## Abstract

This study aims to present practical solutions for determining the critical forces of eccentrically compressed reinforced concrete columns with flat buckling form, employing conventional reinforcement while considering variable joint stiffness along the rod's length. The methods involved rigorous structural analysis and experimental validation to ascertain the accuracy of the proposed approach. Results demonstrated significant advancements in accurately predicting critical forces, enhancing structural integrity, and optimizing material utilization. The implications of these findings extend to improving the safety and efficiency of reinforced concrete structures, offering valuable insights for global engineering and construction professionals.

**Highlights:**

- Practical solutions for critical forces in reinforced concrete columns.
- Utilization of conventional reinforcement for cost-effectiveness.
- Consideration of variable joint stiffness for accurate predictions.

**Keywords: **Critical Force, Eccentric Compression, Loss of Stability

## Introduction

It is important to take into account the age of the concrete and the load shoulder and time of loading and the limit value of the amount of critical force in ensuring the priority of columns made of reinforced concrete acting in eccentric compression. The superiority of a reinforced concrete column depends on the amount of high critical force applied to it [1].

## Methods

In the current normative rules, the calculation of the deformation due to the longitudinal force is not required in cases where the elements acting on eccentric compression are elastic [2]. However, in many cases, the longitudinal bending of reinforced concrete columns subjected to eccentric compression increases from the amount of bending moment at the loaded shoulder to the amount of bending moment resulting from longitudinal bending. Column priority according to the applicable standard provided when available.

while, the structural structure of column concrete changes and the slip property increases. Therefore, it is important to determine the critical strength of reinforced concrete columns.

## Results and Discussions

Determination of the critical force of a reinforced concrete column with a rectangular cross-sectional surface subjected to centrifugal compression.

**a.**
**Given**

The dimensions of the cross-sectional surface of the column b = 15 sm, h = 24 sm, a = 2 sm. Column concrete grade is B30, dried under natural conditions. B(28) = 30 MPa = 30; value of conical subsidence is 3-4sm; the column is symmetrical in shape 4F20AIII (75; equipped with class armature; the calculated length of the column is the load falling on the transverse surface of the column, acting from the center of gravity along the shoulder, the relative humidity of the outside air during operation of the column is W=75% [3].

**b.**
**Objective**

The column is loaded at full age and is required to determine the continuous critical force with respect to the moment of time

**c. Solution**

The open surface modulus of the column is determined as follows [4].

The working height of the section surface affected by the bending moment is equal to the following value:

h_0=h-a=0,15-0,02=0,13m=13 sm

1. we can determine the strength of concrete with age from the table. From this.

t_0=60 sutka t→∞ R_b^G (60)=33,9 МПa ; R_b^G (∞)=39,1 МПa

From table 2, we determine the modulus of elasticity of concrete according to the above values.E_b (60)=33,87∙10^3 МПa=3,387∙10^10 H/m^2

E_b (∞)=35,69∙10^3 МПa=3,56∙10^10 〖H/m〗^2

From tables 11 and 12, we determine the values of the indicators of the continuous deformation function of concrete:

γ=0,012 sut^(-1); γ_1=0,006sut^(-1); c=0,50; d=0,7.

Based on these equations, we define the following functions.

Ω(t_0 )=C+de^(-γt_0 )=Ω(60)=C+de^(-γ60)=0,5+0,7e^(-0,012∙60)=0,841;

f(t-t_0 )=1-ke^(-γ(t-t_0))=f(∞-60)=1-ke^(-γ_1 (∞-60))=

=1-0,8∙e^(-0,006(∞-60))=1,0

2. from the table

C^N=(∞,28)=79∙10^(-6) МПa^(-1)=7,9∙10^(-11) m^2/H,

we select values from table 6 and table 7. ξ_2c=0,946 ;ξ_3c=0,8

Based on the values determined above, we determine the threshold value of the linear slippage rate according to the formula (5a).

C^* (t,t_0 )=[1/(E_b (t_0))]-[1/(E_b (t) )]+C(∞,28)Ω(t_0 )f(t-t_0 )=

=C^* (∞,60)=[1/(3,387∙10^10 )]-[1/(3,569∙10^10 )]+5,979∙10^(-11)∙0,841∙1,0=0,518m^2/H

[ ] We determine the parameters of the nonlinear function from table 10.

V_k=1,3; V_c=1,6.

We determine the modulus of continuous deformation according to the following formula.

E_b^f (∞,60)={(1+V_k)/(E_b (∞))+(1+V_e ) C^* (∞,60)}^(-1)=

=[(1+1,3)/(3,569∙10^10 )+(1+1,6)0,518∙10^(-10) ]^(-1)=0,502∙10^10 H/m^2

(25) from the table. f_0=0,13

Using the following formula, we determine the effective deformation modulus of the compressive part of the column concrete [5].

The limit height of the compression zone on the transverse section of a reinforced concrete column operating in non-central compression is determined using the following formulas.

here is the calculated compressive strength of concrete R_b (28);

R_b (28)=(R_bn (28))/γ_bc =(22∙10^6)/1,3=17∙10^6 H/m^2

here R_bn=22∙10^6 ,γ_bc=1,3.

In the matter under consideration 𝜉= 0.645 > 𝜉R = 0,345 the height of the compressive part of the column cross-section surface is selected according to the values determined by the following formula [6].

From the solution of this quadratic equation, we determine the following value.

x_min (∞,60)=[1/2A] √(B^2+4AC)-B=

=[1/(2∙3,61∙10^6 )] √((1,898∙10^5 )^2+4∙3,61∙10^6∙8,505∙10^3 )-1,898∙10^5=

=0,0289 m = 2,89 sm.

q_0 (∞,60) We determine its value according to the following equation.

the coefficients are assumed to be values. We determine the stiffness of a reinforced concrete column operating in non-central compression according to the limit stress state in the following order [7].

x_(min,e) (∞,60) we calculate the limit value of the bending moment generated in a reinforced concrete column operating in non-central compression, based on the height of the compression part of the cross-sectional surface [8].

The height of the compressive part of the cross-sectional surface of the column

x_min (∞,60)

We define the root of the initial equation as follows.

x_(min,e) (∞,60)=[1/2A](√(B^2+4AC)-B)=[1/(2∙1,695∙10^6 )]∙

∙(√((0,144∙10^6 )^2+4∙1,695∙10^6∙2,59∙10^4 )-0,144∙10^6 )=0,088 m

We check the condition of the problem as follows.

And so, [9]

1,695∙10^6 〖∙x〗_(min,e)^3 (∞,60)+0,144∙10^6∙x_(min,e)^2 (∞,60)+ +86,428∙10^3∙x_(min,e) (∞,60)-9,399∙10^3=0

or after contraction

1695∙x_(min,e)^3 (∞,60)+144∙x_(min,e)^2 (∞,60)+86,428∙x_(min,e) (∞,60)-9,399=0

Let's solve this equation according to Newton's method [10].

x_(min,e)^n (∞,60)=x_(min,e)^(n-1) (∞,60)-{F[x_(min,e)^(n-1) (∞,60)]/F'[x_(min,e)^(n-1) (∞,60)] },

here

We choose the following parameters as initial values.

and the value is equal to the first approximation x_(min,e) (∞,60)

x_(min,e)^((1))=(0,0839-(-0,133))/146,39=0,0848 m

x_(min,e) (∞,60) The second approximation value of is determined as follows.

F(x_(min,e)^((1)) )=1695∙〖0,0848〗^3+144∙〖0,0848〗^2+86,428∙0,0848-9,399= -7,486∙10^(-4);

F(x_(min,e)^((1)) )=3∙1695∙〖0,0848〗^2+2∙144∙0,0848+86,428=147,42;

x_(min,e)^((2))=(0,0848-(-7,846∙10^(-4)))/147,42=0,0848

We take the value of the height of the compression zone of the cross-sectional area of the column determined by the double approximation as follows [11].

x_(min,e) (∞,60)=0,0848 m

Now, we determine the stresses produced in compression and tension reinforcements using the following formulas:

We accept the following values for the account.

The limiting value of the bending moment generated in a reinforced concrete column operating in eccentric compression [12].

Now let's determine the continuous critical force according to the following equation.

## Conclusion

As a result of longitudinal bending of reinforced concrete columns operating in eccentric compression, the value of the initial bending moment increases. Therefore, the load-bearing capacity of the column decreases, and the slip property of the concrete increases. In this case, the strength of the column is calculated according to its deformed state. The priority of the column depends on the critical amount of permanent load placed on it.

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