59
Journal of Sustainable Architecture and Civil Engineering 2014/4/9

JSACE 4/9

Analysis of Cracking 
Moment of Flexural 
Elements Reinforced 
by Steel and FRP 
Reinforcement

Received  
2014/06/30

Accepted after  
revision 
2014/10/02 

Journal of Sustainable 
Architecture and Civil Engineering
Vol. 4 / No. 9 / 2014
pp. 59-66
DOI 10.5755/j01.sace.9.4.7428 
© Kaunas University of Technology

Mindaugas Augonis*, Mindaugas Kasiulevičius, Rėda Bistrickaitė
Kaunas University of Technology, Faculty of Civil Engineering and  
Architecture, Department of building structures  
Studentu st. 48, LT-51367 Kaunas, Lithuania

*Corresponding author: mindaugas.augonis@ktu.lt

Analysis of Cracking 
Moment of Flexural Elements 
Reinforced by Steel and FRP 
Reinforcement 

http://dx.doi.org/10.5755/j01.sace.9.4.7428

Robertas Balevičius
Vilnius Gediminas Technical University, Faculty of Civil Engineering,  
Department of reinforced concrete and masonry structures  
Sauletekio al. 11, LT-10223 Vilnius, Lithuania

The application of FRP reinforcement in concrete structures is quite popular nowadays. In this article 
the cracking moment of flexural concrete structures reinforced by steel and FRP reinforcement bars is 
investigated. There is estimated that the behaviour of tensile concrete is nonlinear when crack opens. 
The nonlinearity was estimated by two different cases. In the first one the coefficient of elastic and plastic 
strain of tensile concrete were applied and in the second the stress-strain relationship were expressed 
by polynomial function. That function prepared according to compressed concrete stress-strain diagram 
described in design standard EN 1992-1-1. Those two different stress-strain relationships were used for 
cracking moment calculation. Also the cracking moment calculations were performed using mentioned 
standard EN and Technical building code STR 2.05.05:2005. In all cases the calculations were performed 
in assumption that the behaviour of compressed concrete is elastic when crack opens. The obtained 
results were analysed and presented base conclusions.

KEYWORDS: FRP reinforcement, cracking moment, stress-strain relationship, nonlinear behaviour, 
tensile concrete.

Introduction
The steel as base reinforcement in reinforced concrete structures is used many years, however it 
properties satisfy not all requirements to reinforcement, especially the requirements of resistance 
to corrosion. For this reason the durability of concrete reinforced by steel reinforcement is not so 
high in aggressive condition to corrosion (Abdala 2002, Al-Sunna, 2012, Toutanji 2003).

Nowadays the low durability problem could be solving applying the new type of reinforcement in 
concrete structures i.e. the FRP (fiber reinforced polymer) reinforcement which has higher tensile 
strength and higher resistance to corrosion (Barrs el.al. 2012, Barris et.al 2009, Ashour 2006). 

Such type of reinforcement could be used in concrete structures as reinforcement bars, meshes or 
reinforcing cages. The application of FRP reinforcement in concrete structures of buildings, engineering 
works or underground works increase the durability of them and resistance to various damages.



Journal of Sustainable Architecture and Civil Engineering 2014/4/9
60

However such type of reinforcement has lower elasticity and worse anchorage with concrete. The 
modulus elasticity of FRP reinforcement is several or more times less than modulus of steel. For 
this reason the stiffness and cracking problem arises to the flexural structures because into the 
cracked section the tensile forces are permitted only by tensile reinforcement. In this case into the 
concrete structures reinforced by FRP reinforcement the cracking moment will be reached earlier 
and the deflection will be higher in comparing with simply reinforced structures. So, it could be 
concluded that high tensile strength of FRP reinforcement in flexural structures could not be exhaust 
because of too high deformability and application so not effective economically and structurally.

In that case FRP reinforcement could be used with steel reinforcement together in order to increase 
the durability of structures. Because of different elasticity the stress in steel reinforcement will be 
higher than in FRP and the strength of structures significantly depend on amount and strength of 
steel reinforcement. 

But amount of steel reinforcement has no such significantly effect on cracking moment because 
it more depends on the concrete tensile strength and dimensions of cross section. For example 
according to principles of design standard EN 1992-1-1:2004 the amount of reinforcement has no 
effect on cracking moment (Mosley 2007) 

Also the FRP reinforcement is more brittle in compare with steel. So, the application of such 
reinforcement into the bridges structures could be limited.

The FRP reinforcement bar consists from the material (glass or basalt) strands of fibers connected 
by polymers (http://www.schoeck-combar.com, http://galen.su). Such structure of fibers 
materials describes the high strength that could be several or more times higher than strength of 
solid cross section materials. The FRP reinforcement has high strength in longitudinal direction. 
Because of high resistance to alkalinity medium in concrete the FRP reinforcement durability 
is about 100 year and more. It should be mentioned that the bond between concrete and steel 
reinforcement is better than FRP reinforcement but in this article it is not investigated.

The reinforcement could be in different types. The most popular is GFRP (Glass fiber reinforced 
polymer) and Basalt fiber reinforcement, but GFRP reinforcement more use in concrete structures 
because of less price. 

The main properties of GFRP and Basalt fiber reinforcement are presented in table 1. 

Review 
of main 

properties 
of FRP 

reinforcement

Characteristics of 
materials 

Steel 
reinforcement 

S400 class

GFRP 
reinforcement

„Schőck ComBAR“

GFRP 
reinforcement
„ROCKBAR“

Bazalt fiber 
reinforcement
„ROCKBAR“

Tensile strength MPa 360 1200 1000 1200

Density, t/m3 7 2,2 2,0 2,0

Corrosion Corrosion 
materals

No corrosion material satisfy to the first group of chemical 
resistance 

Length of reinforcement 
product, m

Bars of 6-12m 
length

Bars of 10-14m 
length

Bars until 12m length

Elastic Modulus, N/mm2 200000 60000 45000 50000-55000

Cover, mm According to 
codes, STR

Ø+10mm,
For precast reinforced concrete Ø+5mm

Table 1
The main propeties 

of steel and FRP 
reinforcement (http://

www.schoeck-combar.
com, http://galen.su)



61
Journal of Sustainable Architecture and Civil Engineering 2014/4/9

In this investigation the concrete stress-strain relationship were expressed in two cases.

1 case. Analysing the elastic plastic behaviour of tensile concrete should be note that 
c c

σ ε−  
relationship could have quite significant influence to the crack moment crcM . The EC2 standard 
presents 

c c
σ ε−  relation for compressed concrete but not describe the tensile concrete. If to 

assume that ultimate strain of tensile concrete correspond the ultimate tensile stress ( )ctmf , and 
the form of tensile concrete 

c c
σ ε−  curve correspond to the form of compressed concrete until 

the peak of stress, it is possible to apply the EC2 standard expression of stress-strain relationship. 

In all figures the font of text are changed in “times new roman” and in fig. 1 and fig. 2 are presented 
units of tensile stress

So, if to calculate the tensile stress 

Basalt fiber reinforcement, but GFRP reinforcement more 

use in concrete structures because of less price.  

The main properties of GFRP and Basalt fiber 

reinforcement are presented in table 1.  

 

 

1 lentelė. The main propeties of steel and FRP reinforcement (http://www.schoeck-combar.com, http://galen.su) 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3. The tensile concrete stress-strain relationship  
 

In this investigation the concrete stress-strain 

relationship were expressed in two cases. 

1 case. Analysing the elastic plastic behaviour of 

tensile concrete should be note that 
c

c

σ ε−  relationship 

could have quite significant influence to the crack 

moment
crc

M . The EC2 standard presents 
c

c

σ ε−  relation 

for compressed concrete but not describe the tensile 

concrete. If to assume that ultimate strain of tensile concrete 

correspond the ultimate tensile stress ( )
ctm

f , and the form of 

tensile concrete 
c

c

σ ε−  curve correspond to the form of 

compressed concrete until the peak of stress, it is possible to 

apply the EC2 standard expression of stress-strain 

relationship.  

In all figures the font of text are changed in “times new 

roman” and in fig. 1 and fig. 2 are presented units of tensile 

stress 

 

Fig. 1. The tensile concrete stress-strain relationship 

when the EC2 standard expression is used. Concrete class 

C40/50. 

 

So, if to calculate the tensile stress 
ct

σ  directly 

according to the EC2 standard expression rewriting the 
ctm

f  

instead of 
cm

f  only, we will obtain: 

 

( )

2

1 2

ct ctm

k

f

k

η η

σ

η

⎛ ⎞−

= ⎜ ⎟
⎜ ⎟

+ −
⎝ ⎠

,     (1) 

where 
1

1 05
cm ct, ctm

k , E / fε= , 
0 31

1
0 0007

ctm

.

ct,
, fε = . 

According such way drawing up the 
c

c

σ ε−  

relationship of tensile concrete we will obtain the curve that 

form at all will not be similar to the form of curve of 

compressed concrete (fig. 2). In this case the curves forms 

not correspond each other because of different values of the 

ratios 

1

cm

c,

f

ε

 and 

1

ctm

ct,

f

ε

 . For example to the concrete 

class 
20

25

C  the numerical values is: 28
cm

f MPa= ; 

1
0 001967

c,

,ε = , 2 21
ctm

f , MPa= , 
4

1
8 95 10

ct,
,ε

−

= ⋅ . Then 

we obtain that the ratio of 

1

14234 875
cm

c,

f
,

ε

=  and 

1

2469 273
ctm

ct,

f
,

ε

=  differ more than six times. Also, 

according such case the secant modulus of compressed 

concrete at the peak stress ( )
ct ctm

fσ =  was 5,765 times 

bigger than to the tensile concrete. If to express the ultimate 

elastic plastic coefficient at the stress peak it would obtained 

the 0 475,λ =  to the compressed concrete and only the 

0 0842
ct

,λ =  to the tensile concrete. If to assume the one 

and the same elastic plastic coefficient to the compressed 

and tensile concrete, the strain at the peak stress 
1ct,

ε  should 

be expressed with different factor as 
0 31

1
0 000121

ctm

,

ct, f
,ε =  . 

This factor, of course, depend on the 
ctm

f , so, to the 

Characteristics of 

materials  

Steel 

reinforcement 

S400 class 

GFRP 

reinforcement 

„Schőck 

ComBAR“ 

GFRP 

reinforcement 

„ROCKBAR“ 

Bazalt fiber 

reinforcement 

„ROCKBAR“ 

Tensile strength 

MPa 

360 1200 1000 1200 

Density, t/m
3 

7 2,2 2,0 2,0 

Corrosion Corrosion 

materals 

No corrosion material satisfy to the first group of chemical 

resistance  

 

Length of 

reinforcement 

product, m 

Bars of 6-12m 

length 

Bars of 10-14m 

length 

 

Bars until 12m length 

 

Elastic Modulus, 

N/mm
2 

200000 60000 45000 50000-55000 

Cover, mm According to 

codes, STR 

Ø+10mm, 

For precast reinforced concrete Ø+5mm 

 

 directly according to the EC2 standard expression rewriting 
the ctmf  instead of cmf  only, we will obtain:

The tensile 
concrete 
stress-strain 
relationship 

According such way drawing up the 
c c

σ ε−  relationship of tensile concrete we will obtain the 
curve that form at all will not be similar to the form of curve of compressed concrete (fig. 2). In 
this case the curves forms not correspond each other because of different values of the ratios 

1
cm

c ,

f
ε  and 1

ctm
ct ,

f
ε . For example to the concrete class 

20
25C  the numerical values is: 

28cmf MPa= ; 1 0 001967c , ,ε = , 2 21ctmf , MPa= , 
4

1 8 95 10ct , ,ε
−= ⋅ . Then we obtain that the ratio

of 
1

14234 875cm
c ,

f ,ε =  and 1
2469 273ctm

ct ,

f ,ε =  differ more than six times. Also, according

where: 

(1)

Basalt fiber reinforcement, but GFRP reinforcement more 

use in concrete structures because of less price.  

The main properties of GFRP and Basalt fiber 

reinforcement are presented in table 1.  

 

 

1 lentelė. The main propeties of steel and FRP reinforcement (http://www.schoeck-combar.com, http://galen.su) 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3. The tensile concrete stress-strain relationship  
 

In this investigation the concrete stress-strain 

relationship were expressed in two cases. 

1 case. Analysing the elastic plastic behaviour of 

tensile concrete should be note that 
c

c

σ ε−  relationship 

could have quite significant influence to the crack 

moment
crc

M . The EC2 standard presents 
c

c

σ ε−  relation 

for compressed concrete but not describe the tensile 

concrete. If to assume that ultimate strain of tensile concrete 

correspond the ultimate tensile stress ( )
ctm

f , and the form of 

tensile concrete 
c

c

σ ε−  curve correspond to the form of 

compressed concrete until the peak of stress, it is possible to 

apply the EC2 standard expression of stress-strain 

relationship.  

In all figures the font of text are changed in “times new 

roman” and in fig. 1 and fig. 2 are presented units of tensile 

stress 

 

Fig. 1. The tensile concrete stress-strain relationship 

when the EC2 standard expression is used. Concrete class 

C40/50. 

 

So, if to calculate the tensile stress 
ct

σ  directly 

according to the EC2 standard expression rewriting the 
ctm

f  

instead of 
cm

f  only, we will obtain: 

 

( )

2

1 2

ct ctm

k

f

k

η η

σ

η

⎛ ⎞−

= ⎜ ⎟
⎜ ⎟

+ −
⎝ ⎠

,     (1) 

where 
1

1 05
cm ct, ctm

k , E / fε= , 
0 31

1
0 0007

ctm

.

ct,
, fε = . 

According such way drawing up the 
c

c

σ ε−  

relationship of tensile concrete we will obtain the curve that 

form at all will not be similar to the form of curve of 

compressed concrete (fig. 2). In this case the curves forms 

not correspond each other because of different values of the 

ratios 

1

cm

c,

f

ε

 and 

1

ctm

ct,

f

ε

 . For example to the concrete 

class 
20

25

C  the numerical values is: 28
cm

f MPa= ; 

1
0 001967

c,

,ε = , 2 21
ctm

f , MPa= , 
4

1
8 95 10

ct,
,ε

−

= ⋅ . Then 

we obtain that the ratio of 

1

14234 875
cm

c,

f
,

ε

=  and 

1

2469 273
ctm

ct,

f
,

ε

=  differ more than six times. Also, 

according such case the secant modulus of compressed 

concrete at the peak stress ( )
ct ctm

fσ =  was 5,765 times 

bigger than to the tensile concrete. If to express the ultimate 

elastic plastic coefficient at the stress peak it would obtained 

the 0 475,λ =  to the compressed concrete and only the 

0 0842
ct

,λ =  to the tensile concrete. If to assume the one 

and the same elastic plastic coefficient to the compressed 

and tensile concrete, the strain at the peak stress 
1ct,

ε  should 

be expressed with different factor as 
0 31

1
0 000121

ctm

,

ct, f
,ε =  . 

This factor, of course, depend on the 
ctm

f , so, to the 

Characteristics of 

materials  

Steel 

reinforcement 

S400 class 

GFRP 

reinforcement 

„Schőck 

ComBAR“ 

GFRP 

reinforcement 

„ROCKBAR“ 

Bazalt fiber 

reinforcement 

„ROCKBAR“ 

Tensile strength 

MPa 

360 1200 1000 1200 

Density, t/m
3 

7 2,2 2,0 2,0 

Corrosion Corrosion 

materals 

No corrosion material satisfy to the first group of chemical 

resistance  

 

Length of 

reinforcement 

product, m 

Bars of 6-12m 

length 

Bars of 10-14m 

length 

 

Bars until 12m length 

 

Elastic Modulus, 

N/mm
2 

200000 60000 45000 50000-55000 

Cover, mm According to 

codes, STR 

Ø+10mm, 

For precast reinforced concrete Ø+5mm 

 

Basalt fiber reinforcement, but GFRP reinforcement more 

use in concrete structures because of less price.  

The main properties of GFRP and Basalt fiber 

reinforcement are presented in table 1.  

 

 

1 lentelė. The main propeties of steel and FRP reinforcement (http://www.schoeck-combar.com, http://galen.su) 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3. The tensile concrete stress-strain relationship  
 

In this investigation the concrete stress-strain 

relationship were expressed in two cases. 

1 case. Analysing the elastic plastic behaviour of 

tensile concrete should be note that 
c

c

σ ε−  relationship 

could have quite significant influence to the crack 

moment
crc

M . The EC2 standard presents 
c

c

σ ε−  relation 

for compressed concrete but not describe the tensile 

concrete. If to assume that ultimate strain of tensile concrete 

correspond the ultimate tensile stress ( )
ctm

f , and the form of 

tensile concrete 
c

c

σ ε−  curve correspond to the form of 

compressed concrete until the peak of stress, it is possible to 

apply the EC2 standard expression of stress-strain 

relationship.  

In all figures the font of text are changed in “times new 

roman” and in fig. 1 and fig. 2 are presented units of tensile 

stress 

 

Fig. 1. The tensile concrete stress-strain relationship 

when the EC2 standard expression is used. Concrete class 

C40/50. 

 

So, if to calculate the tensile stress 
ct

σ  directly 

according to the EC2 standard expression rewriting the 
ctm

f  

instead of 
cm

f  only, we will obtain: 

 

( )

2

1 2

ct ctm

k

f

k

η η

σ

η

⎛ ⎞−

= ⎜ ⎟
⎜ ⎟

+ −
⎝ ⎠

,     (1) 

where 
1

1 05
cm ct, ctm

k , E / fε= , 
0 31

1
0 0007

ctm

.

ct,
, fε = . 

According such way drawing up the 
c

c

σ ε−  

relationship of tensile concrete we will obtain the curve that 

form at all will not be similar to the form of curve of 

compressed concrete (fig. 2). In this case the curves forms 

not correspond each other because of different values of the 

ratios 

1

cm

c,

f

ε

 and 

1

ctm

ct,

f

ε

 . For example to the concrete 

class 
20

25

C  the numerical values is: 28
cm

f MPa= ; 

1
0 001967

c,

,ε = , 2 21
ctm

f , MPa= , 
4

1
8 95 10

ct,
,ε

−

= ⋅ . Then 

we obtain that the ratio of 

1

14234 875
cm

c,

f
,

ε

=  and 

1

2469 273
ctm

ct,

f
,

ε

=  differ more than six times. Also, 

according such case the secant modulus of compressed 

concrete at the peak stress ( )
ct ctm

fσ =  was 5,765 times 

bigger than to the tensile concrete. If to express the ultimate 

elastic plastic coefficient at the stress peak it would obtained 

the 0 475,λ =  to the compressed concrete and only the 

0 0842
ct

,λ =  to the tensile concrete. If to assume the one 

and the same elastic plastic coefficient to the compressed 

and tensile concrete, the strain at the peak stress 
1ct,

ε  should 

be expressed with different factor as 
0 31

1
0 000121

ctm

,

ct, f
,ε =  . 

This factor, of course, depend on the 
ctm

f , so, to the 

Characteristics of 

materials  

Steel 

reinforcement 

S400 class 

GFRP 

reinforcement 

„Schőck 

ComBAR“ 

GFRP 

reinforcement 

„ROCKBAR“ 

Bazalt fiber 

reinforcement 

„ROCKBAR“ 

Tensile strength 

MPa 

360 1200 1000 1200 

Density, t/m
3 

7 2,2 2,0 2,0 

Corrosion Corrosion 

materals 

No corrosion material satisfy to the first group of chemical 

resistance  

 

Length of 

reinforcement 

product, m 

Bars of 6-12m 

length 

Bars of 10-14m 

length 

 

Bars until 12m length 

 

Elastic Modulus, 

N/mm
2 

200000 60000 45000 50000-55000 

Cover, mm According to 

codes, STR 

Ø+10mm, 

For precast reinforced concrete Ø+5mm 

 

Basalt fiber reinforcement, but GFRP reinforcement more 

use in concrete structures because of less price.  

The main properties of GFRP and Basalt fiber 

reinforcement are presented in table 1.  

 

 

1 lentelė. The main propeties of steel and FRP reinforcement (http://www.schoeck-combar.com, http://galen.su) 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3. The tensile concrete stress-strain relationship  
 

In this investigation the concrete stress-strain 

relationship were expressed in two cases. 

1 case. Analysing the elastic plastic behaviour of 

tensile concrete should be note that 
c

c

σ ε−  relationship 

could have quite significant influence to the crack 

moment
crc

M . The EC2 standard presents 
c

c

σ ε−  relation 

for compressed concrete but not describe the tensile 

concrete. If to assume that ultimate strain of tensile concrete 

correspond the ultimate tensile stress ( )
ctm

f , and the form of 

tensile concrete 
c

c

σ ε−  curve correspond to the form of 

compressed concrete until the peak of stress, it is possible to 

apply the EC2 standard expression of stress-strain 

relationship.  

In all figures the font of text are changed in “times new 

roman” and in fig. 1 and fig. 2 are presented units of tensile 

stress 

 

Fig. 1. The tensile concrete stress-strain relationship 

when the EC2 standard expression is used. Concrete class 

C40/50. 

 

So, if to calculate the tensile stress 
ct

σ  directly 

according to the EC2 standard expression rewriting the 
ctm

f  

instead of 
cm

f  only, we will obtain: 

 

( )

2

1 2

ct ctm

k

f

k

η η

σ

η

⎛ ⎞−

= ⎜ ⎟
⎜ ⎟

+ −
⎝ ⎠

,     (1) 

where 
1

1 05
cm ct, ctm

k , E / fε= , 
0 31

1
0 0007

ctm

.

ct,
, fε = . 

According such way drawing up the 
c

c

σ ε−  

relationship of tensile concrete we will obtain the curve that 

form at all will not be similar to the form of curve of 

compressed concrete (fig. 2). In this case the curves forms 

not correspond each other because of different values of the 

ratios 

1

cm

c,

f

ε

 and 

1

ctm

ct,

f

ε

 . For example to the concrete 

class 
20

25

C  the numerical values is: 28
cm

f MPa= ; 

1
0 001967

c,

,ε = , 2 21
ctm

f , MPa= , 
4

1
8 95 10

ct,
,ε

−

= ⋅ . Then 

we obtain that the ratio of 

1

14234 875
cm

c,

f
,

ε

=  and 

1

2469 273
ctm

ct,

f
,

ε

=  differ more than six times. Also, 

according such case the secant modulus of compressed 

concrete at the peak stress ( )
ct ctm

fσ =  was 5,765 times 

bigger than to the tensile concrete. If to express the ultimate 

elastic plastic coefficient at the stress peak it would obtained 

the 0 475,λ =  to the compressed concrete and only the 

0 0842
ct

,λ =  to the tensile concrete. If to assume the one 

and the same elastic plastic coefficient to the compressed 

and tensile concrete, the strain at the peak stress 
1ct,

ε  should 

be expressed with different factor as 
0 31

1
0 000121

ctm

,

ct, f
,ε =  . 

This factor, of course, depend on the 
ctm

f , so, to the 

Characteristics of 

materials  

Steel 

reinforcement 

S400 class 

GFRP 

reinforcement 

„Schőck 

ComBAR“ 

GFRP 

reinforcement 

„ROCKBAR“ 

Bazalt fiber 

reinforcement 

„ROCKBAR“ 

Tensile strength 

MPa 

360 1200 1000 1200 

Density, t/m
3 

7 2,2 2,0 2,0 

Corrosion Corrosion 

materals 

No corrosion material satisfy to the first group of chemical 

resistance  

 

Length of 

reinforcement 

product, m 

Bars of 6-12m 

length 

Bars of 10-14m 

length 

 

Bars until 12m length 

 

Elastic Modulus, 

N/mm
2 

200000 60000 45000 50000-55000 

Cover, mm According to 

codes, STR 

Ø+10mm, 

For precast reinforced concrete Ø+5mm 

 

;

Basalt fiber reinforcement, but GFRP reinforcement more 

use in concrete structures because of less price.  

The main properties of GFRP and Basalt fiber 

reinforcement are presented in table 1.  

 

 

1 lentelė. The main propeties of steel and FRP reinforcement (http://www.schoeck-combar.com, http://galen.su) 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3. The tensile concrete stress-strain relationship  
 

In this investigation the concrete stress-strain 

relationship were expressed in two cases. 

1 case. Analysing the elastic plastic behaviour of 

tensile concrete should be note that 
c

c

σ ε−  relationship 

could have quite significant influence to the crack 

moment
crc

M . The EC2 standard presents 
c

c

σ ε−  relation 

for compressed concrete but not describe the tensile 

concrete. If to assume that ultimate strain of tensile concrete 

correspond the ultimate tensile stress ( )
ctm

f , and the form of 

tensile concrete 
c

c

σ ε−  curve correspond to the form of 

compressed concrete until the peak of stress, it is possible to 

apply the EC2 standard expression of stress-strain 

relationship.  

In all figures the font of text are changed in “times new 

roman” and in fig. 1 and fig. 2 are presented units of tensile 

stress 

 

Fig. 1. The tensile concrete stress-strain relationship 

when the EC2 standard expression is used. Concrete class 

C40/50. 

 

So, if to calculate the tensile stress 
ct

σ  directly 

according to the EC2 standard expression rewriting the 
ctm

f  

instead of 
cm

f  only, we will obtain: 

 

( )

2

1 2

ct ctm

k

f

k

η η

σ

η

⎛ ⎞−

= ⎜ ⎟
⎜ ⎟

+ −
⎝ ⎠

,     (1) 

where 
1

1 05
cm ct, ctm

k , E / fε= , 
0 31

1
0 0007

ctm

.

ct,
, fε = . 

According such way drawing up the 
c

c

σ ε−  

relationship of tensile concrete we will obtain the curve that 

form at all will not be similar to the form of curve of 

compressed concrete (fig. 2). In this case the curves forms 

not correspond each other because of different values of the 

ratios 

1

cm

c,

f

ε

 and 

1

ctm

ct,

f

ε

 . For example to the concrete 

class 
20

25

C  the numerical values is: 28
cm

f MPa= ; 

1
0 001967

c,

,ε = , 2 21
ctm

f , MPa= , 
4

1
8 95 10

ct,
,ε

−

= ⋅ . Then 

we obtain that the ratio of 

1

14234 875
cm

c,

f
,

ε

=  and 

1

2469 273
ctm

ct,

f
,

ε

=  differ more than six times. Also, 

according such case the secant modulus of compressed 

concrete at the peak stress ( )
ct ctm

fσ =  was 5,765 times 

bigger than to the tensile concrete. If to express the ultimate 

elastic plastic coefficient at the stress peak it would obtained 

the 0 475,λ =  to the compressed concrete and only the 

0 0842
ct

,λ =  to the tensile concrete. If to assume the one 

and the same elastic plastic coefficient to the compressed 

and tensile concrete, the strain at the peak stress 
1ct,

ε  should 

be expressed with different factor as 
0 31

1
0 000121

ctm

,

ct, f
,ε =  . 

This factor, of course, depend on the 
ctm

f , so, to the 

Characteristics of 

materials  

Steel 

reinforcement 

S400 class 

GFRP 

reinforcement 

„Schőck 

ComBAR“ 

GFRP 

reinforcement 

„ROCKBAR“ 

Bazalt fiber 

reinforcement 

„ROCKBAR“ 

Tensile strength 

MPa 

360 1200 1000 1200 

Density, t/m
3 

7 2,2 2,0 2,0 

Corrosion Corrosion 

materals 

No corrosion material satisfy to the first group of chemical 

resistance  

 

Length of 

reinforcement 

product, m 

Bars of 6-12m 

length 

Bars of 10-14m 

length 

 

Bars until 12m length 

 

Elastic Modulus, 

N/mm
2 

200000 60000 45000 50000-55000 

Cover, mm According to 

codes, STR 

Ø+10mm, 

For precast reinforced concrete Ø+5mm 

 

concrete class from 
8

10

C  to 
50

60

C  the strain could be 

expressed with average factor: 

 

cm

ctm

ct

f

f
31.1

1,

00156.0

=ε .     (2) 

 

Then the member k to the tensile concrete would be 

obtained from the other equation 

 

0.3

,1

1.31

0.31 0.7

1.05

1.05 22000

10

0.00156 18.06

cmcm ct

ctm

ctm

ctm cm

cm ctm

fE

k
f

f

f f

f f

ε

−

⎛ ⎞
= = ⋅ ⋅

⎜ ⎟

⎝ ⎠

⋅ =

⋅

.   (3) 

When the ultimate concrete strain it is known the tensile 

stress could be expressed according to the adjusted EC2 

equation: 

 

( ) ( )
ctctmcmcmctm

ctctmcmctcmctm

ctm

ct

ffff

ffff

k

k

f ε

εε

η

ηησ

31.17.031.0

262.223.012

206.186411

41091411577

21
−−

−−

−⋅+

−

=

−+

−

= .        (4) 

 

 
 

The tensile stress of concrete (class 40/50) obtained 

according to the equation (4) is presented in fig. 2 where the 

peak of stress is reached at the strain equal to 0.000165. 

 

 

Fig. 2. The stress-strain relationship expressed according to 

the equation (4) (1-curve) and equation (1) (2-curve). 

 

The obtained expression (4) is not so convenient to the 

resultant calculation of the tensile and compressed concrete 

zones and to the crack moment also, because the strain 

member is in the denominator. Such not well on calculation 

of integrals. In this case more convenient the tensile stress 

expressed by polynomials (Židonis 2007, 2009, 2010). Then 

the stress of tensile concrete will be equal  

 

2

1 2 3ct ct ct
a a aσ ε ε= ⋅ + ⋅ + ,    (5) 

 

 where 
1
a , 

2
a , 

3
a  – free members.  

 

The free members corresponded the stress-strain 

relationship of the C30/37 class concrete is equal to: 

1 105003697.6a = − , 2 34987.8a = , 3 0.0127a = . 

2 case. It is also convenient to apply the elastic plastic 

coefficient λ  to express the σ ε−  relationship. If to 

assume that this coefficient linear vary from the beginning 

of elastic plastic zone until to concrete fracture it could be 

expressed as (Augonis 2013)  

 

1
i el

lim

lim el

ε ε

λ λ

ε ε

−

= −

−

,      (6) 

 

where 
lim

λ  – ultimate value of the elastic plastic coefficient 

at the moment of failure, 
i

ε  –strain, 
lim
ε  – ultimate strain 

(strain at the moment of failure) 
( )

c

lim

lim c

f

E

ε

λ

= , 
el

ε  – 

ultimate elastic strain 

0 4
c

el

c

, f

E

ε ≈ . 

 

Then the tensile stress could be expressed as 

 

ctctcct

E ελσ = .        (7) 

 

The stress-strain relationships expressed according to 

the equation (5) and equation (7) to the C30/37 class 

concrete is presented in fig. 3 where the are used not the 

average values of stress (
ctm

f ) but characteristic values 

(fctk0,05). For this reason the factor 0,7 were applied to the 

values obtained by equation (5). According to that figure the 

curves is differ but it should be note that the ultimate strain 

is differ also. 

 

Fig. 3. The stress-strain relationship of tensile concrete 

(
005ct ctk,

f f= ). 1– when the elastic plastic coefficient 

0 5
ct,lim

,λ =  is applied, 2– when the equation (5) is applied with 

the factor 0,7. 

 

4. The crack moment of members reinforced by 
steel reinforcement and FRP reinforcement 

 

In this section the cracking moments are calculated 

with assumption that the behaviour of compressed concrete 

is elastic because not high the amount of reinforcement was 

predicted. For calculations the stress and strain schemes 

presented in fig. 4 were used. 

Calculating the cracking moment of elements 

reinforced by FRP reinforcement, were assumed that the 

bond between reinforcement and concrete is quite enough.  

ε

ct,lim

 

such case the secant modulus of 
compressed concrete at the peak 
stress ( )ct ctmfσ =  was 5,765 times 
bigger than to the tensile concrete. 
If to express the ultimate elastic 
plastic coefficient at the stress peak 
it would obtained the 0 475,λ =  
to the compressed concrete and 
only the 0 0842ct ,λ =  to the tensile 
concrete. If to assume the one and 
the same elastic plastic coefficient to 
the compressed and tensile concrete, 
the strain at the peak stress 1ct ,ε  
should be expressed with different 

factor as 0 311 0 000121 ctm
,

ct , f,ε = . This
factor, of course, depend on 
the ctmf , so, to the concrete
class from 810C  to 

50
60C  the 

strain could be expressed with average 
factor:

(2)

concrete class from 
8

10

C  to 
50

60

C  the strain could be 

expressed with average factor: 

 

cm

ctm

ct

f

f
31.1

1,

00156.0

=ε .     (2) 

 

Then the member k to the tensile concrete would be 

obtained from the other equation 

 

0.3

,1

1.31

0.31 0.7

1.05

1.05 22000

10

0.00156 18.06

cmcm ct

ctm

ctm

ctm cm

cm ctm

fE

k
f

f

f f

f f

ε

−

⎛ ⎞
= = ⋅ ⋅

⎜ ⎟

⎝ ⎠

⋅ =

⋅

.   (3) 

When the ultimate concrete strain it is known the tensile 

stress could be expressed according to the adjusted EC2 

equation: 

 

( ) ( )
ctctmcmcmctm

ctctmcmctcmctm

ctm

ct

ffff

ffff

k

k

f ε

εε

η

ηησ

31.17.031.0

262.223.012

206.186411

41091411577

21
−−

−−

−⋅+

−

=

−+

−

= .        (4) 

 

 
 

The tensile stress of concrete (class 40/50) obtained 

according to the equation (4) is presented in fig. 2 where the 

peak of stress is reached at the strain equal to 0.000165. 

 

 

Fig. 2. The stress-strain relationship expressed according to 

the equation (4) (1-curve) and equation (1) (2-curve). 

 

The obtained expression (4) is not so convenient to the 

resultant calculation of the tensile and compressed concrete 

zones and to the crack moment also, because the strain 

member is in the denominator. Such not well on calculation 

of integrals. In this case more convenient the tensile stress 

expressed by polynomials (Židonis 2007, 2009, 2010). Then 

the stress of tensile concrete will be equal  

 

2

1 2 3ct ct ct
a a aσ ε ε= ⋅ + ⋅ + ,    (5) 

 

 where 
1
a , 

2
a , 

3
a  – free members.  

 

The free members corresponded the stress-strain 

relationship of the C30/37 class concrete is equal to: 

1 105003697.6a = − , 2 34987.8a = , 3 0.0127a = . 

2 case. It is also convenient to apply the elastic plastic 

coefficient λ  to express the σ ε−  relationship. If to 

assume that this coefficient linear vary from the beginning 

of elastic plastic zone until to concrete fracture it could be 

expressed as (Augonis 2013)  

 

1
i el

lim

lim el

ε ε

λ λ

ε ε

−

= −

−

,      (6) 

 

where 
lim

λ  – ultimate value of the elastic plastic coefficient 

at the moment of failure, 
i

ε  –strain, 
lim
ε  – ultimate strain 

(strain at the moment of failure) 
( )

c

lim

lim c

f

E

ε

λ

= , 
el

ε  – 

ultimate elastic strain 

0 4
c

el

c

, f

E

ε ≈ . 

 

Then the tensile stress could be expressed as 

 

ctctcct

E ελσ = .        (7) 

 

The stress-strain relationships expressed according to 

the equation (5) and equation (7) to the C30/37 class 

concrete is presented in fig. 3 where the are used not the 

average values of stress (
ctm

f ) but characteristic values 

(fctk0,05). For this reason the factor 0,7 were applied to the 

values obtained by equation (5). According to that figure the 

curves is differ but it should be note that the ultimate strain 

is differ also. 

 

Fig. 3. The stress-strain relationship of tensile concrete 

(
005ct ctk,

f f= ). 1– when the elastic plastic coefficient 

0 5
ct,lim

,λ =  is applied, 2– when the equation (5) is applied with 

the factor 0,7. 

 

4. The crack moment of members reinforced by 
steel reinforcement and FRP reinforcement 

 

In this section the cracking moments are calculated 

with assumption that the behaviour of compressed concrete 

is elastic because not high the amount of reinforcement was 

predicted. For calculations the stress and strain schemes 

presented in fig. 4 were used. 

Calculating the cracking moment of elements 

reinforced by FRP reinforcement, were assumed that the 

bond between reinforcement and concrete is quite enough.  

ε

ct,lim

 

Fig. 1
The tensile concrete 
stress-strain relationship 
when the EC2 standard 
expression is used. 
Concrete class C40/50

Fig. 2
The stress-strain 
relationship expressed 
according to the equation 
(4) (1-curve) and equation 
(1) (2-curve)



Journal of Sustainable Architecture and Civil Engineering 2014/4/9
62

Then the member k to the tensile concrete would be obtained from the other equation

When the ultimate concrete strain it is known the tensile stress could be expressed according to 
the adjusted EC2 equation:

The tensile stress of concrete (class 40/50) obtained according to the equation (4) is presented in 
fig. 2 where the peak of stress is reached at the strain equal to 0.000165.

The obtained expression (4) is not so convenient to the resultant calculation of the tensile and 
compressed concrete zones and to the crack moment also, because the strain member is in the 
denominator. Such not well on calculation of integrals. In this case more convenient the tensile stress 
expressed by polynomials (Židonis 2007, 2009, 2010). Then the stress of tensile concrete will be equal 

(3)

concrete class from 
8

10

C  to 
50

60

C  the strain could be 

expressed with average factor: 

 

cm

ctm

ct

f

f
31.1

1,

00156.0

=ε .     (2) 

 

Then the member k to the tensile concrete would be 

obtained from the other equation 

 

0.3

,1

1.31

0.31 0.7

1.05

1.05 22000

10

0.00156 18.06

cmcm ct

ctm

ctm

ctm cm

cm ctm

fE

k
f

f

f f

f f

ε

−

⎛ ⎞
= = ⋅ ⋅

⎜ ⎟

⎝ ⎠

⋅ =

⋅

.   (3) 

When the ultimate concrete strain it is known the tensile 

stress could be expressed according to the adjusted EC2 

equation: 

 

( ) ( )
ctctmcmcmctm

ctctmcmctcmctm

ctm

ct

ffff

ffff

k

k

f ε

εε

η

ηησ

31.17.031.0

262.223.012

206.186411

41091411577

21
−−

−−

−⋅+

−

=

−+

−

= .        (4) 

 

 
 

The tensile stress of concrete (class 40/50) obtained 

according to the equation (4) is presented in fig. 2 where the 

peak of stress is reached at the strain equal to 0.000165. 

 

 

Fig. 2. The stress-strain relationship expressed according to 

the equation (4) (1-curve) and equation (1) (2-curve). 

 

The obtained expression (4) is not so convenient to the 

resultant calculation of the tensile and compressed concrete 

zones and to the crack moment also, because the strain 

member is in the denominator. Such not well on calculation 

of integrals. In this case more convenient the tensile stress 

expressed by polynomials (Židonis 2007, 2009, 2010). Then 

the stress of tensile concrete will be equal  

 

2

1 2 3ct ct ct
a a aσ ε ε= ⋅ + ⋅ + ,    (5) 

 

 where 
1
a , 

2
a , 

3
a  – free members.  

 

The free members corresponded the stress-strain 

relationship of the C30/37 class concrete is equal to: 

1 105003697.6a = − , 2 34987.8a = , 3 0.0127a = . 

2 case. It is also convenient to apply the elastic plastic 

coefficient λ  to express the σ ε−  relationship. If to 

assume that this coefficient linear vary from the beginning 

of elastic plastic zone until to concrete fracture it could be 

expressed as (Augonis 2013)  

 

1
i el

lim

lim el

ε ε

λ λ

ε ε

−

= −

−

,      (6) 

 

where 
lim

λ  – ultimate value of the elastic plastic coefficient 

at the moment of failure, 
i

ε  –strain, 
lim
ε  – ultimate strain 

(strain at the moment of failure) 
( )

c

lim

lim c

f

E

ε

λ

= , 
el

ε  – 

ultimate elastic strain 

0 4
c

el

c

, f

E

ε ≈ . 

 

Then the tensile stress could be expressed as 

 

ctctcct

E ελσ = .        (7) 

 

The stress-strain relationships expressed according to 

the equation (5) and equation (7) to the C30/37 class 

concrete is presented in fig. 3 where the are used not the 

average values of stress (
ctm

f ) but characteristic values 

(fctk0,05). For this reason the factor 0,7 were applied to the 

values obtained by equation (5). According to that figure the 

curves is differ but it should be note that the ultimate strain 

is differ also. 

 

Fig. 3. The stress-strain relationship of tensile concrete 

(
005ct ctk,

f f= ). 1– when the elastic plastic coefficient 

0 5
ct,lim

,λ =  is applied, 2– when the equation (5) is applied with 

the factor 0,7. 

 

4. The crack moment of members reinforced by 
steel reinforcement and FRP reinforcement 

 

In this section the cracking moments are calculated 

with assumption that the behaviour of compressed concrete 

is elastic because not high the amount of reinforcement was 

predicted. For calculations the stress and strain schemes 

presented in fig. 4 were used. 

Calculating the cracking moment of elements 

reinforced by FRP reinforcement, were assumed that the 

bond between reinforcement and concrete is quite enough.  

ε

ct,lim

 

concrete class from 
8

10

C  to 
50

60

C  the strain could be 

expressed with average factor: 

 

cm

ctm

ct

f

f
31.1

1,

00156.0

=ε .     (2) 

 

Then the member k to the tensile concrete would be 

obtained from the other equation 

 

0.3

,1

1.31

0.31 0.7

1.05

1.05 22000

10

0.00156 18.06

cmcm ct

ctm

ctm

ctm cm

cm ctm

fE

k
f

f

f f

f f

ε

−

⎛ ⎞
=

⎜ ⎟

⎝ ⎠

⋅ =

⋅

.   (3) 

When the ultimate concrete strain it is known the tensile 

stress could be expressed according to the adjusted EC2 

equation: 

 

( ) ( )
ctctmcmcmctm

ctctmcmctcmctm

ctm

ct

ffff

ffff

k

k

f ε

εε

η

ηησ

31.17.031.0

262.223.012

206.186411

41091411577

21
−−

−−

−⋅+

−

=

−+

−

= .        (4) 

 

 
 

The tensile stress of concrete (class 40/50) obtained 

according to the equation (4) is presented in fig. 2 where the 

peak of stress is reached at the strain equal to 0.000165. 

 

 

Fig. 2. The stress-strain relationship expressed according to 

the equation (4) (1-curve) and equation (1) (2-curve). 

 

The obtained expression (4) is not so convenient to the 

resultant calculation of the tensile and compressed concrete 

zones and to the crack moment also, because the strain 

member is in the denominator. Such not well on calculation 

of integrals. In this case more convenient the tensile stress 

expressed by polynomials (Židonis 2007, 2009, 2010). Then 

the stress of tensile concrete will be equal  

 

2

1 2 3ct ct ct
a a aσ ε ε= ⋅ + ⋅ + ,    (5) 

 

 where 
1
a , 

2
a , 

3
a  – free members.  

 

The free members corresponded the stress-strain 

relationship of the C30/37 class concrete is equal to: 

1 105003697.6a = − , 2 34987.8a = , 3 0.0127a = . 

2 case. It is also convenient to apply the elastic plastic 

coefficient λ  to express the σ ε−  relationship. If to 

assume that this coefficient linear vary from the beginning 

of elastic plastic zone until to concrete fracture it could be 

expressed as (Augonis 2013)  

 

1
i el

lim

lim el

ε ε

λ λ

ε ε

−

= −

−

,      (6) 

 

where 
lim

λ  – ultimate value of the elastic plastic coefficient 

at the moment of failure, 
i

ε  –strain, 
lim
ε  – ultimate strain 

(strain at the moment of failure) 
( )

c

lim

lim c

f

E

ε

λ

= , 
el

ε  – 

ultimate elastic strain 

0 4
c

el

c

, f

E

ε ≈ . 

 

Then the tensile stress could be expressed as 

 

ctctcct

E ελσ = .        (7) 

 

The stress-strain relationships expressed according to 

the equation (5) and equation (7) to the C30/37 class 

concrete is presented in fig. 3 where the are used not the 

average values of stress (
ctm

f ) but characteristic values 

(fctk0,05). For this reason the factor 0,7 were applied to the 

values obtained by equation (5). According to that figure the 

curves is differ but it should be note that the ultimate strain 

is differ also. 

 

Fig. 3. The stress-strain relationship of tensile concrete 

(
005ct ctk,

f f= ). 1– when the elastic plastic coefficient 

0 5
ct,lim

,λ =  is applied, 2– when the equation (5) is applied with 

the factor 0,7. 

 

4. The crack moment of members reinforced by 
steel reinforcement and FRP reinforcement 

 

In this section the cracking moments are calculated 

with assumption that the behaviour of compressed concrete 

is elastic because not high the amount of reinforcement was 

predicted. For calculations the stress and strain schemes 

presented in fig. 4 were used. 

Calculating the cracking moment of elements 

reinforced by FRP reinforcement, were assumed that the 

bond between reinforcement and concrete is quite enough.  

ε

ct,lim

 

concrete class from 
8

10

C  to 
50

60

C  the strain could be 

expressed with average factor: 

 

cm

ctm

ct

f

f
31.1

1,

00156.0

=ε .     (2) 

 

Then the member k to the tensile concrete would be 

obtained from the other equation 

 

0.3

,1

1.31

0.31 0.7

1.05

1.05 22000

10

0.00156 18.06

cmcm ct

ctm

ctm

ctm cm

cm ctm

fE

k
f

f

f f

f f

ε

−

⎛ ⎞
= = ⋅ ⋅

⎜ ⎟

⎝ ⎠

⋅ =

⋅

.   (3) 

When the ultimate concrete strain it is known the tensile 

stress could be expressed according to the adjusted EC2 

equation: 

 

( ) ( )
ctctmcmcmctm

ctctmcmctcmctm

ctm

ct

ffff

ffff

k

k

f ε

εε

η

ηησ

31.17.031.0

262.223.012

206.186411

41091411577

21
−−

−−

−⋅+

−

=

−+

−

= .        (4) 

 

 
 

The tensile stress of concrete (class 40/50) obtained 

according to the equation (4) is presented in fig. 2 where the 

peak of stress is reached at the strain equal to 0.000165. 

 

 

Fig. 2. The stress-strain relationship expressed according to 

the equation (4) (1-curve) and equation (1) (2-curve). 

 

The obtained expression (4) is not so convenient to the 

resultant calculation of the tensile and compressed concrete 

zones and to the crack moment also, because the strain 

member is in the denominator. Such not well on calculation 

of integrals. In this case more convenient the tensile stress 

expressed by polynomials (Židonis 2007, 2009, 2010). Then 

the stress of tensile concrete will be equal  

 

2

1 2 3ct ct ct
a a aσ ε ε= ⋅ + ⋅ + ,    (5) 

 

 where 
1
a , 

2
a , 

3
a  – free members.  

 

The free members corresponded the stress-strain 

relationship of the C30/37 class concrete is equal to: 

1 105003697.6a = − , 2 34987.8a = , 3 0.0127a = . 

2 case. It is also convenient to apply the elastic plastic 

coefficient λ  to express the σ ε−  relationship. If to 

assume that this coefficient linear vary from the beginning 

of elastic plastic zone until to concrete fracture it could be 

expressed as (Augonis 2013)  

 

1
i el

lim

lim el

ε ε

λ λ

ε ε

−

= −

−

,      (6) 

 

where 
lim

λ  – ultimate value of the elastic plastic coefficient 

at the moment of failure, 
i

ε  –strain, 
lim
ε  – ultimate strain 

(strain at the moment of failure) 
( )

c

lim

lim c

f

E

ε

λ

= , 
el

ε  – 

ultimate elastic strain 

0 4
c

el

c

, f

E

ε ≈ . 

 

Then the tensile stress could be expressed as 

 

ctctcct

E ελσ = .        (7) 

 

The stress-strain relationships expressed according to 

the equation (5) and equation (7) to the C30/37 class 

concrete is presented in fig. 3 where the are used not the 

average values of stress (
ctm

f ) but characteristic values 

(fctk0,05). For this reason the factor 0,7 were applied to the 

values obtained by equation (5). According to that figure the 

curves is differ but it should be note that the ultimate strain 

is differ also. 

 

Fig. 3. The stress-strain relationship of tensile concrete 

(
005ct ctk,

f f= ). 1– when the elastic plastic coefficient 

0 5
ct,lim

,λ =  is applied, 2– when the equation (5) is applied with 

the factor 0,7. 

 

4. The crack moment of members reinforced by 
steel reinforcement and FRP reinforcement 

 

In this section the cracking moments are calculated 

with assumption that the behaviour of compressed concrete 

is elastic because not high the amount of reinforcement was 

predicted. For calculations the stress and strain schemes 

presented in fig. 4 were used. 

Calculating the cracking moment of elements 

reinforced by FRP reinforcement, were assumed that the 

bond between reinforcement and concrete is quite enough.  

ε

ct,lim

 

.

(4)

concrete class from 
8

10

C  to 
50

60

C  the strain could be 

expressed with average factor: 

 

cm

ctm

ct

f

f
31.1

1,

00156.0

=ε .     (2) 

 

Then the member k to the tensile concrete would be 

obtained from the other equation 

 

0.3

,1

1.31

0.31 0.7

1.05

1.05 22000

10

0.00156 18.06

cmcm ct

ctm

ctm

ctm cm

cm ctm

fE

k
f

f

f f

f f

ε

−

⎛ ⎞
= = ⋅ ⋅

⎜ ⎟

⎝ ⎠

⋅ =

⋅

.   (3) 

When the ultimate concrete strain it is known the tensile 

stress could be expressed according to the adjusted EC2 

equation: 

 

( ) ( )
ctctmcmcmctm

ctctmcmctcmctm

ctm

ct

ffff

ffff

k

k

f ε

εε

η

ηησ

31.17.031.0

262.223.012

206.186411

41091411577

21
−−

−−

−⋅+

−

=

−+

−

= .        (4) 

 

 
 

The tensile stress of concrete (class 40/50) obtained 

according to the equation (4) is presented in fig. 2 where the 

peak of stress is reached at the strain equal to 0.000165. 

 

 

Fig. 2. The stress-strain relationship expressed according to 

the equation (4) (1-curve) and equation (1) (2-curve). 

 

The obtained expression (4) is not so convenient to the 

resultant calculation of the tensile and compressed concrete 

zones and to the crack moment also, because the strain 

member is in the denominator. Such not well on calculation 

of integrals. In this case more convenient the tensile stress 

expressed by polynomials (Židonis 2007, 2009, 2010). Then 

the stress of tensile concrete will be equal  

 

2

1 2 3ct ct ct
a a aσ ε ε= ⋅ + ⋅ + ,    (5) 

 

 where 
1
a , 

2
a , 

3
a  – free members.  

 

The free members corresponded the stress-strain 

relationship of the C30/37 class concrete is equal to: 

1 105003697.6a = − , 2 34987.8a = , 3 0.0127a = . 

2 case. It is also convenient to apply the elastic plastic 

coefficient λ  to express the σ ε−  relationship. If to 

assume that this coefficient linear vary from the beginning 

of elastic plastic zone until to concrete fracture it could be 

expressed as (Augonis 2013)  

 

1
i el

lim

lim el

ε ε

λ λ

ε ε

−

= −

−

,      (6) 

 

where 
lim

λ  – ultimate value of the elastic plastic coefficient 

at the moment of failure, 
i

ε  –strain, 
lim
ε  – ultimate strain 

(strain at the moment of failure) 
( )

c

lim

lim c

f

E

ε

λ

= , 
el

ε  – 

ultimate elastic strain 

0 4
c

el

c

, f

E

ε ≈ . 

 

Then the tensile stress could be expressed as 

 

ctctcct

E ελσ = .        (7) 

 

The stress-strain relationships expressed according to 

the equation (5) and equation (7) to the C30/37 class 

concrete is presented in fig. 3 where the are used not the 

average values of stress (
ctm

f ) but characteristic values 

(fctk0,05). For this reason the factor 0,7 were applied to the 

values obtained by equation (5). According to that figure the 

curves is differ but it should be note that the ultimate strain 

is differ also. 

 

Fig. 3. The stress-strain relationship of tensile concrete 

(
005ct ctk,

f f= ). 1– when the elastic plastic coefficient 

0 5
ct,lim

,λ =  is applied, 2– when the equation (5) is applied with 

the factor 0,7. 

 

4. The crack moment of members reinforced by 
steel reinforcement and FRP reinforcement 

 

In this section the cracking moments are calculated 

with assumption that the behaviour of compressed concrete 

is elastic because not high the amount of reinforcement was 

predicted. For calculations the stress and strain schemes 

presented in fig. 4 were used. 

Calculating the cracking moment of elements 

reinforced by FRP reinforcement, were assumed that the 

bond between reinforcement and concrete is quite enough.  

ε

ct,lim

 

where: 

where 1a , 2a , 3a  – free members.
(5)

concrete class from 
8

10

C  to 
50

60

C  the strain could be 

expressed with average factor: 

 

cm

ctm

ct

f

f
31.1

1,

00156.0

=ε .     (2) 

 

Then the member k to the tensile concrete would be 

obtained from the other equation 

 

0.3

,1

1.31

0.31 0.7

1.05

1.05 22000

10

0.00156 18.06

cmcm ct

ctm

ctm

ctm cm

cm ctm

fE

k
f

f

f f

f f

ε

−

⎛ ⎞
= = ⋅ ⋅

⎜ ⎟

⎝ ⎠

⋅ =

⋅

.   (3) 

When the ultimate concrete strain it is known the tensile 

stress could be expressed according to the adjusted EC2 

equation: 

 

( ) ( )
ctctmcmcmctm

ctctmcmctcmctm

ctm

ct

ffff

ffff

k

k

f ε

εε

η

ηησ

31.17.031.0

262.223.012

206.186411

41091411577

21
−−

−−

−⋅+

−

=

−+

−

= .        (4) 

 

 
 

The tensile stress of concrete (class 40/50) obtained 

according to the equation (4) is presented in fig. 2 where the 

peak of stress is reached at the strain equal to 0.000165. 

 

 

Fig. 2. The stress-strain relationship expressed according to 

the equation (4) (1-curve) and equation (1) (2-curve). 

 

The obtained expression (4) is not so convenient to the 

resultant calculation of the tensile and compressed concrete 

zones and to the crack moment also, because the strain 

member is in the denominator. Such not well on calculation 

of integrals. In this case more convenient the tensile stress 

expressed by polynomials (Židonis 2007, 2009, 2010). Then 

the stress of tensile concrete will be equal  

 

2

1 2 3ct ct ct
a a aσ ε ε= ⋅ + ⋅ + ,    (5) 

 

 where 
1
a , 

2
a , 

3
a  – free members.  

 

The free members corresponded the stress-strain 

relationship of the C30/37 class concrete is equal to: 

1 105003697.6a = − , 2 34987.8a = , 3 0.0127a = . 

2 case. It is also convenient to apply the elastic plastic 

coefficient λ  to express the σ ε−  relationship. If to 

assume that this coefficient linear vary from the beginning 

of elastic plastic zone until to concrete fracture it could be 

expressed as (Augonis 2013)  

 

1
i el

lim

lim el

ε ε

λ λ

ε ε

−

= −

−

,      (6) 

 

where 
lim

λ  – ultimate value of the elastic plastic coefficient 

at the moment of failure, 
i

ε  –strain, 
lim
ε  – ultimate strain 

(strain at the moment of failure) 
( )

c

lim

lim c

f

E

ε

λ

= , 
el

ε  – 

ultimate elastic strain 

0 4
c

el

c

, f

E

ε ≈ . 

 

Then the tensile stress could be expressed as 

 

ctctcct

E ελσ = .        (7) 

 

The stress-strain relationships expressed according to 

the equation (5) and equation (7) to the C30/37 class 

concrete is presented in fig. 3 where the are used not the 

average values of stress (
ctm

f ) but characteristic values 

(fctk0,05). For this reason the factor 0,7 were applied to the 

values obtained by equation (5). According to that figure the 

curves is differ but it should be note that the ultimate strain 

is differ also. 

 

Fig. 3. The stress-strain relationship of tensile concrete 

(
005ct ctk,

f f= ). 1– when the elastic plastic coefficient 

0 5
ct,lim

,λ =  is applied, 2– when the equation (5) is applied with 

the factor 0,7. 

 

4. The crack moment of members reinforced by 
steel reinforcement and FRP reinforcement 

 

In this section the cracking moments are calculated 

with assumption that the behaviour of compressed concrete 

is elastic because not high the amount of reinforcement was 

predicted. For calculations the stress and strain schemes 

presented in fig. 4 were used. 

Calculating the cracking moment of elements 

reinforced by FRP reinforcement, were assumed that the 

bond between reinforcement and concrete is quite enough.  

ε

ct,lim

 

The free members corresponded the stress-strain relationship of the C30/37 class concrete is ual 
to: 1 105003697.6a = − , 2 34987.8a = , 3 0.0127a = .
2 case. It is also convenient to apply the elastic plastic coefficient λ  to express the σ ε−  
relationship. If to assume that this coefficient linear vary from the beginning of elastic plastic zone 
until to concrete fracture it could be expressed as (Augonis 2013) 

concrete class from 
8

10

C  to 
50

60

C  the strain could be 

expressed with average factor: 

 

cm

ctm

ct

f

f
31.1

1,

00156.0

=ε .     (2) 

 

Then the member k to the tensile concrete would be 

obtained from the other equation 

 

0.3

,1

1.31

0.31 0.7

1.05

1.05 22000

10

0.00156 18.06

cmcm ct

ctm

ctm

ctm cm

cm ctm

fE

k
f

f

f f

f f

ε

−

⎛ ⎞
= = ⋅ ⋅

⎜ ⎟

⎝ ⎠

⋅ =

⋅

.   (3) 

When the ultimate concrete strain it is known the tensile 

stress could be expressed according to the adjusted EC2 

equation: 

 

( ) ( )
ctctmcmcmctm

ctctmcmctcmctm

ctm

ct

ffff

ffff

k

k

f ε

εε

η

ηησ

31.17.031.0

262.223.012

206.186411

41091411577

21
−−

−−

−⋅+

−

=

−+

−

= .        (4) 

 

 
 

The tensile stress of concrete (class 40/50) obtained 

according to the equation (4) is presented in fig. 2 where the 

peak of stress is reached at the strain equal to 0.000165. 

 

 

Fig. 2. The stress-strain relationship expressed according to 

the equation (4) (1-curve) and equation (1) (2-curve). 

 

The obtained expression (4) is not so convenient to the 

resultant calculation of the tensile and compressed concrete 

zones and to the crack moment also, because the strain 

member is in the denominator. Such not well on calculation 

of integrals. In this case more convenient the tensile stress 

expressed by polynomials (Židonis 2007, 2009, 2010). Then 

the stress of tensile concrete will be equal  

 

2

1 2 3ct ct ct
a a aσ ε ε= ⋅ + ⋅ + ,    (5) 

 

 where 
1
a , 

2
a , 

3
a  – free members.  

 

The free members corresponded the stress-strain 

relationship of the C30/37 class concrete is equal to: 

1 105003697.6a = − , 2 34987.8a = , 3 0.0127a = . 

2 case. It is also convenient to apply the elastic plastic 

coefficient λ  to express the σ ε−  relationship. If to 

assume that this coefficient linear vary from the beginning 

of elastic plastic zone until to concrete fracture it could be 

expressed as (Augonis 2013)  

 

1
i el

lim

lim el

ε ε

λ λ

ε ε

−

= −

−

,      (6) 

 

where 
lim

λ  – ultimate value of the elastic plastic coefficient 

at the moment of failure, 
i

ε  –strain, 
lim
ε  – ultimate strain 

(strain at the moment of failure) 
( )

c

lim

lim c

f

E

ε

λ

= , 
el

ε  – 

ultimate elastic strain 

0 4
c

el

c

, f

E

ε ≈ . 

 

Then the tensile stress could be expressed as 

 

ctctcct

E ελσ = .        (7) 

 

The stress-strain relationships expressed according to 

the equation (5) and equation (7) to the C30/37 class 

concrete is presented in fig. 3 where the are used not the 

average values of stress (
ctm

f ) but characteristic values 

(fctk0,05). For this reason the factor 0,7 were applied to the 

values obtained by equation (5). According to that figure the 

curves is differ but it should be note that the ultimate strain 

is differ also. 

 

Fig. 3. The stress-strain relationship of tensile concrete 

(
005ct ctk,

f f= ). 1– when the elastic plastic coefficient 

0 5
ct,lim

,λ =  is applied, 2– when the equation (5) is applied with 

the factor 0,7. 

 

4. The crack moment of members reinforced by 
steel reinforcement and FRP reinforcement 

 

In this section the cracking moments are calculated 

with assumption that the behaviour of compressed concrete 

is elastic because not high the amount of reinforcement was 

predicted. For calculations the stress and strain schemes 

presented in fig. 4 were used. 

Calculating the cracking moment of elements 

reinforced by FRP reinforcement, were assumed that the 

bond between reinforcement and concrete is quite enough.  

ε

ct,lim

 
where 

concrete class from 
8

10

C  to 
50

60

C  the strain could be 

expressed with average factor: 

 

cm

ctm

ct

f

f
31.1

1,

00156.0

=ε .     (2) 

 

Then the member k to the tensile concrete would be 

obtained from the other equation 

 

0.3

,1

1.31

0.31 0.7

1.05

1.05 22000

10

0.00156 18.06

cmcm ct

ctm

ctm

ctm cm

cm ctm

fE

k
f

f

f f

f f

ε

−

⎛ ⎞
= = ⋅ ⋅

⎜ ⎟

⎝ ⎠

⋅ =

⋅

.   (3) 

When the ultimate concrete strain it is known the tensile 

stress could be expressed according to the adjusted EC2 

equation: 

 

( ) ( )
ctctmcmcmctm

ctctmcmctcmctm

ctm

ct

ffff

ffff

k

k

f ε

εε

η

ηησ

31.17.031.0

262.223.012

206.186411

41091411577

21
−−

−−

−⋅+

−

=

−+

−

= .        (4) 

 

 
 

The tensile stress of concrete (class 40/50) obtained 

according to the equation (4) is presented in fig. 2 where the 

peak of stress is reached at the strain equal to 0.000165. 

 

 

Fig. 2. The stress-strain relationship expressed according to 

the equation (4) (1-curve) and equation (1) (2-curve). 

 

The obtained expression (4) is not so convenient to the 

resultant calculation of the tensile and compressed concrete 

zones and to the crack moment also, because the strain 

member is in the denominator. Such not well on calculation 

of integrals. In this case more convenient the tensile stress 

expressed by polynomials (Židonis 2007, 2009, 2010). Then 

the stress of tensile concrete will be equal  

 

2

1 2 3ct ct ct
a a aσ ε ε= ⋅ + ⋅ + ,    (5) 

 

 where 
1
a , 

2
a , 

3
a  – free members.  

 

The free members corresponded the stress-strain 

relationship of the C30/37 class concrete is equal to: 

1 105003697.6a = − , 2 34987.8a = , 3 0.0127a = . 

2 case. It is also convenient to apply the elastic plastic 

coefficient λ  to express the σ ε−  relationship. If to 

assume that this coefficient linear vary from the beginning 

of elastic plastic zone until to concrete fracture it could be 

expressed as (Augonis 2013)  

 

1
i el

lim

lim el

ε ε

λ λ

ε ε

−

= −

−

,      (6) 

 

where 
lim

λ  – ultimate value of the elastic plastic coefficient 

at the moment of failure, 
i

ε  –strain, 
lim
ε  – ultimate strain 

(strain at the moment of failure) 
( )

c

lim

lim c

f

E

ε

λ

= , 
el

ε  – 

ultimate elastic strain 

0 4
c

el

c

, f

E

ε ≈ . 

 

Then the tensile stress could be expressed as 

 

ctctcct

E ελσ = .        (7) 

 

The stress-strain relationships expressed according to 

the equation (5) and equation (7) to the C30/37 class 

concrete is presented in fig. 3 where the are used not the 

average values of stress (
ctm

f ) but characteristic values 

(fctk0,05). For this reason the factor 0,7 were applied to the 

values obtained by equation (5). According to that figure the 

curves is differ but it should be note that the ultimate strain 

is differ also. 

 

Fig. 3. The stress-strain relationship of tensile concrete 

(
005ct ctk,

f f= ). 1– when the elastic plastic coefficient 

0 5
ct,lim

,λ =  is applied, 2– when the equation (5) is applied with 

the factor 0,7. 

 

4. The crack moment of members reinforced by 
steel reinforcement and FRP reinforcement 

 

In this section the cracking moments are calculated 

with assumption that the behaviour of compressed concrete 

is elastic because not high the amount of reinforcement was 

predicted. For calculations the stress and strain schemes 

presented in fig. 4 were used. 

Calculating the cracking moment of elements 

reinforced by FRP reinforcement, were assumed that the 

bond between reinforcement and concrete is quite enough.  

ε

ct,lim

 

 – ultimate value of 
the elastic plastic coefficient at the 
moment of failure; (6)

concrete class from 
8

10

C  to 
50

60

C  the strain could be 

expressed with average factor: 

 

cm

ctm

ct

f

f
31.1

1,

00156.0

=ε .     (2) 

 

Then the member k to the tensile concrete would be 

obtained from the other equation 

 

0.3

,1

1.31

0.31 0.7

1.05

1.05 22000

10

0.00156 18.06

cmcm ct

ctm

ctm

ctm cm

cm ctm

fE

k
f

f

f f

f f

ε

−

⎛ ⎞
= = ⋅ ⋅

⎜ ⎟

⎝ ⎠

⋅ =

⋅

.   (3) 

When the ultimate concrete strain it is known the tensile 

stress could be expressed according to the adjusted EC2 

equation: 

 

( ) ( )
ctctmcmcmctm

ctctmcmctcmctm

ctm

ct

ffff

ffff

k

k

f ε

εε

η

ηησ

31.17.031.0

262.223.012

206.186411

41091411577

21
−−

−−

−⋅+

−

=

−+

−

= .        (4) 

 

 
 

The tensile stress of concrete (class 40/50) obtained 

according to the equation (4) is presented in fig. 2 where the 

peak of stress is reached at the strain equal to 0.000165. 

 

 

Fig. 2. The stress-strain relationship expressed according to 

the equation (4) (1-curve) and equation (1) (2-curve). 

 

The obtained expression (4) is not so convenient to the 

resultant calculation of the tensile and compressed concrete 

zones and to the crack moment also, because the strain 

member is in the denominator. Such not well on calculation 

of integrals. In this case more convenient the tensile stress 

expressed by polynomials (Židonis 2007, 2009, 2010). Then 

the stress of tensile concrete will be equal  

 

2

1 2 3ct ct ct
a a aσ ε ε= ⋅ + ⋅ + ,    (5) 

 

 where 
1
a , 

2
a , 

3
a  – free members.  

 

The free members corresponded the stress-strain 

relationship of the C30/37 class concrete is equal to: 

1 105003697.6a = − , 2 34987.8a = , 3 0.0127a = . 

2 case. It is also convenient to apply the elastic plastic 

coefficient λ  to express the σ ε−  relationship. If to 

assume that this coefficient linear vary from the beginning 

of elastic plastic zone until to concrete fracture it could be 

expressed as (Augonis 2013)  

 

1
i el

lim

lim el

ε ε

λ λ

ε ε

−

= −

−

,      (6) 

 

where 
lim

λ  – ultimate value of the elastic plastic coefficient 

at the moment of failure, 
i

ε  –strain, 
lim
ε  – ultimate strain 

(strain at the moment of failure) 
( )

c

lim

lim c

f

E

ε

λ

= , 
el

ε  – 

ultimate elastic strain 

0 4
c

el

c

, f

E

ε ≈ . 

 

Then the tensile stress could be expressed as 

 

ctctcct

E ελσ = .        (7) 

 

The stress-strain relationships expressed according to 

the equation (5) and equation (7) to the C30/37 class 

concrete is presented in fig. 3 where the are used not the 

average values of stress (
ctm

f ) but characteristic values 

(fctk0,05). For this reason the factor 0,7 were applied to the 

values obtained by equation (5). According to that figure the 

curves is differ but it should be note that the ultimate strain 

is differ also. 

 

Fig. 3. The stress-strain relationship of tensile concrete 

(
005ct ctk,

f f= ). 1– when the elastic plastic coefficient 

0 5
ct,lim

,λ =  is applied, 2– when the equation (5) is applied with 

the factor 0,7. 

 

4. The crack moment of members reinforced by 
steel reinforcement and FRP reinforcement 

 

In this section the cracking moments are calculated 

with assumption that the behaviour of compressed concrete 

is elastic because not high the amount of reinforcement was 

predicted. For calculations the stress and strain schemes 

presented in fig. 4 were used. 

Calculating the cracking moment of elements 

reinforced by FRP reinforcement, were assumed that the 

bond between reinforcement and concrete is quite enough.  

ε

ct,lim

 

iε  – strain;

limε  – ultimate strain (strain at the 

moment of failure) ( )
c

lim
lim c

f
E

ε
λ

= , 

elε  – ultimate elastic strain
0 4 c

el
c

, f
E

ε ≈ .

Then the tensile stress could be 
expressed as

(7)

concrete class from 
8

10

C  to 
50

60

C  the strain could be 

expressed with average factor: 

 

cm

ctm

ct

f

f
31.1

1,

00156.0

=ε .     (2) 

 

Then the member k to the tensile concrete would be 

obtained from the other equation 

 

0.3

,1

1.31

0.31 0.7

1.05

1.05 22000

10

0.00156 18.06

cmcm ct

ctm

ctm

ctm cm

cm ctm

fE

k
f

f

f f

f f

ε

−

⎛ ⎞
= = ⋅ ⋅

⎜ ⎟

⎝ ⎠

⋅ =

⋅

.   (3) 

When the ultimate concrete strain it is known the tensile 

stress could be expressed according to the adjusted EC2 

equation: 

 

( ) ( )
ctctmcmcmctm

ctctmcmctcmctm

ctm

ct

ffff

ffff

k

k

f ε

εε

η

ηησ

31.17.031.0

262.223.012

206.186411

41091411577

21
−−

−−

−⋅+

−

=

−+

−

= .        (4) 

 

 
 

The tensile stress of concrete (class 40/50) obtained 

according to the equation (4) is presented in fig. 2 where the 

peak of stress is reached at the strain equal to 0.000165. 

 

 

Fig. 2. The stress-strain relationship expressed according to 

the equation (4) (1-curve) and equation (1) (2-curve). 

 

The obtained expression (4) is not so convenient to the 

resultant calculation of the tensile and compressed concrete 

zones and to the crack moment also, because the strain 

member is in the denominator. Such not well on calculation 

of integrals. In this case more convenient the tensile stress 

expressed by polynomials (Židonis 2007, 2009, 2010). Then 

the stress of tensile concrete will be equal  

 

2

1 2 3ct ct ct
a a aσ ε ε= ⋅ + ⋅ + ,    (5) 

 

 where 
1
a , 

2
a , 

3
a  – free members.  

 

The free members corresponded the stress-strain 

relationship of the C30/37 class concrete is equal to: 

1 105003697.6a = − , 2 34987.8a = , 3 0.0127a = . 

2 case. It is also convenient to apply the elastic plastic 

coefficient λ  to express the σ ε−  relationship. If to 

assume that this coefficient linear vary from the beginning 

of elastic plastic zone until to concrete fracture it could be 

expressed as (Augonis 2013)  

 

1
i el

lim

lim el

ε ε

λ λ

ε ε

−

= −

−

,      (6) 

 

where 
lim

λ  – ultimate value of the elastic plastic coefficient 

at the moment of failure, 
i

ε  –strain, 
lim
ε  – ultimate strain 

(strain at the moment of failure) 
( )

c

lim

lim c

f

E

ε

λ

= , 
el

ε  – 

ultimate elastic strain 

0 4
c

el

c

, f

E

ε ≈ . 

 

Then the tensile stress could be expressed as 

 

ctctcct

E ελσ = .        (7) 

 

The stress-strain relationships expressed according to 

the equation (5) and equation (7) to the C30/37 class 

concrete is presented in fig. 3 where the are used not the 

average values of stress (
ctm

f ) but characteristic values 

(fctk0,05). For this reason the factor 0,7 were applied to the 

values obtained by equation (5). According to that figure the 

curves is differ but it should be note that the ultimate strain 

is differ also. 

 

Fig. 3. The stress-strain relationship of tensile concrete 

(
005ct ctk,

f f= ). 1– when the elastic plastic coefficient 

0 5
ct,lim

,λ =  is applied, 2– when the equation (5) is applied with 

the factor 0,7. 

 

4. The crack moment of members reinforced by 
steel reinforcement and FRP reinforcement 

 

In this section the cracking moments are calculated 

with assumption that the behaviour of compressed concrete 

is elastic because not high the amount of reinforcement was 

predicted. For calculations the stress and strain schemes 

presented in fig. 4 were used. 

Calculating the cracking moment of elements 

reinforced by FRP reinforcement, were assumed that the 

bond between reinforcement and concrete is quite enough.  

ε

ct,lim

 

Fig. 3
The stress-strain 

relationship of tensile 
concrete ( 005ct ctk ,f f= ). 
1 – when the elastic plastic 
coefficient 0 5ct ,lim ,λ =  
is applied, 2 – when the 
equation (5) is applied with 
the factor 0,7



63
Journal of Sustainable Architecture and Civil Engineering 2014/4/9

The stress-strain relationships expressed according to the equation (5) and equation (7) to the 
C30/37 class concrete is presented in fig. 3 where the are used not the average values of stress 
( ctmf ) but characteristic values (fctk0,05). For this reason the factor 0,7 were applied to the values 
obtained by equation (5). According to that figure the curves is differ but it should be note that the 
ultimate strain is differ also.

In this section the cracking moments are calculated with assumption that the behaviour of 
compressed concrete is elastic because not high the amount of reinforcement was predicted. For 
calculations the stress and strain schemes presented in fig. 4 were used.

Calculating the cracking moment of elements reinforced by FRP reinforcement, were assumed 
that the bond between reinforcement and concrete is quite enough. 

When the cracking moment is calculating according to the elastic plastic coefficient the ultimate 

tensile strain is equal to ctct ,lim c
f

Eε λ=  (Baikov 1991). In this case it is important right to evaluate 
the member limλ  when crack opens. 
For tensile concrete at the short term failure case this coefficient approximately equal to ~0.5, i.e., 

0 5lim .λ =  (Baikov 1991). Of course it is average value that could vary. But if to admit that value, 

the ultimate strain will be equal to 2 ctct ,lim
c

f
Eε = . Then the resultant of tensile concrete zone 

would be equal:

The crack 
moment of 
members 
reinforced 
by steel 
reinforcement 
and FRP 
reinforcement

When the cracking moment is calculating according to 

the elastic plastic coefficient the ultimate tensile strain is 

equal to 
ct

ct,lim
c

f

E
ε

λ

=  (Baikov 1991). In this case it is 

important right to evaluate the member 
lim

λ  when crack 

opens.  

For tensile concrete at the short term failure case this 

coefficient approximately equal to ~0.5, i.e., 0 5
lim

.λ =  

(Baikov 1991). Of course it is average value that could vary. 

But if to admit that value, the ultimate strain will be equal to 

2
ct

ct,lim

c

f

E
ε = . Then the resultant of tensile concrete zone 

would be equal: 

 

ε
x

b

A
s

ct,el

ε
ct

ε
s

y

xpl

el

x

h

c

d

M

x

ciε

F
ct,el

σ

σct

ct,el

F
s

ct,pl
F

c

σ
ci

h

F
c

 

 

Fig. 4. Strain and stress of element when compressive concrete 

behaviour is elastic. 

 

1

2
el

h x

ct el c el ct ct c

x

F E b x E bε ε λ

−

= ⋅ ⋅ + ∫ .    (8) 

 

So, the cracking moment will be reached then the 

strain of the tensile concrete will reach the ultimate strain 

value 
ct ct,lim

ε ε= . According this the cracking moment 

could be expressed by the next equation (Augonis, 2013) 

 

( )

( )

( ) ( )( )

( )

ss

c

c

elct

ccc

elctctelct

elelctctelelct

ccrc

AE

x

xd

xbExbE

kxhkxh

xx

kxhbEM

2

2

2

3

,2

,lim,,

,lim,,

3

1

3

1

24

5

3

34

3

−

+++

+

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

−−

+

−

+

+

−

+

−=

ε

ε

ε

ε

εεε

εεε

, (9) 

 

where 1
ct,el

c

k

ε

ε

= + ,  
ct,el

el

c

x

x

ε

ε

= ,  
ct,lim

c

x

h x

ε

ε =

−

.  

 

The height of the compressed zone is calculated 

according to the force equilibrium equation.  

According to the equation (9) obtained cracking 

moment results were compared with results according to the 

methods of standard EC2 and STR For calculations were 

accepted the rectangular section beam with parameters: 

0 2b . m= , 
1

0 05d . m= , 200
s

E GPa= , 50
k

E GPa= , 

30
c

E GPa= , 
2

10
s k

A A cm= = , C30/37. When the 

reinforcement area is constant and varies the height of cross 

section, the relationship of cracking moment and 

reinforcement ratio is presented in fig. 5 where could be 

seen that the results obtained according to the equation (9) is 

between the results of EC2 and STR methods. In this figure 

the marking: “Mcrc,F” means the cracking moment of 

element reinforced by FRP reinforcement, “Mcrc,F” – 

reinforced by steel reinforcement, “Mec2” – cracking 

moment calculated according to EC2 method and “Mstr” – 

cracking moment calculated according to STR method. 

When the cross section area is constant and varies the 

reinforcement area, the relationship of cracking moment and 

reinforcement ratio is presented in fig. 6. At this case also 

the results obtained according to the equation (8) is between 

the results of EC2 and STR methods but there is higher 

increase of cracking moment when the amount of 

reinforcement arises. 

 

 

Fig. 5. The relationship of the cracking moment and the 

reinforcement coefficient when the area of the reinforcement is 

constant and varies the height of cross section.  

 

 

Fig. 6. The relationship of the cracking moment and the 

reinforcement coefficient when the area of cross section is constant 

and varies the area of reinforcement.  

 

When the cracking moment is calculating according to 

the stress-strain expression (5) the resultant of the tensile 

concrete zone will be equal: 

 

( )
2

1 2 3

1

2
el

h x

ct el c el ct ct

x

F E b x b a a aε ε ε

−

= ⋅ ⋅ + ⋅ + ⋅ +∫ . (10) 

 

The calculation of cracking moment was performed 

analogically as before using the scheme presented in 

figure 4. The calculated cracking moment values according 

to the stress-strain expression (5) and expression (7) is 

presented in figure 7 and figure 8.  

The cracking moment obtained some higher when the 

polynomial stress-strain relationship were used. This 

increase is evaluated by higher value of the ultimate strain 

that is 
,1

0,0001656
ct

ε = . In case when the expression (7) 

Fig. 4
Strain and stress 
of element when 
compressive concrete 
behaviour is elastic

So, the cracking moment will be reached then the 
strain of the tensile concrete will reach the ultimate 
strain value ct ct ,limε ε= . According this the cracking 
moment could be expressed by the next equation 
(Augonis, 2013)

(8)

When the cracking moment is calculating according to 

the elastic plastic coefficient the ultimate tensile strain is 

equal to 
ct

ct,lim
c

f

E
ε

λ

=  (Baikov 1991). In this case it is 

important right to evaluate the member 
lim

λ  when crack 

opens.  

For tensile concrete at the short term failure case this 

coefficient approximately equal to ~0.5, i.e., 0 5
lim

.λ =  

(Baikov 1991). Of course it is average value that could vary. 

But if to admit that value, the ultimate strain will be equal to 

2
ct

ct,lim

c

f

E
ε = . Then the resultant of tensile concrete zone 

would be equal: 

 

ε
x

b

A
s

ct,el

ε
ct

ε
s

y

xpl

el

x

h

c

d

M

x

ciε

F
ct,el

σ

σct

ct,el

F
s

ct,pl
F

c

σ
ci

h

F
c

 

 

Fig. 4. Strain and stress of element when compressive concrete 

behaviour is elastic. 

 

1

2
el

h x

ct el c el ct ct c

x

F E b x E bε ε λ

−

= ⋅ ⋅ + ∫ .    (8) 

 

So, the cracking moment will be reached then the 

strain of the tensile concrete will reach the ultimate strain 

value 
ct ct,lim

ε ε= . According this the cracking moment 

could be expressed by the next equation (Augonis, 2013) 

 

( )

( )

( ) ( )( )

( )

ss

c

c

elct

ccc

elctctelct

elelctctelelct

ccrc

AE

x

xd

xbExbE

kxhkxh

xx

kxhbEM

2

2

2

3

,2

,lim,,

,lim,,

3

1

3

1

24

5

3

34

3

−

+++

+

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

−−

+

−

+

+

−

+

−=

ε

ε

ε

ε

εεε

εεε

, (9) 

 

where 1
ct,el

c

k

ε

ε

= + ,  
ct,el

el

c

x

x

ε

ε

= ,  
ct,lim

c

x

h x

ε

ε =

−

.  

 

The height of the compressed zone is calculated 

according to the force equilibrium equation.  

According to the equation (9) obtained cracking 

moment results were compared with results according to the 

methods of standard EC2 and STR For calculations were 

accepted the rectangular section beam with parameters: 

0 2b . m= , 
1

0 05d . m= , 200
s

E GPa= , 50
k

E GPa= , 

30
c

E GPa= , 
2

10
s k

A A cm= = , C30/37. When the 

reinforcement area is constant and varies the height of cross 

section, the relationship of cracking moment and 

reinforcement ratio is presented in fig. 5 where could be 

seen that the results obtained according to the equation (9) is 

between the results of EC2 and STR methods. In this figure 

the marking: “Mcrc,F” means the cracking moment of 

element reinforced by FRP reinforcement, “Mcrc,F” – 

reinforced by steel reinforcement, “Mec2” – cracking 

moment calculated according to EC2 method and “Mstr” – 

cracking moment calculated according to STR method. 

When the cross section area is constant and varies the 

reinforcement area, the relationship of cracking moment and 

reinforcement ratio is presented in fig. 6. At this case also 

the results obtained according to the equation (8) is between 

the results of EC2 and STR methods but there is higher 

increase of cracking moment when the amount of 

reinforcement arises. 

 

 

Fig. 5. The relationship of the cracking moment and the 

reinforcement coefficient when the area of the reinforcement is 

constant and varies the height of cross section.  

 

 

Fig. 6. The relationship of the cracking moment and the 

reinforcement coefficient when the area of cross section is constant 

and varies the area of reinforcement.  

 

When the cracking moment is calculating according to 

the stress-strain expression (5) the resultant of the tensile 

concrete zone will be equal: 

 

( )
2

1 2 3

1

2
el

h x

ct el c el ct ct

x

F E b x b a a aε ε ε

−

= ⋅ ⋅ + ⋅ + ⋅ +∫ . (10) 

 

The calculation of cracking moment was performed 

analogically as before using the scheme presented in 

figure 4. The calculated cracking moment values according 

to the stress-strain expression (5) and expression (7) is 

presented in figure 7 and figure 8.  

The cracking moment obtained some higher when the 

polynomial stress-strain relationship were used. This 

increase is evaluated by higher value of the ultimate strain 

that is 
,1

0,0001656
ct

ε = . In case when the expression (7) 

The height of the compressed zone is calculated according to the force equilibrium equation. 

According to the equation (9) obtained cracking moment results were compared with results 

according to the methods of standard EC2 and STR For calculations were accepted the 

rectangular section beam with parameters: 0 2b . m= , 1 0 05d . m= , 200sE GPa= , 50kE GPa= ,
30cE GPa= , 

210s kA A cm= = , C30/37. 
When the reinforcement area is constant 
and varies the height of cross section, 
the relationship of cracking moment and 
reinforcement ratio is presented in fig. 5 
where could be seen that the results 
obtained according to the equation (9) 
is between the results of EC2 and STR 
methods. In this figure the marking: 

(9)

When the cracking moment is calculating according to 

the elastic plastic coefficient the ultimate tensile strain is 

equal to 
ct

ct,lim
c

f

E
ε

λ

=  (Baikov 1991). In this case it is 

important right to evaluate the member 
lim

λ  when crack 

opens.  

For tensile concrete at the short term failure case this 

coefficient approximately equal to ~0.5, i.e., 0 5
lim

.λ =  

(Baikov 1991). Of course it is average value that could vary. 

But if to admit that value, the ultimate strain will be equal to 

2
ct

ct,lim

c

f

E
ε = . Then the resultant of tensile concrete zone 

would be equal: 

 

ε
x

b

A
s

ct,el

ε
ct

ε
s

y

xpl

el

x

h

c

d

M

x

ciε

F
ct,el

σ

σct

ct,el

F
s

ct,pl
F

c

σ
ci

h

F
c

 

 

Fig. 4. Strain and stress of element when compressive concrete 

behaviour is elastic. 

 

1

2
el

h x

ct el c el ct ct c

x

F E b x E bε ε λ

−

= ⋅ ⋅ + ∫ .    (8) 

 

So, the cracking moment will be reached then the 

strain of the tensile concrete will reach the ultimate strain 

value 
ct ct,lim

ε ε= . According this the cracking moment 

could be expressed by the next equation (Augonis, 2013) 

 

( )

( )

( ) ( )( )

( )

ss

c

c

elct

ccc

elctctelct

elelctctelelct

ccrc

AE

x

xd

xbExbE

kxhkxh

xx

kxhbEM

2

2

2

3

,2

,lim,,

,lim,,

3

1

3

1

24

5

3

34

3

−

+++

+

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

−−

+

−

+

+

−

+

−=

ε

ε

ε

ε

εεε

εεε

, (9) 

 

where 1
ct,el

c

k

ε

ε

= + ,  
ct,el

el

c

x

x

ε

ε

= ,  
ct,lim

c

x

h x

ε

ε =

−

.  

 

The height of the compressed zone is calculated 

according to the force equilibrium equation.  

According to the equation (9) obtained cracking 

moment results were compared with results according to the 

methods of standard EC2 and STR For calculations were 

accepted the rectangular section beam with parameters: 

0 2b . m= , 
1

0 05d . m= , 200
s

E GPa= , 50
k

E GPa= , 

30
c

E GPa= , 
2

10
s k

A A cm= = , C30/37. When the 

reinforcement area is constant and varies the height of cross 

section, the relationship of cracking moment and 

reinforcement ratio is presented in fig. 5 where could be 

seen that the results obtained according to the equation (9) is 

between the results of EC2 and STR methods. In this figure 

the marking: “Mcrc,F” means the cracking moment of 

element reinforced by FRP reinforcement, “Mcrc,F” – 

reinforced by steel reinforcement, “Mec2” – cracking 

moment calculated according to EC2 method and “Mstr” – 

cracking moment calculated according to STR method. 

When the cross section area is constant and varies the 

reinforcement area, the relationship of cracking moment and 

reinforcement ratio is presented in fig. 6. At this case also 

the results obtained according to the equation (8) is between 

the results of EC2 and STR methods but there is higher 

increase of cracking moment when the amount of 

reinforcement arises. 

 

 

Fig. 5. The relationship of the cracking moment and the 

reinforcement coefficient when the area of the reinforcement is 

constant and varies the height of cross section.  

 

 

Fig. 6. The relationship of the cracking moment and the 

reinforcement coefficient when the area of cross section is constant 

and varies the area of reinforcement.  

 

When the cracking moment is calculating according to 

the stress-strain expression (5) the resultant of the tensile 

concrete zone will be equal: 

 

( )
2

1 2 3

1

2
el

h x

ct el c el ct ct

x

F E b x b a a aε ε ε

−

= ⋅ ⋅ + ⋅ + ⋅ +∫ . (10) 

 

The calculation of cracking moment was performed 

analogically as before using the scheme presented in 

figure 4. The calculated cracking moment values according 

to the stress-strain expression (5) and expression (7) is 

presented in figure 7 and figure 8.  

The cracking moment obtained some higher when the 

polynomial stress-strain relationship were used. This 

increase is evaluated by higher value of the ultimate strain 

that is 
,1

0,0001656
ct

ε = . In case when the expression (7) 

When the cracking moment is calculating according to 

the elastic plastic coefficient the ultimate tensile strain is 

equal to 
ct

ct,lim
c

f

E
ε

λ

=  (Baikov 1991). In this case it is 

important right to evaluate the member 
lim

λ  when crack 

opens.  

For tensile concrete at the short term failure case this 

coefficient approximately equal to ~0.5, i.e., 0 5
lim

.λ =  

(Baikov 1991). Of course it is average value that could vary. 

But if to admit that value, the ultimate strain will be equal to 

2
ct

ct,lim

c

f

E
ε = . Then the resultant of tensile concrete zone 

would be equal: 

 

ε
x

b

A
s

ct,el

ε
ct

ε
s

y

xpl

el

x

h

c

d

M

x

ciε

F
ct,el

σ

σct

ct,el

F
s

ct,pl
F

c

σ
ci

h

F
c

 

 

Fig. 4. Strain and stress of element when compressive concrete 

behaviour is elastic. 

 

1

2
el

h x

ct el c el ct ct c

x

F E b x E bε ε λ

−

= ⋅ ⋅ + ∫ .    (8) 

 

So, the cracking moment will be reached then the 

strain of the tensile concrete will reach the ultimate strain 

value 
ct ct,lim

ε ε= . According this the cracking moment 

could be expressed by the next equation (Augonis, 2013) 

 

( )

( )

( ) ( )( )

( )

ss

c

c

elct

ccc

elctctelct

elelctctelelct

ccrc

AE

x

xd

xbExbE

kxhkxh

xx

kxhbEM

2

2

2

3

,2

,lim,,

,lim,,

3

1

3

1

24

5

3

34

3

−

+++

+

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

−−

+

−

+

+

−

+

−=

ε

ε

ε

ε

εεε

εεε

, (9) 

 

where 1
ct,el

c

k

ε

ε

= + ,  
ct,el

el

c

x

x

ε

ε

= ,  
ct,lim

c

x

h x

ε

ε =

−

.  

 

The height of the compressed zone is calculated 

according to the force equilibrium equation.  

According to the equation (9) obtained cracking 

moment results were compared with results according to the 

methods of standard EC2 and STR For calculations were 

accepted the rectangular section beam with parameters: 

0 2b . m= , 
1

0 05d . m= , 200
s

E GPa= , 50
k

E GPa= , 

30
c

E GPa= , 
2

10
s k

A A cm= = , C30/37. When the 

reinforcement area is constant and varies the height of cross 

section, the relationship of cracking moment and 

reinforcement ratio is presented in fig. 5 where could be 

seen that the results obtained according to the equation (9) is 

between the results of EC2 and STR methods. In this figure 

the marking: “Mcrc,F” means the cracking moment of 

element reinforced by FRP reinforcement, “Mcrc,F” – 

reinforced by steel reinforcement, “Mec2” – cracking 

moment calculated according to EC2 method and “Mstr” – 

cracking moment calculated according to STR method. 

When the cross section area is constant and varies the 

reinforcement area, the relationship of cracking moment and 

reinforcement ratio is presented in fig. 6. At this case also 

the results obtained according to the equation (8) is between 

the results of EC2 and STR methods but there is higher 

increase of cracking moment when the amount of 

reinforcement arises. 

 

 

Fig. 5. The relationship of the cracking moment and the 

reinforcement coefficient when the area of the reinforcement is 

constant and varies the height of cross section.  

 

 

Fig. 6. The relationship of the cracking moment and the 

reinforcement coefficient when the area of cross section is constant 

and varies the area of reinforcement.  

 

When the cracking moment is calculating according to 

the stress-strain expression (5) the resultant of the tensile 

concrete zone will be equal: 

 

( )
2

1 2 3

1

2
el

h x

ct el c el ct ct

x

F E b x b a a aε ε ε

−

= ⋅ ⋅ + ⋅ + ⋅ +∫ . (10) 

 

The calculation of cracking moment was performed 

analogically as before using the scheme presented in 

figure 4. The calculated cracking moment values according 

to the stress-strain expression (5) and expression (7) is 

presented in figure 7 and figure 8.  

The cracking moment obtained some higher when the 

polynomial stress-strain relationship were used. This 

increase is evaluated by higher value of the ultimate strain 

that is 
,1

0,0001656
ct

ε = . In case when the expression (7) 

;

where:

1 ct ,el
c

k
ε
ε

= + ;     
ct ,el

el
c

x
x

ε
ε

= ;    ct ,limc
x

h x
ε

ε =
−

. 

,



Journal of Sustainable Architecture and Civil Engineering 2014/4/9
64

When the cracking moment is calculating according to 

the elastic plastic coefficient the ultimate tensile strain is 

equal to 
ct

ct,lim
c

f

E
ε

λ

=  (Baikov 1991). In this case it is 

important right to evaluate the member 
lim

λ  when crack 

opens.  

For tensile concrete at the short term failure case this 

coefficient approximately equal to ~0.5, i.e., 0 5
lim

.λ =  

(Baikov 1991). Of course it is average value that could vary. 

But if to admit that value, the ultimate strain will be equal to 

2
ct

ct,lim

c

f

E
ε = . Then the resultant of tensile concrete zone 

would be equal: 

 

ε
x

b

A
s

ct,el

ε
ct

ε
s

y

xpl

el

x

h

c

d

M

x

ciε

F
ct,el

σ

σct

ct,el

F
s

ct,pl
F

c

σ
ci

h

F
c

 

 

Fig. 4. Strain and stress of element when compressive concrete 

behaviour is elastic. 

 

1

2
el

h x

ct el c el ct ct c

x

F E b x E bε ε λ

−

= ⋅ ⋅ + ∫ .    (8) 

 

So, the cracking moment will be reached then the 

strain of the tensile concrete will reach the ultimate strain 

value 
ct ct,lim

ε ε= . According this the cracking moment 

could be expressed by the next equation (Augonis, 2013) 

 

( )

( )

( ) ( )( )

( )

ss

c

c

elct

ccc

elctctelct

elelctctelelct

ccrc

AE

x

xd

xbExbE

kxhkxh

xx

kxhbEM

2

2

2

3

,2

,lim,,

,lim,,

3

1

3

1

24

5

3

34

3

−

+++

+

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

−−

+

−

+

+

−

+

−=

ε

ε

ε

ε

εεε

εεε

, (9) 

 

where 1
ct,el

c

k

ε

ε

= + ,  
ct,el

el

c

x

x

ε

ε

= ,  
ct,lim

c

x

h x

ε

ε =

−

.  

 

The height of the compressed zone is calculated 

according to the force equilibrium equation.  

According to the equation (9) obtained cracking 

moment results were compared with results according to the 

methods of standard EC2 and STR For calculations were 

accepted the rectangular section beam with parameters: 

0 2b . m= , 
1

0 05d . m= , 200
s

E GPa= , 50
k

E GPa= , 

30
c

E GPa= , 
2

10
s k

A A cm= = , C30/37. When the 

reinforcement area is constant and varies the height of cross 

section, the relationship of cracking moment and 

reinforcement ratio is presented in fig. 5 where could be 

seen that the results obtained according to the equation (9) is 

between the results of EC2 and STR methods. In this figure 

the marking: “Mcrc,F” means the cracking moment of 

element reinforced by FRP reinforcement, “Mcrc,F” – 

reinforced by steel reinforcement, “Mec2” – cracking 

moment calculated according to EC2 method and “Mstr” – 

cracking moment calculated according to STR method. 

When the cross section area is constant and varies the 

reinforcement area, the relationship of cracking moment and 

reinforcement ratio is presented in fig. 6. At this case also 

the results obtained according to the equation (8) is between 

the results of EC2 and STR methods but there is higher 

increase of cracking moment when the amount of 

reinforcement arises. 

 

 

Fig. 5. The relationship of the cracking moment and the 

reinforcement coefficient when the area of the reinforcement is 

constant and varies the height of cross section.  

 

 

Fig. 6. The relationship of the cracking moment and the 

reinforcement coefficient when the area of cross section is constant 

and varies the area of reinforcement.  

 

When the cracking moment is calculating according to 

the stress-strain expression (5) the resultant of the tensile 

concrete zone will be equal: 

 

( )
2

1 2 3

1

2
el

h x

ct el c el ct ct

x

F E b x b a a aε ε ε

−

= ⋅ ⋅ + ⋅ + ⋅ +∫ . (10) 

 

The calculation of cracking moment was performed 

analogically as before using the scheme presented in 

figure 4. The calculated cracking moment values according 

to the stress-strain expression (5) and expression (7) is 

presented in figure 7 and figure 8.  

The cracking moment obtained some higher when the 

polynomial stress-strain relationship were used. This 

increase is evaluated by higher value of the ultimate strain 

that is 
,1

0,0001656
ct

ε = . In case when the expression (7) 

When the cracking moment is calculating according to 

the elastic plastic coefficient the ultimate tensile strain is 

equal to 
ct

ct,lim
c

f

E
ε

λ

=  (Baikov 1991). In this case it is 

important right to evaluate the member 
lim

λ  when crack 

opens.  

For tensile concrete at the short term failure case this 

coefficient approximately equal to ~0.5, i.e., 0 5
lim

.λ =  

(Baikov 1991). Of course it is average value that could vary. 

But if to admit that value, the ultimate strain will be equal to 

2
ct

ct,lim

c

f

E
ε = . Then the resultant of tensile concrete zone 

would be equal: 

 

ε
x

b

A
s

ct,el

ε
ct

ε
s

y

xpl

el

x

h

c

d

M

x

ciε

F
ct,el

σ

σct

ct,el

F
s

ct,pl
F

c

σ
ci

h

F
c

 

 

Fig. 4. Strain and stress of element when compressive concrete 

behaviour is elastic. 

 

1

2
el

h x

ct el c el ct ct c

x

F E b x E bε ε λ

−

= ⋅ ⋅ + ∫ .    (8) 

 

So, the cracking moment will be reached then the 

strain of the tensile concrete will reach the ultimate strain 

value 
ct ct,lim

ε ε= . According this the cracking moment 

could be expressed by the next equation (Augonis, 2013) 

 

( )

( )

( ) ( )( )

( )

ss

c

c

elct

ccc

elctctelct

elelctctelelct

ccrc

AE

x

xd

xbExbE

kxhkxh

xx

kxhbEM

2

2

2

3

,2

,lim,,

,lim,,

3

1

3

1

24

5

3

34

3

−

+++

+

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

−−

+

−

+

+

−

+

−=

ε

ε

ε

ε

εεε

εεε

, (9) 

 

where 1
ct,el

c

k

ε

ε

= + ,  
ct,el

el

c

x

x

ε

ε

= ,  
ct,lim

c

x

h x

ε

ε =

−

.  

 

The height of the compressed zone is calculated 

according to the force equilibrium equation.  

According to the equation (9) obtained cracking 

moment results were compared with results according to the 

methods of standard EC2 and STR For calculations were 

accepted the rectangular section beam with parameters: 

0 2b . m= , 
1

0 05d . m= , 200
s

E GPa= , 50
k

E GPa= , 

30
c

E GPa= , 
2

10
s k

A A cm= = , C30/37. When the 

reinforcement area is constant and varies the height of cross 

section, the relationship of cracking moment and 

reinforcement ratio is presented in fig. 5 where could be 

seen that the results obtained according to the equation (9) is 

between the results of EC2 and STR methods. In this figure 

the marking: “Mcrc,F” means the cracking moment of 

element reinforced by FRP reinforcement, “Mcrc,F” – 

reinforced by steel reinforcement, “Mec2” – cracking 

moment calculated according to EC2 method and “Mstr” – 

cracking moment calculated according to STR method. 

When the cross section area is constant and varies the 

reinforcement area, the relationship of cracking moment and 

reinforcement ratio is presented in fig. 6. At this case also 

the results obtained according to the equation (8) is between 

the results of EC2 and STR methods but there is higher 

increase of cracking moment when the amount of 

reinforcement arises. 

 

 

Fig. 5. The relationship of the cracking moment and the 

reinforcement coefficient when the area of the reinforcement is 

constant and varies the height of cross section.  

 

 

Fig. 6. The relationship of the cracking moment and the 

reinforcement coefficient when the area of cross section is constant 

and varies the area of reinforcement.  

 

When the cracking moment is calculating according to 

the stress-strain expression (5) the resultant of the tensile 

concrete zone will be equal: 

 

( )
2

1 2 3

1

2
el

h x

ct el c el ct ct

x

F E b x b a a aε ε ε

−

= ⋅ ⋅ + ⋅ + ⋅ +∫ . (10) 

 

The calculation of cracking moment was performed 

analogically as before using the scheme presented in 

figure 4. The calculated cracking moment values according 

to the stress-strain expression (5) and expression (7) is 

presented in figure 7 and figure 8.  

The cracking moment obtained some higher when the 

polynomial stress-strain relationship were used. This 

increase is evaluated by higher value of the ultimate strain 

that is 
,1

0,0001656
ct

ε = . In case when the expression (7) 

were used the ultimate strain value is 0 000133
ct,lim

,ε = . 

i.e., some lower. 

 

 

Fig. 7. The relationship of the cracking moment and the 

reinforcement coefficient when the area of the reinforcement is 

constant and varies the height of cross section. (Mcrc,Fo and 

Mcrc,So curves when the equation (5) is applied and Mcrc,F and 

Mcrc,S curves when the equation (7) is applied).  

 

 

Fig. 8. The relationship of the cracking moment and the 

reinforcement coefficient when the area of cross section is constant 

and varies the area of reinforcement. (Mcrc,Fo and Mcrc,So 

curves when the equation (5) is applied and Mcrc,F and Mcrc,S 

curves when the equation (7) is applied). 

 

Analysing the influence of stress-strain relationship to 

the cracking moment it is important to evaluate the elastic 

modulus of reinforcement. Calculated the cracking moment 

values according to the standard EC2 and STR methods and 

equation (9) with vary elastic modulus of reinforcement is 

presented in figure 9.  

 

 

 

Fig. 9. The relationship of the cracking moment and the 

elastic modulus of reinforcement. 

 

 

In this figure could be seen that curves obtained using 

different expressions of stress-strain diagram of tensile is 

quite similar.  

In these calculations were accepted such parameters.: 

30 37C / , 0 2b . m= , 0 4h . m= , 
1

0 05d . m= , 
2

10
s

A cm= .  

If some part of steel reinforcement to change by FRP 

reinforcement the cracking moment will be obtained 

between values of element reinforced only by steel 

reinforcement and reinforced only by FRP reinforcement. 

 

4. Conclusions 

1. The application of FRP reinforcement has 

significant influence to the cracking moment of flexural 

elements in compare with steel reinforcement and it 

influence increase when the area of FRP reinforcement 

arises. 

2. The application of different stress-strain diagrams 

has no significant influence to the cracking moment 

calculated to the flexural elements reinforced by FRP and 

steel reinforcement. 

3. In all investigated cases the calculated cracking 

moment values varied between values obtained by standard 

EC2 and STR methods. This is because of rectangular 

EC2 method describes highest reserve. 

 

References 

Structures, Moscow: Stroiizdat, 767p. 

EN 1992-1-1:2004. Eurocode 2: Design of concrete structures – 

Part 1-1: General rules and rules for buildings. 225 p. 

Mosley W.H., Bungey J.H., Hulse R. 2007. Reinforced concrete 

design to Eurocode 2. 6th ed. Basingstoke, Palgrave Macmillan. 

STR 2.05.05:2005. Design of Concrete and Reinforced Concrete 

Structures. 123 p. (in Lithuanian). 

Abdala H. A. Evaluation of deflection in concrete members 

reinforced with fibre reinforced polymer (FRP) bars. 

Composite Structures 56 (2002) 63-71. 

http://dx.doi.org/10.1016/S0263-8223(01)00188-X 

Al-Sunna R., Pilakoutas K., Hajirasouliha I., Guadagnini M. 

Deflection behaviour of FRP reinforced concrete beams and 

slabs: An experimental investigation. Composites Part B 43 

(2012) 2125-2134. 

http://dx.doi.org/10.1016/j.compositesb.2012.03.007 

Ashour A. F. Flexural and shear capacities of concrete beams 

reinforced with GFRP bars. Construction and buildng materials 

20 (2006) 1005-1015. 

Barris C., Torres L., Turon A., Baena M., Catalan A. An 

experimental study of the flexural behaviour of GFRP RC 

beams and comparison with prediction models. Composite 

Structures 91 (2009) 286-295. 

http://dx.doi.org/10.1016/j.compstruct.2009.05.0

05 

Barris C., Torres L., Mias C., Vilanova I. .Design of FRP 

Reinforced Concrete Beams for Serviceability Requirements. 

Journal of Civil Engineering and Management 2012 Volume 

18(6): 843–857. 

http://dx.doi.org/10.3846/13923730.2012.72093

4 

were used the ultimate strain value is 0 000133
ct,lim

,ε = . 

i.e., some lower. 

 

 

Fig. 7. The relationship of the cracking moment and the 

reinforcement coefficient when the area of the reinforcement is 

constant and varies the height of cross section. (Mcrc,Fo and 

Mcrc,So curves when the equation (5) is applied and Mcrc,F and 

Mcrc,S curves when the equation (7) is applied).  

 

 

 

curves when the equation (7) is applied). 

 

Analysing the influence of stress-strain relationship to 

the cracking moment it is important to evaluate the elastic 

modulus of reinforcement. Calculated the cracking moment 

values according to the standard EC2 and STR methods and 

equation (9) with vary elastic modulus of reinforcement is 

presented in figure 9.  

 

 

 

Fig. 9. The relationship of the cracking moment and the 

elastic modulus of reinforcement. 

 

 

In this figure could be seen that curves obtained using 

different expressions of stress-strain diagram of tensile is 

quite similar.  

In these calculations were accepted such parameters.: 

30 37C / , 0 2b . m= , 0 4h . m= , 
1

0 05d . m= , 
2

10
s

A cm= .  

If some part of steel reinforcement to change by FRP 

reinforcement the cracking moment will be obtained 

between values of element reinforced only by steel 

reinforcement and reinforced only by FRP reinforcement. 

 

4. Conclusions 

1. The application of FRP reinforcement has 

significant influence to the cracking moment of flexural 

elements in compare with steel reinforcement and it 

influence increase when the area of FRP reinforcement 

arises. 

2. The application of different stress-strain diagrams 

has no significant influence to the cracking moment 

calculated to the flexural elements reinforced by FRP and 

steel reinforcement. 

3. In all investigated cases the calculated cracking 

moment values varied between values obtained by standard 

EC2 and STR methods. This is because of rectangular 

diagram that is described in STR. Also it could be state that 

EC2 method describes highest reserve. 

 

References 

11. 

design to Eurocode 2. 6th ed. Basingstoke, Palgrave Macmillan. 

STR 2.05.05:2005. Design of Concrete and Reinforced Concrete 

Structures. 123 p. (in Lithuanian). 

Abdala H. A. Evaluation of deflection in concrete members 

reinforced with fibre reinforced polymer (FRP) bars. 

Composite Structures 56 (2002) 63-71. 

http://dx.doi.org/10.1016/S0263-8223(01)00188-X 

Al-Sunna R., Pilakoutas K., Hajirasouliha I., Guadagnini M. 

Deflection behaviour of FRP reinforced concrete beams and 

slabs: An experimental investigation. Composites Part B 43 

(2012) 2125-2134. 

http://dx.doi.org/10.1016/j.compositesb.2012.03.007 

Ashour A. F. Flexural and shear capacities of concrete beams 

reinforced with GFRP bars. Construction and buildng materials 

20 (2006) 1005-1015. 

Barris C., Torres L., Turon A., Baena M., Catalan A. An 

experimental study of the flexural behaviour of GFRP RC 

beams and comparison with prediction models. Composite 

Structures 91 (2009) 286-295. 

http://dx.doi.org/10.1016/j.compstruct.2009.05.0

05 

Barris C., Torres L., Mias C., Vilanova I. .Design of FRP 

Reinforced Concrete Beams for Serviceability Requirements. 

Journal of Civil Engineering and Management 2012 Volume 

18(6): 843–857. 

http://dx.doi.org/10.3846/13923730.2012.72093

4 

“Mcrc,F” means the cracking moment 
of element reinforced by FRP 
reinforcement, “Mcrc,F” – reinforced by 
steel reinforcement, “Mec2” – cracking 
moment calculated according to EC2 
method and “Mstr” – cracking moment 
calculated according to STR method.

When the cross section area is constant 
and varies the reinforcement area, 
the relationship of cracking moment 
and reinforcement ratio is presented 
in fig. 6. At this case also the results 
obtained according to the equation (8) 
is between the results of EC2 and STR 
methods but there is higher increase 
of cracking moment when the amount 
of reinforcement arises.

When the cracking moment is 
calculating according to the stress-
strain expression (5) the resultant of 
the tensile concrete zone will be equal:

The calculation of cracking moment 
was performed analogically as before 
using the scheme presented in figure 4. 
The calculated cracking moment 
values according to the stress-strain 
expression (5) and expression (7) is 
presented in figure 7 and figure 8. 

The cracking moment obtained 
some higher when the polynomial 
stress-strain relationship were used. 
This increase is evaluated by higher 
value of the ultimate strain that is 

,1 0, 0001656ctε = . In case when the 
expression (7) were used the ultimate 
strain value is 0 000133ct ,lim ,ε = . i.e., 
some lower.

Analysing the influence of stress-
strain relationship to the cracking 
moment it is important to evaluate 
the elastic modulus of reinforcement. 

(10)

When the cracking moment is calculating according to 

the elastic plastic coefficient the ultimate tensile strain is 

equal to 
ct

ct,lim
c

f

E
ε

λ

=  (Baikov 1991). In this case it is 

important right to evaluate the member 
lim

λ  when crack 

opens.  

For tensile concrete at the short term failure case this 

coefficient approximately equal to ~0.5, i.e., 0 5
lim

.λ =  

(Baikov 1991). Of course it is average value that could vary. 

But if to admit that value, the ultimate strain will be equal to 

2
ct

ct,lim

c

f

E
ε = . Then the resultant of tensile concrete zone 

would be equal: 

 

ε
x

b

A
s

ct,el

ε
ct

ε
s

y

xpl

el

x

h

c

d

M

x

ciε

F
ct,el

σ

σct

ct,el

F
s

ct,pl
F

c

σ
ci

h

F
c

 

 

Fig. 4. Strain and stress of element when compressive concrete 

behaviour is elastic. 

 

1

2
el

h x

ct el c el ct ct c

x

F E b x E bε ε λ

−

= ⋅ ⋅ + ∫ .    (8) 

 

So, the cracking moment will be reached then the 

strain of the tensile concrete will reach the ultimate strain 

value 
ct ct,lim

ε ε= . According this the cracking moment 

could be expressed by the next equation (Augonis, 2013) 

 

( )

( )

( ) ( )( )

( )

ss

c

c

elct

ccc

elctctelct

elelctctelelct

ccrc

AE

x

xd

xbExbE

kxhkxh

xx

kxhbEM

2

2

2

3

,2

,lim,,

,lim,,

3

1

3

1

24

5

3

34

3

−

+++

+

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

−−

+

−

+

+

−

+

−=

ε

ε

ε

ε

εεε

εεε

, (9) 

 

where 1
ct,el

c

k

ε

ε

= + ,  
ct,el

el

c

x

x

ε

ε

= ,  
ct,lim

c

x

h x

ε

ε =

−

.  

 

The height of the compressed zone is calculated 

according to the force equilibrium equation.  

0 2b . m= , 
1

0 05d . m= , 200
s

E GPa= , 50
k

E =

30
c

E GPa= , 
2

10
s k

A A cm= = , C30/37. When the 

reinforcement area is constant and varies the height of cross 

section, the relationship of cracking moment and 

reinforcement ratio is presented in fig. 5 where could be 

seen that the results obtained according to the equation (9) is 

between the results of EC2 and STR methods. In this figure 

the marking: “Mcrc,F” means the cracking moment of 

element reinforced by FRP reinforcement, “Mcrc,F” – 

reinforced by steel reinforcement, “Mec2” – cracking 

moment calculated according to EC2 method and “Mstr” – 

cracking moment calculated according to STR method. 

When the cross section area is constant and varies the 

reinforcement area, the relationship of cracking moment and 

reinforcement ratio is presented in fig. 6. At this case also 

the results obtained according to the equation (8) is between 

the results of EC2 and STR methods but there is higher 

increase of cracking moment when the amount of 

reinforcement arises. 

 

 

Fig. 5. 

constant and varies the height of cross section.  

 

 

Fig. 6. The relationship of the cracking moment and the 

 

( )
2

1 2 3

1

2
el

h x

ct el c el

x

F E b xε

−

= ⋅ ⋅ + ∫ . (10) 

polynomial stress-strain relationship were used. This 

increase is evaluated by higher value of the ultimate strain 

that is 
,1

0,0001656
ct

ε = . In case when the expression (7) 

When the cracking moment is calculating according to 

the elastic plastic coefficient the ultimate tensile strain is 

equal to 
ct

ct,lim
c

f

E
ε

λ

=  (Baikov 1991). In this case it is 

important right to evaluate the member 
lim

λ  when crack 

opens.  

For tensile concrete at the short term failure case this 

coefficient approximately equal to ~0.5, i.e., 0 5
lim

.λ =  

(Baikov 1991). Of course it is average value that could vary. 

But if to admit that value, the ultimate strain will be equal to 

2
ct

ct,lim

c

f

E
ε = . Then the resultant of tensile concrete zone 

would be equal: 

 

ε
x

b

A
s

ct,el

ε
ct

ε
s

y

xpl

el

x

h

c

d

M

x

ciε

F
ct,el

σ

σct

ct,el

F
s

ct,pl
F

c

σ
ci

h

F
c

 

 

Fig. 4. Strain and stress of element when compressive concrete 

behaviour is elastic. 

 

1

2
el

h x

ct el c el ct ct c

x

F E b x E bε ε λ

−

= ⋅ ⋅ + ∫ .    (8) 

 

So, the cracking moment will be reached then the 

strain of the tensile concrete will reach the ultimate strain 

value 
ct ct,lim

ε ε= . According this the cracking moment 

could be expressed by the next equation (Augonis, 2013) 

 

( )

( )

( ) ( )( )

( )

ss

c

c

elct

ccc

elctctelct

elelctctelelct

ccrc

AE

x

xd

xbExbE

kxhkxh

xx

kxhbEM

2

2

2

3

,2

,lim,,

,lim,,

3

1

3

1

24

5

3

34

3

−

+++

+

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

−−

+

−

+

+

−

+

−=

ε

ε

ε

ε

εεε

εεε

, (9) 

 

where 1
ct,el

c

k

ε

ε

= + ,  
ct,el

el

c

x

x

ε

ε

= ,  
ct,lim

c

x

h x

ε

ε =

−

.  

 

The height of the compressed zone is calculated 

according to the force equilibrium equation.  

0 2b . m= , 
1

0 05d . m= , 200
s

E GPa= , 50
k

E =

30
c

E GPa= , 
2

10
s k

A A cm= = , C30/37. When the 

reinforcement area is constant and varies the height of cross 

section, the relationship of cracking moment and 

reinforcement ratio is presented in fig. 5 where could be 

seen that the results obtained according to the equation (9) is 

between the results of EC2 and STR methods. In this figure 

the marking: “Mcrc,F” means the cracking moment of 

element reinforced by FRP reinforcement, “Mcrc,F” – 

reinforced by steel reinforcement, “Mec2” – cracking 

moment calculated according to EC2 method and “Mstr” – 

cracking moment calculated according to STR method. 

When the cross section area is constant and varies the 

reinforcement area, the relationship of cracking moment and 

reinforcement ratio is presented in fig. 6. At this case also 

the results obtained according to the equation (8) is between 

the results of EC2 and STR methods but there is higher 

increase of cracking moment when the amount of 

reinforcement arises. 

 

 

Fig. 5. The relationship of the cracking moment and the 

reinforcement coefficient when the area of the reinforcement is 

constant and varies the height of cross section.  

 

 

Fig. 6. The relationship of the cracking moment and the 

 

( )
2

1 2 3

1

2
el

h x

ct el c el ct ct

x

F E b x b a a aε ε ε

−

= ⋅ ⋅ + ⋅ + ⋅ +∫ . (10) 

polynomial stress-strain relationship were used. This 

increase is evaluated by higher value of the ultimate strain 

that is 
,1

0,0001656
ct

ε = . In case when the expression (7) 

.

Fig. 5
The relationship of the 

cracking moment and the 
reinforcement coefficient 

when the area of the 
reinforcement is constant 

and varies the height of 
cross section

Fig. 6
The relationship of the 

cracking moment and the 
reinforcement coefficient 

when the area of cross 
section is constant 

and varies the area of 
reinforcement

Fig. 7
The relationship of the 

cracking moment and the 
reinforcement coefficient 

when the area of the 
reinforcement is constant 

and varies the height of 
cross section. (Mcrc,Fo 

and Mcrc,So curves when 
the equation (5) is applied 

and Mcrc,F and Mcrc,S 
curves when the equation 

(7) is applied)

Fig. 8
The relationship of the 

cracking moment and the 
reinforcement coefficient 

when the area of cross 
section is constant 

and varies the area of 
reinforcement. (Mcrc,Fo 

and Mcrc,So curves when 
the equation (5) is applied 

and Mcrc,F and Mcrc,S 
curves when the equation 

(7) is applied)



65
Journal of Sustainable Architecture and Civil Engineering 2014/4/9

Calculated the cracking moment 
values according to the standard EC2 
and STR methods and equation (9) with 
vary elastic modulus of reinforcement 
is presented in figure 9. 

In this figure could be seen that curves 
obtained using different expressions 
of stress-strain diagram of tensile is 
quite similar. 

In these calculations were accepted 
such parameters: 30 37C / , 0 2b . m= , 

0 4h . m= , 1 0 05d . m= , 
210sA cm= . 

were used the ultimate strain value is 0 000133
ct,lim

,ε = . 

i.e., some lower. 

 

 

Fig. 7. The relationship of the cracking moment and the 

reinforcement coefficient when the area of the reinforcement is 

constant and varies the height of cross section. (Mcrc,Fo and 

Mcrc,So curves when the equation (5) is applied and Mcrc,F and 

Mcrc,S curves when the equation (7) is applied).  

 

 

 

Fig. 8. The relationship of the cracking moment and the 

reinforcement coefficient when the area of cross section is constant 

and varies the area of reinforcement. (Mcrc,Fo and Mcrc,So 

curves when the equation (5) is applied and Mcrc,F and Mcrc,S 

the cracking moment it is important to evaluate the elastic 

modulus of reinforcement. Calculated the cracking moment 

values according to the standard EC2 and STR methods and 

equation (9) with vary elastic modulus of reinforcement is 

presented in figure 9.  

 

 

 

Fig. 9. The relationship of the cracking moment and the 

elastic modulus of reinforcement. 

 

 

In this figure could be seen that curves obtained using 

different expressions of stress-strain diagram of tensile is 

quite similar.  

In these calculations were accepted such parameters.: 

30 37C / , 0 2b . m= , 0 4h . m= , 
1

0 05d . m= , 
2

10
s

A cm= .  

If some part of steel reinforcement to change by FRP 

reinforcement the cracking moment will be obtained 

between values of element reinforced only by steel 

reinforcement and reinforced only by FRP reinforcement. 

 

4. Conclusions 

1. The application of FRP reinforcement has 

significant influence to the cracking moment of flexural 

elements in compare with steel reinforcement and it 

influence increase when the area of FRP reinforcement 

arises. 

2. The application of different stress-strain diagrams 

has no significant influence to the cracking moment 

calculated to the flexural elements reinforced by FRP and 

steel reinforcement. 

3. In all investigated cases the calculated cracking 

moment values varied between values obtained by standard 

EC2 and STR methods. This is because of rectangular 

diagram that is described in STR. Also it could be state that 

EC2 method describes highest reserve. 

 

References 

Augonis A., Zadlauskas S. 2013. The elastoplastic concrete strain 

influence on the cracking moment and deformation of 

rectangular reinforced concrete elements. Mechanika, 19(1), 5-

11. http://dx.doi.org/10.5755/j01.mech.19.1.3619 

Baikov, V.; Sigalov, E. 1991. Reinforced Concrete 

Structures, Moscow: Stroiizdat, 767p. 

EN 1992-1-1:2004. Eurocode 2: Design of concrete structures – 

Part 1-1: General rules and rules for buildings. 225 p. 

Mosley W.H., Bungey J.H., Hulse R. 2007. Reinforced concrete 

reinforced with fibre reinforced polymer (FRP) bars. 

Composite Structures 56 (2002) 63-71. 

 

Al-Sunna R., Pilakoutas K., Hajirasouliha I., Guadagnini M. 

Deflection behaviour of FRP reinforced concrete beams and 

slabs: An experimental investigation. Composites Part B 43 

(2012) 2125-2134. 

 

reinforced with GFRP bars. Construction and buildng materials 

20 (2006) 1005-1015. 

Barris C., Torres L., Turon A., Baena M., Catalan A. An 

experimental study of the flexural behaviour of GFRP RC 

beams and comparison with prediction models. Composite 

843–857. 

Fig. 9
The relationship of the 
cracking moment and 
the elastic modulus of 
reinforcement

If some part of steel reinforcement to change by FRP reinforcement the cracking moment will be 
obtained between values of element reinforced only by steel reinforcement and reinforced only by 
FRP reinforcement.

1 The application of FRP reinforcement has significant influence to the cracking moment of flexural elements in compare with steel reinforcement and it influence increase when the 
area of FRP reinforcement arises.

2 The application of different stress-strain diagrams has no significant influence to the cracking moment calculated to the flexural elements reinforced by FRP and steel 
reinforcement.

3 In all investigated cases the calculated cracking moment values varied between values obtained by standard EC2 and STR methods. This is because of rectangular diagram that is 
described in STR. Also it could be state that EC2 method describes highest reserve.

Conclusions

Augonis A., Zadlauskas S. 2013. The elastoplastic 
concrete strain influence on the cracking moment 
and deformation of rectangular reinforced concrete 
elements. Mechanika, 19(1), 5-11. http://dx.doi.
org/10.5755/j01.mech.19.1.3619

Baikov, V.; Sigalov, E. 1991. Reinforced Concrete 
Structures, Moscow: Stroiizdat, 767p.

EN 1992-1-1:2004. Eurocode 2: Design of concrete 
structures – Part 1-1: General rules and rules for 
buildings. 225 p.

Mosley W.H., Bungey J.H., Hulse R. 2007. Reinforced 
concrete design to Eurocode 2. 6th ed. Basingstoke, 
Palgrave Macmillan.

STR 2.05.05:2005. Design of Concrete and Reinforced 
Concrete Structures. 123 p. (in Lithuanian).

Abdala H. A. Evaluation of deflection in concrete 
members reinforced with fibre reinforced polymer 
(FRP) bars. Composite Structures 56 (2002) 63-71. 
http://dx.doi.org/10.1016/S0263-8223(01)00188-X

Al-Sunna R., Pilakoutas K., Hajirasouliha I., 
Guadagnini M. Deflection behaviour of FRP 
reinforced concrete beams and slabs: An 

experimental investigation. Composites Part B 
43 (2012) 2125-2134. http://dx.doi.org/10.1016/j.
compositesb.2012.03.007

Ashour A. F. Flexural and shear capacities of concrete 
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h t t p : / / g a l e n . s u / p ro d u k t s i y a / k o m p o z i t n a y a -
a r m a t u ra - ro c k b a r / f i z i k o - m e k h a n i c h e s k i ye -
ispytaniya

MINDAUGAS 
AUGONIS

Assoc. Prof.

Kaunas University of 
Technology, Faculty 
of Civil Engineering 
and Architecture, 
Department of Building 
Structures

Main research area
Durability of 
Engineering Structures, 
Strength and Stability 
of Reinforced Concrete 
Structures

Address
Studentu st. 48 
LT-51367, Kaunas 
Lithuania
Tel. +370 37 3004731
E-mail: 
mindaugas.augonis@
ktu.lt

ROBERTAS 
BALEVIČIUS  

Assoc. Prof.

Vilnius Gediminas 
Technical University 
Faculty of Civil 
Engineering, 
Department of 
reinforced concrete and 
masonry structures

Main research area
Strength, Stability 
and Serviceability of 
Reinforced Concrete 
Structures

Address
Sauletekio al. 11 
LT-51367, Vilnius 
Lithuania
Tel. +370 37 300473
E-mail: robertas.
balevicius@vgtu.lt

MINDAUGAS 
KASIULEVIČIUS  

Lecturer

Kaunas University of 
Technology, Faculty 
of Civil Engineering 
and Architecture, 
Department of Building 
Structures

Main research area
Strength and durability 
of Timber Structures

Address
Studentu st. 48 
LT-51367, Kaunas 
Lithuania.

RĖDA 
BISTRICKAITĖ    

Lecturer  

Kaunas University of 
Technology, Faculty 
of Civil Engineering 
and Architecture 
Department of Building 
Structures

Main research area
Research of concrete 
and reinforced concrete 
innovative structures, 
mechanics of layerd 
structures, new 
composite materials

Address
Studentu st. 48 
LT-51367, Kaunas 
Lithuania

About the 
authors