilon }_R\right)$ is the coefficient of change of the secant modulus at the top of the diagram $\sigma -\epsilon$.

The tangent modulus can be calculated using the formula:

$$E_{tan}=\frac{d\sigma }{d\epsilon }=\frac{R}{{\epsilon }_R}\frac{\left(k-2\eta \right)\left(1+\left(k-2\right)\eta \right)-\left(k\eta -{\eta }^2\right)\left(k-2\right)}{{\left(1+\left(k-2\right)\eta \right)}^2}.$$

(9)

For reinforcement, a two-line Prandtl diagram is used.

2.2. Deformation Model of Columns with Local Prestress at the Stage of Work under Load

In this article, we consider the case of an eccentrically compressed flexible column hinged at the ends (Figure 5).

In contrast to the previous stage, at this stage the values ${\epsilon }_0$ and $\chi =\frac{d^{2}v}{d{x}^2}$ become functions of the x coordinate. At this stage, we also consider the creep and shrinkage of concrete.

In any cross-section of the column, the following integral dependencies are valid:

$$\begin{gathered} \mathsf{\Delta }{\sigma }_s{A}_s{y}_s-\mathsf{\Delta }{\sigma }_s^{\prime }{A}_s^{\prime }{y}_s^{\prime }-\underset{A}{{\displaystyle \int }}\mathsf{\Delta }{\sigma }_{b}ydA=\mathsf{\Delta }M; \\ -\mathsf{\Delta }F-\mathsf{\Delta }{\sigma }_s{A}_s-\mathsf{\Delta }{\sigma }_s^{\prime }{A}_s^{\prime }-\underset{A}{{\displaystyle \int }}\mathsf{\Delta }{\sigma }_{b}dA=0. \end{gathered}$$

(10)

The increment of the bending moment ΔM will be written as:

$$\mathsf{\Delta }M=\left(F+\mathsf{\Delta }F\right)\left(e_0-\left(v+\mathsf{\Delta }v\right)\right)-F\left(e_0-v\right)=-F\mathsf{\Delta }v-\mathsf{\Delta }Fv+\mathsf{\Delta }F{e}_0-\mathsf{\Delta }F\mathsf{\Delta }v.$$

(11)

The term ΔFΔv can be neglected in comparison with other terms due to its higher order of smallness.

The stress increments in the reinforcement $A_s$ can be written as:

$$\mathsf{\Delta }{\sigma }_s=E_s\left(\mathsf{\Delta }{\epsilon }_0+y_s\mathsf{\Delta }\chi \right).$$

(12)

The stress increments in the reinforcement $A_s^{\prime }$, as before, are determined by Formula (3). The increments of stresses in concrete, considering shrinkage and creep, can take the form:

$$\mathsf{\Delta }{\sigma }_b\left(y\right)=E_b\left(y\right)\left(\mathsf{\Delta }{\epsilon }_0-y\mathsf{\Delta }\chi -\mathsf{\Delta }{\epsilon }^{*}\right),$$

(13)

here $\mathsf{\Delta }{\epsilon }^{*}=\mathsf{\Delta }{\epsilon }_{cr}+\mathsf{\Delta }{\epsilon }_{sh}$ is the sum of increments of creep deformations and shrinkage of concrete.

Substituting (13), (12), and (3) into (10), we obtain the following system of equations:

$$\begin{gathered} EI\frac{d^2\mathsf{\Delta }v}{d{x}^2}-\mathsf{\Delta }{\epsilon }_0\cdot ES=-F\mathsf{\Delta }v-\mathsf{\Delta }Fv+\mathsf{\Delta }F{e}_0-\mathsf{\Delta }{M}^{*}; \\ ES\frac{d^2\mathsf{\Delta }v}{d{x}^2}-\mathsf{\Delta }{\epsilon }_0\cdot EA=\mathsf{\Delta }F-\mathsf{\Delta }{N}^{*}, \end{gathered}$$

(14)

here

$$\mathsf{\Delta }{M}^{*}=\underset{A}{{\displaystyle \int }}{E}_b\left(y\right)\mathsf{\Delta }{\epsilon }^{*}ydA;\mathsf{\Delta }{N}^{*}=\underset{A}{{\displaystyle \int }}{E}_b\left(y\right)\mathsf{\Delta }{\epsilon }^{*}dA.$$

The reduced stiffnesses $EI, EA, ES$ in the $l_0$ zone are determined by the formulas:

$$\begin{gathered} EI=E{I}_b^{\left(1\right)}+E{I}_b^{\left(2\right)}+E_s{A}_s{y}_s^2+E_s^{\prime }{A}_s^{\prime }{\left(y_s^{\prime }\right)}^2; EA=E{A}_b^{\left(1\right)}+E{A}_b^{\left(2\right)}+E_s{A}_s+E_s^{\prime }{A}_s^{\prime }; \\ ES=E{S}_b^{\left(1\right)}+E{S}_b^{\left(2\right)}-E_s{A}_s{y}_s+E_s{A}_s^{\prime }{y}_s^{\prime }; \end{gathered}$$

$$E{I}_b^{\left(1\right)}=b_0\underset{-\frac{h}{2}}{\overset{-\frac{h}{2}+a_0}{{\displaystyle \int }}}{E}_b^{\left(1\right)}\left(y\right)y^{2}dy+b\underset{-\frac{h}{2}+a_0}{\overset{\frac{h}{2}}{{\displaystyle \int }}}{E}_b^{\left(1\right)}\left(y\right)y^{2}dy; E{I}_b^{\left(2\right)}=\left(b-b_0\right)\underset{-\frac{h}{2}}{\overset{-\frac{h}{2}+a_0}{{\displaystyle \int }}}{E}_b^{\left(2\right)}\left(y\right)y^{2}dy;$$

$$E{A}_b^{\left(1\right)}=b_0\underset{-\frac{h}{2}}{\overset{-\frac{h}{2}+a_0}{{\displaystyle \int }}}{E}_b^{\left(1\right)}\left(y\right)dy+b\underset{-\frac{h}{2}+a_0}{\overset{\frac{h}{2}}{{\displaystyle \int }}}{E}_b^{\left(1\right)}\left(y\right)dy; E{A}_b^{\left(2\right)}=\left(b-b_0\right)\underset{-\frac{h}{2}}{\overset{-\frac{h}{2}+a_0}{{\displaystyle \int }}}{E}_b^{\left(2\right)}\left(y\right)dy;$$

$$E{S}_b^{\left(1\right)}=b_0\underset{-\frac{h}{2}}{\overset{-\frac{h}{2}+a_0}{{\displaystyle \int }}}{E}_b^{\left(1\right)}\left(y\right)ydy+b\underset{-\frac{h}{2}+a_0}{\overset{\frac{h}{2}}{{\displaystyle \int }}}{E}_b^{\left(1\right)}\left(y\right)ydy; E{S}_b^{\left(2\right)}=\left(b-b_0\right)\underset{-\frac{h}{2}}{\overset{-\frac{h}{2}+a_0}{{\displaystyle \int }}}{E}_b^{\left(2\right)}\left(y\right)ydy.$$

Indices (1) here correspond to the main concrete, and (2) to the concrete of the hollows.

At the end non-stressed sections, the reduced stiffnesses are determined by the formulas:

$$\begin{gathered} EA=b\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{{\displaystyle \int }}}{E}_b\left(y\right)dy+E_s{A}_s+E_s^{\prime }{A}_s^{\prime }; ES=b\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{{\displaystyle \int }}}{E}_b\left(y\right)ydy-E_s{A}_s{y}_s+E_s^{\prime }{A}_s^{\prime }{y}_s^{\prime }; \\ EI=b\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{{\displaystyle \int }}}{E}_b\left(y\right)y^{2}dy+E_s{A}_s{y}_s^2+E_s^{\prime }{A}_s^{\prime }{\left(y_s^{\prime }\right)}^2. \end{gathered}$$

(15)

Eliminating the value $\mathsf{\Delta }{\epsilon }_0$ from (14), we obtain the resolving equation for the deflection increment:

$$\left(EI-\frac{{\left(ES\right)}^2}{EA}\right)\frac{d^2\left(\mathsf{\Delta }v\right)}{d{x}^2}+F\mathsf{\Delta }v=\left(-\mathsf{\Delta }F+\mathsf{\Delta }{N}^{*}\right)\frac{ES}{EA}-\mathsf{\Delta }Fv+\mathsf{\Delta }F{e}_0-\mathsf{\Delta }{M}^{*}.$$

(16)

The solution of this equation is numerically performed at each load step by the finite difference method. On the interval $0\le x\le l/2$, a uniform grid is introduced with a step $\mathsf{\Delta }x=l/\left(2n_x\right)$, where $n_x$is the number of intervals in $x$. The finite difference approximation of Equation (16) has the form:

$$\begin{gathered} \frac{\mathsf{\Delta }{v}_{i+1}-2\mathsf{\Delta }{v}_i+\mathsf{\Delta }{v}_{i-1}}{\mathsf{\Delta }{x}^2}{\left[EI-\frac{{\left(ES\right)}^2}{EA}\right]}_i+F\mathsf{\Delta }{v}_i=\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\left(-\mathsf{\Delta }F+\mathsf{\Delta }{N}_i^{*}\right)\frac{E{S}_i}{E{A}_i}-\mathsf{\Delta }F{v}_i+\mathsf{\Delta }F{e}_0-\mathsf{\Delta }{M}_i^{*}. \end{gathered}$$

(17)

The boundary conditions have the form:

$$\mathsf{\Delta }v=0 at x=0; \frac{d\left(\mathsf{\Delta }v\right)}{dx}=0 at x=\frac{l}{2}.$$

(18)

Equation (17) is compiled for $i=2\dots n_x.$ For $x=l/2$, the boundary condition is approximated using the formula:

$${\frac{d\left(\mathsf{\Delta }v\right)}{dx}|}_{x=l/2}=\frac{1}{2\mathsf{\Delta }x}\left(\mathsf{\Delta }{v}_{n_x-1}-4\mathsf{\Delta }{v}_{n_x}+3\mathsf{\Delta }{v}_{n_x+1}\right)=0.$$

(19)

Finally, we obtain a system of $n_x$ linear algebraic equations with $n_x$ unknowns. The simplified calculation block diagram at the stage of operation under load in the case of short-term loading is shown in Figure 6.

The criterion for structure failure is a sharp increase in displacements, or the unconvergence of the solution (coefficients matrix of the system of linear algebraic equations is close to singular).

2.3. The Program of Experiments and the Design of Experimental Samples

The program of the experiment included the manufacture and testing of 8 reinforced concrete columns. To establish the influence of local prestressing on the bearing capacity, rigidity, and crack resistance of structures, four samples were made with prestressed tensile reinforcement, and for each prestressed sample, a “twin” was made with reinforcement without prestressing. The cross-section of the samples was assumed to be rectangular b × h 250 × 120 mm. Two prestressed specimens were made with a length of 180 cm and two specimens were made with a length of 240 cm.

The length of the local prestress zone $l_0$ was taken equal to l/2. Ultimate load tests were carried out according to the scheme of hinged support at the ends with two values of the eccentricity of the longitudinal force $e_0=2$cm and $e_0=4$cm. The samples also differed in the prism strength of concrete R. The longitudinal reinforcement for all samples was taken in the form of four rods with a yield strength $R_s=$ 800 MPa, rectangular stirrups with a diameter of 6.5 mm, and a step of 180 mm were used as transverse reinforcement. The distances $y_s$ and $y_s^{\prime }$ from the center line of the cross-section to the centers of gravity of the reinforcement were assumed to be 3 cm. The prestress level for all prestressed specimens was ${\sigma }_{sp}=0.5R_s=400$MPa. Characteristics of prototypes are presented in

Table 1. Characteristics of experimental samples.

Sample Number	Code	Prism Strength R, MPa	Sample Length l, cm	Longitudinal Force Eccentricity ${\mathit{e}}_0,\mathbf{c}\mathbf{m}$	${\mathit{\sigma }}_{\mathit{s}\mathit{p}}, \mathbf{MPa}$
1	180-2P	46.4	180	2	400
2	180-2N	46.4	180	2	0
3	180-4P	36.6	180	4	400
4	180-4N	36.6	180	4	0
5	240-2P	39.2	240	2	400
6	240-2N	39.2	240	2	0
7	240-4P	28.7	240	4	400
8	240-4N	28.7	240	4	0

.

All experimental samples were made in a special mold in the laboratory at a temperature of 18–22 °C and a relative humidity of 80–85%. The concrete was laid in layers with careful vibration of the layers. For manufacturing technological hollows around the prestressing reinforcement, it was wrapped in two layers of pipe insulation and polyethylene film before concreting (Figure 7), which completely excluded the concreting of reinforcing bars subjected to prestress in the $l_0$zone.

Later, when the experimental specimens were stripped from the formwork, after the concrete had gained the required strength, the insulation was easily removed (Figure 8 and Figure 9).

Simultaneously with the concreting of the columns, prisms with size 150 × 150 × 600 mm were made to determine the strength of concrete in accordance with the Russian standard GOST 24452-80.

The value of the initial elasticity modulus of concrete E₀ was determined based on the prismatic compressive strength R, according to the empirical formula [28]:

$$E_0=1000· \frac{0.05·R+57}{1+\frac{29}{3.8+R}}, \mathrm{MPa}$$

(20)

The value of R in Formula (17) should be substituted in MPa.

The value of ${\epsilon }_R$ was determined by the empirical formula [29]:

$${\epsilon }_R=\alpha {\left(\frac{R}{E_0}\right)}^{0.5},$$

(21)

where $\alpha =0.058$.

The tensile strength of concrete $R_t$ was calculated from the compressive strength using the empirical formula [28]:

$$R_t=0.29·R^{0.6}.$$

(22)

Exposure of prototypes was carried out at a temperature of 20 ± 2 °C and a relative humidity of 80 ± 5%. A few hours after pouring, the open surface of the freshly laid concrete was covered with plastic wrap.

The prestressing of the reinforcement was carried out using tension sleeves (Figure 10).

The creation of prestress was carried out 28 days after the concreting of the columns. The magnitude of the prestress was controlled by the deformations of the reinforcement using dial indicators, with a base of 30 cm. To register the strains, metal benchmarks were attached to the reinforcement under the indicators (Figure 11). The rotation of the couplings was performed sequentially in 6 steps to avoid distortion.

Besides the reinforcement strains, the deflection of the columns was measured using three deflection meters installed at the ends and in the middle of the structure. The general scheme of the experiment at the stage of prestressing is shown in Figure 12.

After the creation of pre-stresses, measurements of prestress losses in reinforcement were carried out by reinforcement deformations using dial gauges for 7 days. The average value of prestress losses in reinforcement was 12%. Further, by rotating the couplings, the level of prestress was increased to the initial value. Then, the grooves were sealed with fine-grained concrete from which prisms were also made to control its mechanical characteristics. The average value of the prismatic strength in compression for the concrete of the grooves was 38.4 MPa, tensile strength $R_t$ was 2.96 MPa, the initial elasticity modulus of concrete E₀ was 3.8 × 10⁴ MPa, and the ${\epsilon }_R$ value was 2.06 × 10⁻³.

Testing of the columns for a short-term load effect was carried out 28 days after the concreting hollows. Thus, the age of the base concrete at the time of testing was 63 days. The eccentric compression experiment was carried out according to the hinged supported scheme on a hydraulic press with a maximum force of 5000 kN. The first load step was fixed as 120 kN, which is associated with the design features of the press, and then it was selected in the range from 20 to 70 kN, depending on the theoretical level of the ultimate load and the proximity to the breaking force. The ends of the column were placed in the heads, which had slots that made it possible to set an eccentricity of 2 and 4 cm, then knives were inserted into these slots, to which the force from the press was transmitted.

To determine the concrete strains at the compressed and stretched faces, 3 clock-type indicators, with a base of 30 cm were installed on them in the middle of the span (in the middle and along the edges). The coincidence of the indications on one face showed the absence of skew and eccentricity in two planes. To avoid distortion, indicators were also installed on the side faces. Besides concrete strains, horizontal displacements of points at the ends and in the middle of the span were measured using deflection meters.

A general view of the column prepared for testing is shown in Figure 13. The scheme of dial indicators and deflection meters installation is shown in Figure 14. At each stage, to take readings on indicators, fix the occurrence of cracks and monitor their development, and measure deflections; exposures were given for 10 min.

3. Results

The results of the experiment at the stage of creating pre-stresses are given in

Table 2. Results of the experiment at the stage of prestressing.

Sample	$\mathbf{Deflection}\mathit{f}$, mm		${\mathit{\sigma }}_{\mathit{s}\mathit{p},\mathit{c}\mathit{r}\mathit{c}}, \mathrm{MPa}$
	Experiment	Theory	Experiment	Theory
180-2P	1.05	1.23	400	376
180-4P	1.39	1.56	333	325
240-2P	2.2	2.26	333	339
240-4P	2.87	2.73	267	280

. The deviations of the theoretical values of the column deflections from the experimental at the manufacturing stage lie in the range of 2.7–14.6%. The calculated values of the pre-stress’ levels in the reinforcement ${\sigma }_{sp,crc}$, at which technological cracks are formed in the concrete, also agree well with the experiment.

The results for all tested samples at the stage of compressive force application are summarized in

Table 3. Eccentric compression test results for columns.

Sample	${\mathit{F}}_{\mathit{u}\mathit{l}\mathit{t}}, \mathit{k}\mathit{N}$		$\mathit{\delta }{\mathit{F}}_{\mathit{u}\mathit{l}\mathit{t}},\%$	${\mathit{F}}_{\mathit{c}\mathit{r}\mathit{c}}, \mathit{k}\mathit{N}$		$\mathit{\delta }{\mathit{F}}_{\mathit{c}\mathit{r}\mathit{c}},\%$
	Experiment	Theory		Experiment	Theory
180-2P	860	800	7.5	860	800	7.5
180-2N	740	656	12.8	430	456	5.7
180-4P	450	420	7.1	300	317	5.4
180-4N	345	344	0.3	120	74.8	-
240-2P	636	600	6	400	384	4.2
240-2N	525	456	15.1	310	327	5.2
240-4P	270	300	10	180	142	26.8
240-4N	240	225	6.7	120	63	-

Note: For samples 180-4N and 240-4N, the value of $\delta F_{crc}$ was not determined because the theoretical value of $F_{crc}$ is significantly lower than the minimum possible value of the first load step for the used press (120 kN).

. $F_{ult}$ here is the ultimate load, $\delta F_{ult}$ is the deviation in the percentage of the experimental ultimate load from the theoretical, $F_{crc}$ is the load at which the first crack is formed in the tension zone, and $\delta F_{crc}$ is the deviation in percentage of the experimental from the theoretical load at the onset of cracking.

Experiments on eccentric compression of prestressed columns showed an increase in bearing capacity compared to elements without prestressing from 12.5 to 30%. The crack resistance also increased markedly. The load level at which the first crack is formed in the stretched zone increased by 1.3–2.5 times. The experimental results agree well with theory. The deviation of the experimental values of the ultimate loads for prestressed columns with the results obtained by the author’s method lies in the range from 6 to 10%. Figure 15 shows theoretical and experimental plots of maximum deflection versus load for prestressed specimens 180-2P, 180-4P, 240-2P, and 240-4P. The destruction of the first of these samples occurred along the concrete of the compressed zone (Figure 16), and the remaining samples were destroyed due to the breakage of the reinforcement in the couplings (Figure 17 and Figure 18).

4. Discussion

The differential Equation (16) obtained in Section 3 coincides in structure with the differential equation for the buckling of compressed wooden bars, derived in [30]:

$$\left(EI-\frac{{\left(ES\right)}^2}{EA}\right)\frac{d^2\left(\mathsf{\Delta }v\right)}{d{x}^2}+F\mathsf{\Delta }v=\left(-\mathsf{\Delta }F+\mathsf{\Delta }{N}^{*}\right)\frac{ES}{EA}-\mathsf{\Delta }F\left(v+v_0\right)-\mathsf{\Delta }{M}^{*}.$$

(23)

The difference between Equations (16) and (23) is only that the latter considers the bars with initial deflection $v_0$ without eccentricity $e_0$. In the case of the prestressed reinforced concrete column, the deflection that it received at the manufacturing stage is included in the term $\mathsf{\Delta }Fv$. Term $\mathsf{\Delta }Fv$ partially compensates the bending moment $\mathsf{\Delta }F{e}_0$ caused by the eccentricity of the axial force; and, for wooden bars, stiffnesses $EA, EI$, and $ES$ are defined somewhat differently than for reinforced concrete columns.

Equation (16) is only suitable for the columns hinged at the ends. To satisfy arbitrary boundary conditions, it is sufficient to differentiate it twice with respect to x.

The conducted study showed a noticeable increase due to the creation of prestress not only in stiffness and crack resistance but also in the bearing capacity of the columns. The obtained results on the increase in the bearing capacity are consistent with previously published works [31,32,33] in which columns with prestressing reinforcement along the entire length of the elements were considered.

Note that with the proposed method of creating pre-stresses, columns have a reduced cross-section in the area $l_0$ at the manufacturing stage. This leads to an increase in the initial deflection of the elements and the initial stresses in the concrete compared to structures prestressed to the full length. Since the initial deflection partially compensates for the eccentricity of the longitudinal force, the ultimate load for columns with local prestressing according to the proposed technology can be higher than for prestressed columns manufactured using traditional technology.

Table 4. Comparison of theoretical initial deflections for columns with local prestressing and analogues prestressed for the full length.

Sample	Initial Deflection f, mm		Ultimate Load ${\mathit{F}}_{\mathit{u}\mathit{l}\mathit{t}}, \mathit{k}\mathit{N}$
	Local Prestress	Full Length Prestress	Local Prestress	Full Length Prestress
180-2P	1.23	0.838	800	745
180-4P	1.56	0.931	420	403
240-2P	2.26	1.60	600	588
240-4P	2.73	1.86	300	310

compares the theoretical initial deflections $f$ and ultimate loads $F_{ult}$ for the specimens tested by us with the theoretical values for analog specimens with full length prestressing, calculated by the method given in [34].

Table 4 shows that full length prestressing is slightly more effective than local prestressing only for the sample 240-4P. This can be explained by the fact that the theoretical fracture zone for sample 240-4P is located not in the middle of the span, but at the junction of the stressed and unstressed zones. For specimens 180-2P, 180-4P and 240-2P, the theoretical fracture zone without considering the weakening of the reinforcement section by the thread is in the middle of the span. The bearing capacity of sample 240-4P could be increased by increasing $l_0$. Our further studies will be devoted to determining the optimal value of $l_0$.

Compared to traditionally prestressed elements, columns with local prestressing are also characterized by favorable working conditions at the end sections: they do not require additional reinforcement to prevent negative effects caused by prestressing forces. In areas without prestressing, conventional reinforcement can also be used instead of high-strength reinforcement, which leads to some reduction in the cost of structures. Another important advantage of columns with local prestressing, made according to the considered technology, is the possibility of their manufacture directly on the construction site. The level of prestressing can be controlled by the number of turns of the turnbuckles.

However, the proposed technology is not without drawbacks. In three of the four tested prestressed specimens, the failure occurred along with the reinforcement in the couplings due to the weakening of the threaded section, despite the fact that the couplings were located at a distance from the zone with maximum bending moments. Further research can be aimed at improving the design of tension sleeves to ensure the uniform strength of the joint.

The proposed technology makes it possible to create not only preliminary tensile but also preliminary compressive forces in the reinforcement. Elements with pre-compressed reinforcement on the compressed side and pre-tensioned reinforcement on the opposite side can be even more effective than designs with only pre-tensioned reinforcement. Further research may also be devoted to the development of such structures.

It should be noted that in this study, columns with the same diameter of all longitudinal reinforcing bars were tested. In practice, if the eccentricity is known in advance, the structures will not have symmetrical reinforcement. The proposed model makes it possible to take into account asymmetric reinforcement, and we plan to test such columns in the future.

5. Conclusions

• A new type of reinforced concrete columns with local prestressing of reinforcement is proposed. Resolving equations are obtained and algorithms for calculating the stress-strain state of columns at the stage of manufacture and work under load are developed;
• An experimental study of the stiffness, crack resistance, and bearing capacity of columns with local prestressing has been carried out. As a result, it was found that, compared with elements without prestressing, there was an increase in the bearing capacity from 12.5 to 30%. Crack resistance also noticeably increased: the load at which the first crack forms in the stretched zone increased by 1.3–2.5 times;
• The experimental results are in good agreement with the developed calculation model. The discrepancy in the value of the ultimate load was 6–10%;
• Theoretical calculations have shown that the bearing capacity of columns with local prestressing can be higher than that of columns with full length prestressing.