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Engineering, Technology & Applied Science Research Vol. 13, No. 4, 2023, 11230-11234 11230  
 

www.etasr.com Kumar & Kumar: Bending Analysis of Steel-Concrete Composite Beams with Porosity 

 

Bending Analysis of Steel-Concrete Composite 

Beams with Porosity 
 

Prashant Kumar 

Department of Civil Engineering, National Institute of Technology Patna, India  

pk0895300@gmail.com (corresponding author) 

 

Ajay Kumar 

Department of Civil Engineering, National Institute of Technology Delhi, India 

sajaydce@gmail.com 

Received: 17 May 2023 | Revised: 26 May 2023 | Accepted: 28 May 2023 

Licensed under a CC-BY 4.0 license | Copyright (c) by the authors | DOI: https://doi.org/10.48084/etasr.6050 

ABSTRACT 

This study investigated the bending behavior of a simply supported pervious composite beam using the 

finite element method based on cubic order beam theory. The cubic order axial displacement was assumed 

and solved using shear stress-free conditions on the extreme surfaces of the composite beam. For 

simplicity, this study considered one-dimensional axial displacement. Three nodded finite elements were 

assumed, and each node had eight unknowns. Shear locking was eliminated in the present model by 

numerical integration of the stiffness matrix. A uniform porosity distribution was implemented in the 

upper layer of the composite beam. The damping ratio was also considered for the bending analysis. The 

accuracy of the present model makes it robust for the analysis of composite beams. 

Keywords- porous concrete; finite element method; bending response; shear interface; shear stress 

I. INTRODUCTION  

Lightweight materials with higher strength are always the 
first choice of engineers. Composite materials meet all the 
criteria for complex construction. Over the past forty years, 
high-temperature resistant, more flexible, high-strength, and 
cost-effective composite materials have been created for civil 
engineering, aerospace technology, and the formation of 
household materials. Other applications for these materials 
include electrical appliances, energy transformation, 
biomedical engineering, optics, etc. [1-2]. The advantages of 
these structures are their high strength, improved material 
properties, and lower weight. 

Functionally graded composites can fulfill current and 
distinctive functions that traditional composite materials cannot 
[3]. These are sophisticated composite materials with a 
microscopically inhomogeneous morphology that are created 
using powder metallurgy processes from a combination of 
metal and ceramic. There are fewer stress discontinuities in FG 
materials because the material characteristics change gradually 
and smoothly across the thickness, and interlaminar stress 
discontinuities are removed. Knowing the bending, buckling, 
and free vibration frequencies is vital because structural 
components made of composite laminates, such as aircraft 
wings and panels, are regularly subjected to mechanical loads, 
critical buckling, shocks, and vibrations [4]. 

The static and dynamic analysis of beams has an exciting 
history for its development. Various beam theories have been 

developed to get precise and accurate results. The Euler-
Bernoulli beam theory is used to analyze the beam [5]. As the 
Euler-Bernoulli beam theory is not enough to analyze a thin 
beam, the Timoshenko Beam Theory (TBT) has become 
popular [6-7]. High-order Beam Theory (HBT) is more 
accurate than TBT. Today, HBT is the first choice for the 
analysis of beam behavior [8-10]. 

The strength of a composite material also depends on the 
type and strength of shear studs used to hold the two elements, 
so investigation of shear stiffness parameters and inter-layer 
slip is essential [11-12]. In [13-15], the impact of porosity 
distribution on the vibrational behavior was investigated. In 
[16], the dynamic behavior of a beam was examined under 
various loading conditions using the state-space technique. 

This study investigated the impact of porosity distribution 
on the bending of simply supported two-layered sandwich 
steel-concrete composite beams. The one-dimensional C

0
 finite 

element method was incorporated to model this problem and 
produce accurate results. The effect of the damping ratio was 
also calculated. To the best of our knowledge, the present work 
has not been used in any previous bending analysis study of 
composite beams. The novelty of the present work is the 
uniform distribution of porosity provided along the transverse 
direction of the upper element of the two-layered sandwich 
composite beam. The formation of voids at the time of 
modeling is a major issue for every engineer. So, its analysis is 
useful for future work. 



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www.etasr.com Kumar & Kumar: Bending Analysis of Steel-Concrete Composite Beams with Porosity 

 

II. FORMULATION 

A. Porosity Distribution in the Upper Element 

Porosity is incubated in the concrete element changing the 
material properties of the simply supported composite beam, 
such as Young’s modulus (E), shear modulus (G), and density 
of the concrete element. 

� �� ��� = ���	
 �1 − 
� ��� ��� = ���	
 �1 − 
� ��� ��� = ���	
 �1 − ��� 
�    (1) 
where E is Young's modulus, G is the shear modulus, ρ is mass 
density, and k is the function of porosity.  

B. Mathematical Formulation 

The high-order beam theory was used to solve the cubic 
order axial displacement equation. Figure 1 shows the shear 
interface of the beam. 

 

 
Fig. 1.  Shear flexible interface of the composite beam. 

The cubic order axial displacement equation for the upper 
and lower layer of the beam is written as: 

��� � = �1 −�� �� � �� ������ �� �� �� ��  (2) 
where u is the axial displacement equation along the centroidal 
axis, θ is the bending rotation, ζ and ξ are higher order terms, 
and I = c, s. The transverse equation of both layers is expressed 
as: 

�� ��, �� ,  � = �! ��, �! ,  � = ���� = � (3) 
The partial shear interaction between two layers of the 

composite beam is modeled by taking distributed shear springs 
at their interface. Interfacial stiffness and shear slip at the 
interface are used to determine the shear stress at the interface. 
Interfacial slip (s) is calculated as given below, where uc and us 
are the axial displacements of the upper and lower layer at the 
interface. 

" = ��′� − �′! �    (4) 
The axial displacements in (2) are higher-order equations 

that are concerned with the warping of the transverse sections 
but do not describe the commonly used displacement 
parameters adopted in beam theories. So, higher-order terms 
are eliminated using shear stress-free conditions at the extreme 
surfaces of the composite beam. The shear stress can be 
calculated at any point in the upper layer using (2) and (3). 

�#� � = ��� ��$��  
$� = %&'(&)( + &+&
 ,  

           = �−1 2�� 3�� � 1� /�� �� �� &+&
 0�   (5) 
where γc is the shear strain and Gc is the shear modulus of the 
porous concrete layer. 

1� = %1 − 23�)456(4 7 + 238)956(9 7,,   ;� = %23�)456(4 7 − 238)956(9 7,,   <� = %−� − 2=)456( 7 + 23�)956(4 7,,   >� = − %2�)456( 7 + 2 =)956(47,   1! = %1 − 23�)456?4 7 − 238)956?9 7,,   (6)  ;! = %23�)456?4 7 + 238)956?9 7,,   <! = %−� + 2=)456? 7 + 23�)956?4 7,,   >! = %2�)456? 7 − 2 =)956?47,, and @ = A+A
  
The shear stress-free conditions are applied at the top and 

bottom surface of the beam to find out the displacement 
equation: 

��� � = B1� ;� C� >� D���� �� �� @��  (7) 
The normal stress and normal strain at any point beam are 

calculated using the following equation: 

EF̄HI = �>J �I EK̄HI     (8) 
where σj is the normal stress, τj is the shear stress, Ej is the 
modulus of elasticity, Gj is the shear modulus, εj is the normal 
strain, and γj is the shear strain of the j

th
 layer, and j=c 

represents the upper layer and j=s represents the lower layer of 
the two material composite beams. 

EK̄HI = %KI$I , = �
&'L&
&'L&)L + &+&
 M = �N�I EKHI   (9) 

The strain energy can be given as a function of stress and 
strain using (8) and (9): 

� = 3� O�EKH̅�� EFQH� + EKH̅!� EFQH!�R1R�  
          =

3� O�EKH̅��  E>H� EKH� + EKH̅!� E>H! EKH! �R� (10) 
where: 

�>�� = O�N����>S���N�� R1�  and �>�! = O�N�!� �>S�! �N�! R1!  
Numerical integration was used to evaluate the cross-

section rigidity matrix �>�� and �>�!. The stored strain energy 
was calculated using (4) and distributed shear springs stiffness 
ks. 

T = 3� O U! "�R� = 3� O U! ��� − �! �R�  (11) 
 



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www.etasr.com Kumar & Kumar: Bending Analysis of Steel-Concrete Composite Beams with Porosity 

 

C. Finite Element Formulation 

Three nodded iso-parametric C
0
 elements were selected. 

This theory assumed eight nodal degrees of freedom to solve 
the present problem using a one-dimensional finite element 
approximation. The unknown nodal displacement vector {d} 
on the middle surface of a typical element is given by: 

ERH = ∑ W� ��, ��. ER� H��Y3    (12) 
where ψi is the shape function. The generalized strain vector as 
a function of the nodal displacement vector {d} can be found 
by: 

EKHI = ∑ �Z� �ER� H��Y3     (13) 
where [Pi] is the interpolation function differential operator 
matrix. The strain vectors were calculated using (11) and (13) 
as a function of the stiffness matrix [K

l
]. 

T = 3� ERH� �[ \ �ERH    (14) 
where: 

�[ \ � = O��Z��� �>���ZH� + �Z�!� �>�! �ZH! �R� (15) 
Similarly, the interfacial stiffness and the stiffness due to 

the penalty function approach were calculated. An element's 
mass matrix and its geometric stiffness matrix can be found in 
the same way as the element stiffness matrix calculated above. 
Equations (3) and (12) were used to find the displacement 
component vector at a point in the beam layer as follows: 

] Ĵ_ = %�I�, = B Ì DE^H = B Ì D�a�ERH  (16) 
where Fj is a matrix of order 2×8 which contains the coefficient 
of displacement component expressed in (16), and [X] is a 
shape function matrix of order 8×24. The consistent mass 
matrix of three nodded elements is: 

�b� = ∭�a�� 2B Ì D� �I B Ì DR 7 �a�R�. R� (17) 
Free vibration analysis was performed to determine the 

fundamental natural frequency from: 

B[d − e�]bd_DEfH = 0   (18) 
where ω is the vibration frequency and λ is the eigenvector. 
The dynamic response of the composite beam was analyzed 
using the following equation: 

�[d <d bd � hRRiRj k = E l̀ H   (19) 
where, d, Ri , and Rj  are the displacement, velocity, and 
acceleration vectors. 

III. RESULT AND DISCOUSSION 

A C
0
 finite element method was used to solve the cubic-

order axial displacement equation. These equations were solved 
using FORTRAN. 

A. Comparison Study 

To validate the results, the mid-span deflection of a two-
layered simply supported composite beam was compared with 
a T-cross section, using the results of [16], which used the 
state-step approach to tackle this problem. Table I shows the 
mid-span deflection of a simply supported composite beam, 
which varies with its interfacial stiffness. The percentage error 
mentioned in Table I shows the accuracy of the model. 

TABLE I.  VARIATION OF MID-SPAN DEFLECTION WITH 
INTERLAYER SHEAR STIFFNESS. 

S.N. 
Interfacial shear 

stiffness (MPa) 

Mid-span deflection (mm) 

Present [16] 

1 10
-16

 9.7741 9.7224 (0.529) 

2 2 8.2072 8.1957 (0.140) 
 

B. Steel-Concrete Porous Beam 

This study considered a simply supported composite beam 
having a porous upper layer. Figure 2 shows the cross-section 
of the beam. A 15 m long simply supported beam was made up 
of a porous concrete slab and a steel joist. The dimensions of 
the concrete slab were 2.25×0.15 m

2
. The dimensions of the 

flange and web of the I-shaped steel joist were 0.1780×013 m
2 

and 0.380×0078 m
2
, respectively. The material properties of 

the upper layer were Ec = 13.55 GPa, Gc = 6.775 GPa, and ρc = 
2396.45 Kg/m

3
. Similarly, the material properties of the upper 

layer were Es = 200 GPa, Gs = 100 GPa, and ρs = 7948.89 
Kg/m

3
. A 100 KN moving point load is taken for the analysis, 

moving at a velocity of 16.67 m/s. The interfacial shear 
stiffness (ks) of the partially composite beam was considered as 
100 MPa. 

 

 

Fig. 2.  Cross section of the steel-concrete composite beam. 

C. Porosity and Damping Effect on Mid-span Deflection 

The porosity of the upper layer ranged from 0 to 0.5. The 
effect of damping ratios on mid-span deflection was observed. 
This study used a variation of damping ratios from 0% to 10%. 
Figures 3 and 4 show that the deflection of the partial and 
complete composite beams increased with increasing the 
porosity of the concrete elements.  



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www.etasr.com Kumar & Kumar: Bending Analysis of Steel-Concrete Composite Beams with Porosity 

 

 
Fig. 3.  Variation of deflection with concrete porosity of partial composite 
beam. 

 
Fig. 4.  Variation of deflection with concrete porosity of complete 
composite beam. 

 
Fig. 5.  Variation of deflection of partial composite beam with damping 
ratio. 

 
Fig. 6.  Variation of deflection of complete composite beam with damping 
ratio. 

TABLE II.  VARIATION OF DEFLECTION WITH POROSITY AND DAMPING RATIO. 

Porosity (e) 

Deflection (mm) 

Partial Composite Full Composite 

D=0% D=2% D=5% D=10% D=0% D=2% D=5% D=10% 
0.00 74.789 73.548 71.974 69.931 54.254 53.248 52.138 50.865 

0.10 75.898 74.634 73.028 70.936 55.217 54.310 53.170 51.852 

0.20 77.043 75.758 74.121 71.982 56.383 55.454 54.284 52.920 

0.30 78.239 76.937 75.270 73.085 57.661 56.709 55.506 54.096 

0.40 79.512 78.196 76.499 74.269 59.092 58.114 56.877 55.420 

0.50 80.898 79.574 77.847 75.573 60.740 59.733 58.457 56.950 

 
The proposed trend of deflection variation satisfied the 

mathematical formulation above. The magnitude of the 
material properties of the concrete element decreases with an 
increase in porosity, and the stiffness of the element was 
directly proportional to the material properties. Figures 5 and 6 
show that the deflection of the partial and complete composite 
beams decreased with increasing damping ratio, which satisfied 
the mathematical formulation since the damping effect reduces 
the load effect. Figures 3-6 show that beam deflection varies 
when the moving load passes from the left to the right support. 
Table II satisfies the above pattern of beam deflection. 

This study introduced an innovative investigation of the 
effect of porosity on the bending behavior of a simply 
supported sandwich composite beam. Porosity in concrete has 
long been a concern for civil engineers due to its detrimental 
impact on the strength of concrete elements. In the field of 
construction, steel-concrete beams are extensively used. 
Therefore, this study is of significant importance as it provides 
valuable guidance for future composite material research.   

IV. CONCLUSIONS 

A C
0
 finite element method was used to solve the cubic-

order one-dimensional axial displacement equation. The cubic 
order shear deformation theory was used to analyze the 
bending behavior of a simply supported composite beam. 

 This study took into account the uniform distribution of 
porosity in the upper layer of the composite, proposing a 
novel relationship between porosity, damping, and bending. 

 The deflection of the simply supported composite beam 
increased with an increase in the porosity of the upper layer 
of the partial and complete composite beams. 

 The porosity of the upper layer of the composite beam 
increased from 0.0 to 0.50, and then the beam deflection 
increased up to 8% in the case of the partial composite and 
12% in the case of the complete composite. 



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 The beam deflection in both the partial and complete 
composite cases decreased with an increase in damping. 

 The beam deflection decreased by 6.5% in the partial 
composite and 6% in the complete composite, while the 
damping increased from 0 to 10 %. 

 The presented model was more accurate in the prediction of 
the banding of the composite beam. 

ACKNOWLEDGMENT 

The authors would like to thank the National Institute of 
Technology Patna (India) for its financial support. 

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