Microsoft Word - ETASR_V11_N3_pp7100-7106


Engineering, Technology & Applied Science Research Vol. 11, No. 3, 2021, 7100-7106 7100 
 

www.etasr.com Hamzaoui & Bouzid: The Proposition of an EI Equation of Square and L–Shaped Slender Reinforced … 

 

The Proposition of an EI Equation of Square and L–
Shaped Slender Reinforced Concrete Columns under 

Combined Loading 
 

Leila Hamzaoui 
LGC–ROI, Civil Engineering Laboratory, Risks and 

Structures Interactions 
Faculty of Science 
University of Batna 2 
Batna, Algeria 

l.hamzaoui@univ-batna2.dz 

Tayeb Bouzid 
LGC–ROI, Civil Engineering Laboratory, Risks and 

Structures Interactions 
Faculty of Science 
University of Batna 2 
Batna, Algeria 

tayeb.bouzid@univ-batna2.dz 

 

Abstract-The stability and strength of slender Reinforced 
Concrete (RC) columns depend directly on the flexural stiffness 

EI, which is a major parameter in strain calculations including 

those with bending and axial load. Due to the non-linearity of the 

stress-strain curve of concrete, the effective bending stiffness EI 

always remains variable. Numerical simulations were performed 

for square and L-shaped reinforced concrete sections of slender 

columns subjected to an eccentric axial force to estimate the 
variation of El resulting from the actual behavior of the column, 

based on the moment-curvature relationship. Seventy thousand 

(70000) hypothetical slender columns, each with a different 

combination of variables, were used to investigate the main 

variables that affect the EI of RC slender columns. Using linear 

regression analysis, a new simple and linear expression of EI was 

developed. Slenderness, axial load level, and concrete strength 
have been identified as the most important factors affecting 

effective stiffness. Finally, the comparison between the results of 

the new equation and the methods proposed by ACI-318 and 

Euro Code-2 was carried out in connection with the experimental 

results of the literature. A good agreement of the results was 
found. 

Keywords-flexural stiffness; reduction factor; reinforced 

concrete; rigidity; slender columns 

I. INTRODUCTION  

For a bi-articulated column supporting a load P applied 
with an eccentricity e, the corresponding moment which is 
equal to P.e is called the "first order moment". Under the action 
of this moment the column deforms in simple curvature and the 
bending moment is at its maximum at half height and equal to 
M=P.e+P.∆ .The increase of the latter caused by the 
displacement ∆ is called the P∆ effect or second order moment. 
The longer the column is compared to its cross section, the 
greater the lateral displacements will be and the greater the 
second order moment will be. Buckling failure will occur under 
a lower resistance than that which would have caused the short 
column to fail. In this case, the strength of slender Reinforced 
Concrete (RC) columns is remarkably reduced. Several factors 
contribute to this reduction. In addition to the P∆ effect, 

shrinkage, creep, and cracking occur along the length of the 
column and are directly related to the supported part of the 
compressive stress block in the concrete. Leonhard Euler who 
based his work on Hook’s law and the moment-curvature 
relationship studied the buckling problem. He developed (1) to 
predict the minimum buckling load of a perfect column: 

��� = ����	�     (1) 
This equation did not depend on the strength of the cross-

section but rather on the bending stiffness EI. It was valid only 
if the moment of buckling obeys Hook's law, where the 
retained stiffness is constant. This is the case of a homogeneous 
isotropic and elastic material but this is not the case for 
reinforced concrete columns whose behavior in practice 
becomes clearly non-linear at the level of the descending 
branch of the σ-ɛ curve, therefore the buckling takes place in 
the non-linear domain. Euler's expression remains valid if the 
bending stiffness is replaced by a value representative of the 
effective stiffness at the moment of failure which is either with 
the crushing material (case of short columns) or with buckling 
(case of thin columns). During the loading and due to the 
inelastic behavior depicted by the stress/strain curve of 
concrete the value of the EI varies at each point along the 
section. The difficulty to choose a single value of EI appears 
because some methods consider that the column stiffness is 
constant along the section and the length of the column, when 
in addition it is necessary to consider shrinkage, creep, and 
lateral deflection. If we introduce the effects of concrete 
cracking, this can cause a large variation in bending stiffness. 
The objective of this work is to develop an analytical model for 
the study and analysis of the non-linear material and 
geometrical behavior of slender reinforced concrete columns of 
square and "L" cross-section under axial load and biaxial 
bending in order to find a simple relationship explicitly 
describing the real behavior of the effective bending stiffness 
EI as a function of parameters such as concrete strength ƒc, 
steel yield strength ƒy, cross-sectional dimensions, percentage 
of reinforced steel ρ, and column length L. In addition, this 

Corresponding author: Leila Hamzaoui



Engineering, Technology & Applied Science Research Vol. 11, No. 3, 2021, 7100-7106 7101 
 

www.etasr.com Hamzaoui & Bouzid: The Proposition of an EI Equation of Square and L–Shaped Slender Reinforced … 

 

stiffness is compared with experimental test results from the 
literature and design codes ACI-318 [1] and EC-2 [2]. The 
studied columns were bent into a single symmetrical curvature, 
in non-oscillating frames subjected to short-term loads.  

The behavior of RC slender columns under axial loads and 
bending moments in the two main planes of inertia is of great 
interest for structural analysis and design .The difficulty in this 
case is in the fact that it is not possible to give a general 
mathematical formulation as in the case of uniaxial bending 
because the position of the neutral axis is not known in 
advance. The difficulty increases if the cross-section of the 
column is not symmetrical, and increases even more if the 
behavior law (σ-ɛ) is not linear. Some researchers have 
proposed an iterative method initiated in [3] which consists of 
establishing by successive trials the neutral axis for which the 
internal forces in the section are in equilibrium with the 
external loads. Other methods, based on the determination of 
interaction curves, interaction surfaces or interaction contours 
for loading, were used for calculation and verification [4]. In 
most computer programs for reinforced concrete structures, 
there are indications of the limit state justification of square and 
rectangular columns while the L-shaped cross-section is not 
treated. It is therefore interesting to fill in the gaps in the 
equations presented in the literature because they are valid only 
for ordinary sections, and in practice, there are sections of 
different shapes such as tee (T), simple cross (+), hexagonal or 
thin-walled boxes, and finally L-shaped. In fact, the latter 
presents several difficulties like the absence of vertical and 
horizontal axes of symmetry. 

II. LITERATURE REVIEW 

Flexural rigidity is a necessary component in the estimation 
of the deflection of RC elements subjected to bending moment 
and axial loading, especially slender columns. It should be 
noted that the deflection is important because it has a direct and 
strong influence on the second order moments induced by the 
axial loads. Recently, major progress has been made in the 
development of nonlinear methods for the analysis of slender 
RC columns. Numerous methods are used to evaluate the 
bending stiffness EI taking into account all the factors that 
contribute to its decrease at the level of the concrete stress 
block in compression (non-linear material behavior) and at the 
level of the lateral displacement length of the column (non-
linear geometric behavior) to determine the critical buckling 
load nearest to the effective values. Accordingly, the effective 
bending stiffness of columns under short-term loading can be 
expressed as: EI=α.Ec.Ic+Es.Is where α is the stiffness 
reduction factor, Ec and Es are the elasticity modulus of 
concrete and steel respectively, Ic and Is are the moments of 
inertia of the cross-section of concrete and steel about the 
central axis of the cross-section respectively. After determining 
the factor α, the EI can be easily calculated. The factor α is a 
function of the material properties, the steel ratio, the axial 
load, and other parameters. This reduction factor of bending 
stiffness has been evaluated and studied by a number of 
researchers and some current codes and according to [1], is 
always taken equal to 0.2 for columns. The Eurocode EC-2 [2] 
is provided with a coefficient of flexibility Ke which is a 
function of a set of parameters such as: the relative normal 

force ν, the mechanical slenderness λm, the geometric 
reinforcement ratio ρ, the creep rate φeff, and the characteristic 
strength of the concrete compression cylinder at 28 days fck. 
Many formulations have been proposed in the literature. 
Authors in [5-7] have presented statistical evaluations of the 
flexural rigidity of RC columns and of rectangular cross-
sections of composite RC columns. Authors in [8] proposed a 
reduction coefficient as a function of various parameters such 
as slenderness ratio (L/h), eccentricity ratio (e/h), and axial load 
ratio (P/P0). Authors in [6] proposed a general expression to 
obtain an effective flexural stiffness applicable to any cross-
sectional shape under short or long-term biaxial eccentric 
loading for normal and high concrete strength. Authors in [9] 
proposed a study for the reduction factor of the RC column 
with equi-axial T section and in [10] dealt with the cross-
shaped cross-section. Authors in [11] presented new 
expressions for a beam with rectangular and T-shaped cross-
sections, square, rectangular and circular cross-sections for 
columns in function of concrete strength ƒc, steel percentage ρ, 
and level of charge P/P0. Authors in [12] presented their 
approach to find effective flexural stiffness for circular 
columns. In this research, the stiffness reduction factor was 
suggested to treat slender square and L-section columns with 
equal flanges. Some researchers have conducted studies on 
eccentrically loaded L-sections. Authors in [13] presented an 
experimental and analytical research on the load carrying 
capacity of L-shaped RC columns with equal and unequal 
flanges. Authors in [14] carried out a theoretical study on the 
calculation of L and square sections by the interaction 
approach. Authors in [15] based their study on the finite 
element method to analyze and draw the interaction surfaces of 
L and square sections. An interesting research concerning the 
behavior of L-section and square composite columns under 
biaxial eccentric loading according to different laws of concrete 
behavior is presented in [16]. Authors in [9] studied the L 
section with equal flanges and suggested a formula for the 
bending stiffness reduction factor of slender RC columns under 
different axial load levels and different seismic action levels. 
The current work is a contribution in this field of study and, in 
addition, a proposition of a simple solution for the calculation 
and verification of thin columns is presented. 

III. ANALYSIS METHOD 

In order to study the effective bending stiffness values of 
the slender RC column under combined biaxial bending and 
axial loading, nonlinear computer analysis was developed. 
Material and geometric nonlinearity were included in the 
research study for L and square sections. It should be noted that 
articulated end columns with equal load eccentricities at the 
ends acting in the same plane have been taken in account. The 
analysis can be divided into three parts: (a) the cross-sectional 
strength (analysis of cross section), (b) the effect of slenderness 
on the column strength (analysis of second order), (c) the third 
and main objective is the development of a new formula for 
estimating the EI bending stiffness of slender biaxially loaded 
RC columns in which these factors are included. 

A. Material Behavior and Assumptions 

The studied columns are all made of RC. This composite 
material consists of two elements with different characteristics. 



Engineering, Technology & Applied Science Research Vol. 11, No. 3, 2021, 7100-7106 7102 
 

www.etasr.com Hamzaoui & Bouzid: The Proposition of an EI Equation of Square and L–Shaped Slender Reinforced … 

 

In order to simulate the non-linear analysis of the complete 
response of RC columns of square and L-shaped cross-sections 
with equal flanges under biaxial bending and compression, a 
computer program compiled by an integral numerical method 
was used, based on the following assumptions: 

• Plane sections before bending remain plane after bending. 

• The tensile strength of concrete is neglected, and the stress-
strain curve of concrete in compression refers to the 
unconfined diagram in [17], as shown in Figure 1(a), whose 
curve characteristics are calculated using (2) and (3): 

σ = �� �
2. ������ � 
 �����
��    (2) 

for 0 � ε� � ε�� = 0.002 
σ = ���1 � ��� � �  �0!"    (3) 
for  �# �  � �  �$ = 0.004 

with: 

Zu = �.()(�*+)�,,						ε(�* = ./�.�0(.1234�12+3���, ε(�* 5 ε�,		���	mpa! 
• The stress-strain curve of steel is taken as the usual bilinear 
relationship shown in Figure 1(b). 

The section is divided into a number of small fiber 
elements. The concrete element stress distribution is also 
divided into a number of small fiber elements. Concrete 
shrinkage and creep are ignored. 

 

  
(a) (b) 

Fig. 1.  Concrete and steel σ-ε diagram 

1) Cross Section Analysis 

This work presents an indirect method for the analysis of 
the ultimate limit states of RC columns of L and square cross-
sections subjected to biaxial loading. We consider a square 
section of dimensions B×B, and another square section of 
dimensions B1×B1 with B1<B being away from the corner of the 
original square (B×B). Removing the section B1×B1 results in 
an L-shaped section, as shown in Figure 2(a). For different 
values of B1, we can obtain different dimensions of the L-
section. If (B1=0) the L-section becomes a square section. The 
ratio (B1/B) varies from 0 to 0.6. It is assumed that the steel 
reinforcement is evenly distributed around the perimeter of the 
cross-section (see Figure 2(b))[14]. 

2) Internal Forces 

The cross section is divided into infinitely small slices (see 
Figure 3). For each one of them, the internal forces (concrete, 

compression reinforcement and tensile reinforcement) are 
evaluated to give finally an axial load and an ultimate moment. 
The resulting axial load and the resulting moment must be 
equal respectively to the sum of the internal loads and the sum 
of elementary moments. 

 

  
(a) (b) 

Fig. 2.  L cross section and steel reinforcement distribution. 

 
Fig. 3.  L cross section subjected to biaxial load. 

The resulting axial load and the resulting moment are equal 
respectively to the sum of the loads and elementary moment 
shown in (4), and (5): 

9 = ∑ ��. ;�<. ∑ =�> � ��?. @. ABC3��D<E3D<E3     (4) 
F = ∑ ��. G�H. ;�<. ∑ =�> � ��?. @. GH. ABC3��D<E3D<E3     (5) 

where n is the number of slices, Aci is the concrete area of the 
slice I, ρ=(As/Ac).100 presents the percentage of longitudinal 
reinforcement, As is the area of longitudinal reinforcement, and 
Yi is the distance from the edge of steel to the center of the 
section. 

Analysis is done by sweeping the section parallel to the 
neutral axis. For the L-shaped section, the computer program 
studies different positions of the neutral axis according to the 
angle of inclination θ, where θ=tang-1(ex/ey) with ex and ey 
being the eccentricities along the x and y axis respectively. The 
computer analysis has been prepared by considering the axis of 
bending. Figure 4 explains these cases: 

• Case 1: Uniaxial bending, angle of inclination θ=00. 

• Case 2: Uniaxial bending, angle inclination θ=900. 



Engineering, Technology & Applied Science Research Vol. 11, No. 3, 2021, 7100-7106 7103 
 

www.etasr.com Hamzaoui & Bouzid: The Proposition of an EI Equation of Square and L–Shaped Slender Reinforced … 

 

• Case3: Biaxial bending, angle of inclination 0<θ<450. 

• Case4: Biaxial bending, angle of inclination 450<θ<900.    

 

 
Fig. 4.  The four cases of uniaxial and biaxial load. 

The codes of the above mentioned cases were developed in 
FORTRAN and were used to obtain the ultimate resistance Pu 
and moments Mu as results of the different positions of the 
neutral axis. The input data include the size of the square B1, 
the depth B of the section, the depth of embedding Dc, the 
characteristic strengths of concrete fc and steel fy, the moduli of 
elasticity of steel Es and concrete Ec, the angle of inclination θ, 
and the percentage of reinforcement ρ.  

IV. SECOND ORDER ANALYSIS 

Columns with a moderate slenderness may have their 
secondary moments determined with an approximate analysis 
such as the moment magnifier method which is used in this 
study. This procedure uses a moment magnification factor δ to 
amplify the applied moment Ma sufficiently for the account of 
secondary moments and to ensure safe design. For columns 
subjected to axial loading and equal and opposite end 
moments, the maximum bending moment can be taken as: 

F� = I. FJ    (6) 
where:  

I = 11 � KK�L
 

where Mc is the design bending moment that includes second-
order effects, Ma is the applied column bending moment, P is 
the factored axial load acting on the column, and Pcr is Euler’s 
buckling strength. The secondary moment due to the column 
slenderness is calculated as the difference between the internal 
resisting moment and the applied moment. For the purpose of 
analysis, Mc is replaced by the cross section bending moment 
MCS, and Ma is replaced by the overall column bending moment 
(Mcol) as shown in (7): 

F�M = N 33+ OOBPQ. F�RS    (7) 
V. CALCULATION OF THE FLEXURAL STIFFNES OF THE 

COLUMN 

The design of the flexural stiffness as a function of the 
moment-to-curvature ratio M/Φ is the most reasonable relation 
of the EI representation. The analysis studies show various 

cross-sectional parameters as well as axial loading influencing 
the bending stiffness. The computer program is structured 
according to the well-known (P, M, Φ) relation. A linear strain 
is assumed and the Kent & Park relation [17] relates stress to 
strain. The curvature is calculated directly from the geometry 
(Φ) and it is known as the exact value in this computer program 
and the extreme fiber is assumed to be at crush strain. The 
depth of the neutral axis Xu can be incremented throughout the 
program to provide the necessary strain variation. Therefore, 
this depth starts at a small fraction of the total section depth and 
is increased in defined increments until the maximum of the 
column capacity in pure axial compression is reached (P0). The 
curvature is obtained by dividing the stress of the extreme fiber 
(usually 0.003) by the depth of the compression zone 
(Φ=0.003/Xu). Using this curvature and assuming a linear 
strain distribution, it is possible to find the stress in each fiber. 
For each increment, the axial load P, the bending moment of 
the cross section Mcs, and the curvature Φ are calculated. 
Equation (1) is used to find the critical load Pcr which allows 
calculating the bending moment of the column Mcol through (7) 
whose slenderness effects were taken into account. The 
program uses the formulation Mcol/Φ to calculate the effective 
bending stiffness of a slender column whose nonlinearities due 
to P∆ effects and nonlinear behavior of the material are taken 
into account at all times. 

VI. PROPOSED FORMULA OF THE REDUCTION FACTOR OF 
FLEXURAL STIFFNESS 

In practice, the engineer often has difficulties to give a 
reasonable interpretation of the bending stiffness that 
represents the real behavior of slender RC columns. This 
stiffness is easy to calculate with the M/Φ method, which gives 
satisfactory results, but it is not easy to solve manually making 
the use of a computer recommended which in this case makes it 
mandatory but not preferable. In order to simplify the 
estimation of EI bending stiffness under short-term loads, the 
well-known equation EI=α.EcIc+EsIs is obtained. Research 
and expressions have proposed to evaluate the reduction factor 
or the factor of effective stiffness which is of origin can be 
expressed by: 

T = ��+�U�U����     (8) 
Thus, to evaluate the EI flexural stiffness, a simple 

expression is proposed to avoid sophisticated and complicated 
calculations. Multiple linear regression approaches, taking into 
account the simulated theoretical stiffness data, are conducted 
to evaluate the EI expression. Linear regression is chosen as the 
method of analysis because the objective is to develop a simple 
and accurate equation for the reduction factor α. This method is 
very useful in order to determine the effective behavior of 
certain parameters. For example the work in [18] was based on 
the same principle to give the equation of effective ductility as 
a function of shear reinforcement ratio and compressive 
strength of concrete. In this work the used analysis method 
consists in finding an expression describing the reduction factor 
α for different values of concrete strength ƒc, steel yield ƒy, 
percentage of reinforced steel ρ, steel ultimate strength fy, L 
section dimensions, square or equal flanges presented by the 
ratio B1/B, slenderness ratio L/B, the neutral axis angle θ which 



Engineering, Technology & Applied Science Research Vol. 11, No. 3, 2021, 7100-7106 7104 
 

www.etasr.com Hamzaoui & Bouzid: The Proposition of an EI Equation of Square and L–Shaped Slender Reinforced … 

 

presents the loading angle, and the axial loading ratio P/P0, 
where P0 is the strength of the axial compression column under 
pure axial loading. This regression concerns a column of a 
square and equal L section presents the proprieties illustrated in 
Table I, based on the statistical analysis of the theoretical 
values of α for all simulated columns in which P/P0 ranged 
from 0.1 to 1. Over 70000 isolated reinforced concrete columns 
were simulated to generate the effective stiffness factor. Each 
simulated column had a different combination of cross section, 
geometric, and material proprieties. 

TABLE I.  SPECIFIED PROPERTIES OF STUDIED RC COLUMNS 

Proprieties Specified values Number of specified values 
Fc 15 ;60     incr 5mPa 10 
Fy 200;600  incr 100mPa 5 

B1/B 0 ; 0.6      incr 0.1 7 
L/B 10 ; 30     incr   5 5 
Ρ 0.5 ;4%    incr 0.5% 8 
θ 00;  900     incr 100 8 

Total number =10×5×7×5×8×8=70000 columns 
 

Finally the following equation of reduction factor α is 
obtained:  

α = � 0.074 + 0.0003·ƒc+ 0.488· KK� � 0.002· LB    (9) 
The effect of ƒc has an increasing influence on the flexural 

stiffness of the slender column .This was already predicted and 
proven in [19] which presented the effect and impact of this 
strength on the general behavior of corner columns. In addition 
to the axial load level, it has an increasing effect on the EI of 
the column. The expression also indicates that a decrease in the 
EI value occurs as slenderness ration (L/B) ratio is increased. 

VII. VALIDATION 

For the experimentation of the proposed model a test series 
has been realized to determine the variation of the reduction 
factor according to the influence of different parameters on the 
behavior of the RC slender columns. The proposed method has 
been realized according to the expression of the reduction 
factor for an ultimate compressive concrete strain equal to 
0.003 because it gives the best results among all other values of 
strain levels which varied between 0.002 and 0.004. The square 
cross section has dimensions B1=0 and B=300mm and the L 
section B1=100nm and B=300mm. Five methods are considered 
in this work to predict the nominal capacity of the RC columns. 
The curves in Figure 5 show the comparison between the 
proposed method, the codes [1, 2], and the methods proposed 
in [3, 6, 8]. All respective equations are presented in Table II. 
A comparison between the proposed method, and that 
suggested by Bonet et al. in [6] for the L-shaped cross-section 
is presented in Figure 6(a) showing the variation of the 
reduction factor as a function of the relative eccentricity η. This 
coefficient presented by Bonet has been calculated as η=e/4.ic, 
where ic is the radius of gyration of the concrete section with 
respect to the bending axis and e is the first order eccentricity. 
The effective stiffness factor shows a non-linear behavior as a 
function of η. A second comparison with Bonet equation 
presented in Figure 6(b) shows the effect of ρ on the α of the 
slender column with a different angle loading θ. 

(a) 

 

(b) 

 

(c) 

 

(d) 

 
Fig. 5.  Comparison of α with different authors and design codes for a 
square sections: Effect of (a) e/B, (b) ρ, (c) ƒc, and (d) of the axial loading 
ratio on the α of a slender column. 



Engineering, Technology & Applied Science Research Vol. 11, No. 3, 2021, 7100-7106 7105 
 

www.etasr.com Hamzaoui & Bouzid: The Proposition of an EI Equation of Square and L–Shaped Slender Reinforced … 

 

(a) 

 

(b) 

 
Fig. 6.  Comparison between the Bonet equation and  the proposed α for 
the L shaped section. (a) Effect of on α of the slender column, (b) effect of ρ 
on α with a variation of angle loading θ. 

TABLE II.  EI EQUATION COMPARISON 

Author Effective flexural stiffness EI 

Proposed 
EI=αEcIc+EsIs 

α=-0.074+0.0003fc+0.488(P/P0)-0.002(l/h) 

[1] 
EI=0.2EcIc+EsIs/1+βd 

EI=0.4 EcIc/1+βd 

[2] 

EI = Kc · Ec · Ic + Ks · Es · Is 
If  ρl ≥ 0.2%, Ks = 1, Kc=(K1K2)/(1+ϕeff), K1=√fck/20, K2=ν 

λm/170   
If  ρl ≥ 1%, Ks = 0, Kc=0.3/1+0.5ϕeff 

[17] 

EI=(αEcIc/1+ ϕeff)+(EsIs/1+η) 
α=(1.95-0.035.λm-0.25ϕeff)(η-0.2) +((fc/225)+0.11) ̸< 0.1; 

if    η˂0.2 
α = (fc/225+0.45)(η-0.2)+(fc/225+0.11)  ̸< 0.1 ; if η≥0.2 

[6] 

EI=αEcIc+0.8EsIs 
α=0.47-0.35(e/h)(1/1+β(e/h))+0.003(l/h) 
where β = 7.0 for ρrs ≤ 2%  ; and    β = 8.0 

for ρrs> 2%. 

[8] 

EI=αEcIc+EsIs 
α=0.38-0.011(l/h)-1.3(e/h)+0.45(1-(P/P0)

2) 
0.1≤α≤0.85

 

EI = flexural stiffness of compression member, Ec= modulus of elasticity of concrete, Es= modulus of 
elasticity of reinforcement, Ic= moment of inertia of gross concrete section, Is = moment of inertia of 
reinforcement, P = axial load, P0= maximum load capacity, e/h = eccentricity ratio, l/h = geometrical 

slenderness ratio, βd = ratio of the maximum factored axial sustained load to the maximum factored axial 
load associated with the same load combination, ν=relative normal force, λm =mechanical slenderness 

ratio,ϕeff=effective creep ratio, fc =concrete strength, ρl=geometrical reinforcement ratio, 
fck=characteristic compressive cylinder strength of concrete at 28 days. 

VIII. RESULTS AND DISCUSSION 

This paper presents a statistical evaluation of the parameters 
that affect the flexural stiffness of slender columns subjected to 
short term loads with biaxial bending. Five methods [1-3, 6, 8] 
for this comparison were considered to predict effective 
rigidity. In order to verify (9), variations of the e/B, P/P0, ρ, and 

fc were used. These parameters are the most appeared in the 
expressions proposed in the literature. It can be noticed that 
both design codes propose an effective stiffness factor α 
independent of all the parameters (Figure 5), however other 
authors include the dependence of α. In general an equation of 
α is proposed that decreases with e/B and ρ and increases with 
fc and P/P0. The proposed equation confirms this behavior of α. 
The majority of tests considered the square cross section, with 
the exception of Bonet’s expression which considered a cross 
section shape. A very large correspondence can be noticed 
between the Bonet’s equation and the proposed equation 
(Figure 6) despite that Bonet's is nonlinear and the proposed 
expression is perfectly linear. Liu et al. [9] proposed a 
reduction stiffness factor αe taking all the rigidity EI of the 
material, not only the concrete material of RC slender columns. 
The calculation model of columns has an L section of 
600×600×200mm. The cross section details and dimensions of 
the specimen are shown in the Table III. The stiffness reduction 
factors calculated with Liu formulation and the flexural 
stiffness EI corresponding for each axial load level are shown 
in Table IV. A comparison of the results of Liu with the 
flexural stiffness calculated with the proposed equation shows a 
good degree of accuracy between the results of Liu and the 
proposed equation (Figure 7). An average ratio of 0.983 was 
achieved. 

TABLE III.  GEOMETRY AND DIMENSIOS OF THE SPECIMENS IN [9] 

Column characteristic Values 
B1 (mm) 400 
B (mm) 600 
Dc (mm) 30 
ρ (%) 2.512 

fc (mpa) 14.3 
fy (mpa) 360 
L (mm) 2900 
θ (°) 45 

Ec (mPa) 30000 
Es (mPa) 200000 

EI=EcIc+EsIs (KN.m
2
) 77686 

 

 
Fig. 7.  Comparison of the proposed EI and the one of [9]. 

TABLE IV.  VERIFICATION OF THE PROPOSED EQUATION WITH [9] 

Axial load 

level 
αe 

EIliu=αe.EI 

(KN.m
2
) 

EIprop 
(KN.m

2
) 

EIprop/EILiu 

0.10 0.4915 38182 31587 0.83 
0.15 0.4782 37149 32662 0.87 
0.20 0.4630 35968 33787 0.94 
0.25 0.4464 34679 34790 1.00 
0.30 0.4281 33257 35914 1.07 
0.35 0.3996 31043 36981 1.19 

Average  0.983 



Engineering, Technology & Applied Science Research Vol. 11, No. 3, 2021, 7100-7106 7106 
 

www.etasr.com Hamzaoui & Bouzid: The Proposition of an EI Equation of Square and L–Shaped Slender Reinforced … 

 

IX. CONCLUSION 

An iterative numerical procedure for the stiffness analysis 
and design of slender RC columns with square and L-shaped 
cross-sections under biaxial bending and axial loading using 
the unconfined Kent and Park stress is presented in this paper. 
The computational procedure takes into account the nonlinear 
behavior of the materials and includes the second order effects 
due to the additional eccentricity of the axial loading as applied 
by the moment-amplification method. The new equation 
suggested in this work is among the simplest proposed 
equations and its implementation is very easy to solve by 
manual calculation in a relatively short time. The concrete 
strength, the level of axial load, and the column slender ratio 
have been identified as the most important factors affecting the 
reduction coefficients and the effective bending stiffness EI. 
The formulation of the proposed method has been tested by 
comparison with other reported formulations. The results show 
that a good degree of accuracy and agreement has been 
achieved. 

REFERENCES 
[1] ACI 318-08: Building Code Requirements for Structural Concrete and 

Commentary. ACI, 2008. 

[2] EN 1992-1-1:2004 - Eurocode 2: Design of concrete structures - Part 1-
1: General rules and rules for. Brussels, Belgium: CEN, 2004. 

[3] C. S. Whitney and E. Cohen, "Guide for Ultimate Strength Design of 
Reinforced Concrete," Journal Proceedings, vol. 53, no. 11, pp. 455–
490, Nov. 1956, https://doi.org/10.14359/11524. 

[4] B. Bresler, "Design Criteria for Reinforced Concrete Columns under 
Axial Load and Biaxial Bending," ACI Journal, Proceedings, vol. 10, 
no. 6, pp. 481–490, Jan. 2005. 

[5] M. Khuntia and S. K. Ghosh, "Flexural Stiffness of Reinforced Concrete 
Columns and Beams: Experimental Verification," ACI Structural 
Journal, vol. 10, no. 3, May 2004. 

[6] J. L. Bonet, M. L. Romero, and P. F. Miguel, "Effective flexural 
stiffness of slender reinforced concrete columns under axial forces and 
biaxial bending," Engineering Structures, vol. 33, no. 3, pp. 881–893, 
Mar. 2011, https://doi.org/10.1016/j.engstruct.2010.12.009. 

[7] M. Resheidat, M. Ghanma, C. Sutton, and W.-F. Chen, "Flexural rigidity 
of biaxially loaded reinforced concrete rectangular column sections," 
Computers & Structures, vol. 55, no. 4, pp. 601–614, May 1995, 
https://doi.org/10.1016/0045-7949(94)00493-M. 

[8] S. Z. Al-Sarraf, I. A. S. Al-Shaarbaf, B. R. Al-Bakri, and K. F. Sarsam, 
"Flexural Rigidity of Slender RC Columns," Engineering and 
Technology Journal, vol. 27, no. 1, pp. 96–116, 2009. 

[9] J. Liu, Y. Zhang, and P. Luo, "Flexural Stiffness Reduction Factor of 
Reinforced Concrete Column with Equal L-Shaped Section," in The 
Twelfth COTA International Conference of Transportation 
Professionals, Dec. 2012, pp. 3187–3193, https://doi.org/10.1061/ 
9780784412442.325. 

[10] X. Feng, M. Shen, C. Sun, J. Chen, and P. Luo, "Research on Flexural 
Stiffness Reduction Factor of Reinforced Concrete Column with 
Equiaxial + Shaped Section," Procedia - Social and Behavioral 
Sciences, vol. 96, pp. 168–174, Nov. 2013, https://doi.org/10.1016/ 
j.sbspro.2013.08.022. 

[11] Ö. Avşar, B. Bayhan, and A. Yakut, "Effective flexural rigidities for 
ordinary reinforced concrete columns and beams," The Structural 
Design of Tall and Special Buildings, vol. 23, no. 6, pp. 463–482, 2014, 
https://doi.org/10.1002/tal.1056. 

[12] N. Caglar, A. Demir, H. Ozturk, and A. Akkaya, "A simple formulation 
for effective flexural stiffness of circular reinforced concrete columns," 
Engineering Applications of Artificial Intelligence, vol. 38, pp. 79–87, 
Feb. 2015, https://doi.org/10.1016/j.engappai.2014.10.011. 

[13] L. N. Ramamurthy and T. A. H. Khan, "L‐Shaped Column Design for 
Biaxial Eccentricity," Journal of Structural Engineering, vol. 109, no. 8, 
pp. 1903–1917, Aug. 1983, https://doi.org/10.1061/(ASCE)0733-
9445(1983)109:8(1903). 

[14] Mallikarjuna and P. Mahadevappa, "Computer aided analysis of 
reinforced concrete columns subjected to axial compression and 
bending—I L-shaped sections," Computers & Structures, vol. 44, no. 5, 
pp. 1121–1138, Aug. 1992, https://doi.org/10.1016/0045-7949(92) 
90333-U. 

[15] W. H. Taso and C.-T. T. Hsu, "Behaviour of biaxially loaded square and 
L-shaped slender reinforced concrete columns," Magazine of Concrete 
Research, vol. 46, no. 169, pp. 257–267, Dec. 1994, https://doi.org/ 
10.1680/macr.1994.46.169.257. 

[16] C. Dundar, S. Tokgoz, A. K. Tanrikulu, and T. Baran, "Behaviour of 
reinforced and concrete-encased composite columns subjected to biaxial 
bending and axial load," Building and Environment, vol. 43, no. 6, pp. 
1109–1120, Jun. 2008, https://doi.org/10.1016/j.buildenv.2007.02.010. 

[17] D. C. Kent and R. Park, "Flexural Members with Confined Concrete," 
Journal of the Structural Division, vol. 97, no. 7, pp. 1969–1990, 1971. 

[18] H. Wang, Y. Zhang, and S. Qin, "A Study on Ductility of Prestressed 
Concrete Pier Based on Response Surface Methodology," Engineering, 
Technology & Applied Science Research, vol. 6, no. 6, pp. 1253–1257, 
Dec. 2016, https://doi.org/10.48084/etasr.855. 

[19] A. Ali, Z. Soomro, S. Iqbal, N. Bhatti, and A. F. Abro, "Prediction of 
Corner Columns’ Load Capacity Using Composite Material Analogy," 
Engineering, Technology & Applied Science Research, vol. 8, no. 2, pp. 
2745–2749, Apr. 2018, https://doi.org/10.48084/etasr.1879.