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

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.


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.( ) where 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 According to codes, STR Ø+10mm, For precast reinforced concrete Ø+5mm directly according to the EC2 standard expression rewriting the ctm f instead of cm f 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 differ more than six times.Also, according where: (1) So, if to calculate the tensile stress ct σ directly cording to the EC2 standard expression rewriting the ctm f stead of cm f only, we will obtain: here According such way drawing up the c c σ ε − lationship of tensile concrete we will obtain the curve that rm at all will not be similar to the form of curve of mpressed concrete (fig.2).In this case the curves forms t correspond each other because of different values of the ( ) where 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 where 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    ( ) 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  (2) en the member k to the tensile concrete would be tained from the other equation When the ultimate concrete strain it is known the tensile stress could be expressed according to the adjusted EC2 equation: 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) 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 .
(2) ile concrete would be When the ultimate concrete strain it is known the tensile stress could be expressed according to the adjusted EC2 equation: (class 40/50) obtained ented in fig. 2 where the equal to 0.000165.
not so convenient to the nd compressed concrete lso, because the strain not well on calculation enient the tensile stress 007, 2009, 2010).Then equal nded the stress-strain concrete is equal to: 3 0.0127 a = .apply the elastic plastic ε − relationship.If to ary from the beginning rete fracture it could be (strain at the moment of failure) ( ) Then the tensile stress could be expressed as 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. .
(2) sile concrete would be When the ultimate concrete strain it is known the tensile stress could be expressed according to the adjusted EC2 equation: (class 40/50) obtained sented in fig. 2 where the n equal to 0.000165.
not so convenient to the and compressed concrete also, because the strain h not well on calculation venient the tensile stress 2007,2009,2010).Then equal onded the stress-strain concrete is equal to: , 3 0.0127 a = .apply the elastic plastic ε − relationship.If to vary from the beginning crete fracture it could be (strain at the moment of failure) ( ) Then the tensile stress could be expressed as 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.).1-when the elastic plastic coefficient 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 concrete class from 8 10 C to 50 60 C the strain could be expressed with average factor: Then the member k to the tensile concrete would be obtained from the other equation (3) 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 where a -free members.
The free members corresponded the stress-strain relationship of the C30/37 class concrete is equal to: 1 105003697.6a = − , 2 34987.8 a = , 3 0.0127 a = .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) (strain at the moment of failure) , f E ε ≈ .
Then the tensile stress could be expressed as 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.C the strain could be essed with average factor: the member k to the tensile concrete would be ined from the other equation When the ultimate concrete strain it is known the tensile stress could be expressed according to the adjusted EC2 equation: ( ) tensile stress of concrete (class 40/50) obtained rding to the equation ( 4) is presented in fig. 2 where the of stress is reached at the strain equal to 0.000165.The obtained expression (4) is not so convenient to the tant calculation of the tensile and compressed concrete s and to the crack moment also, because the strain ber is in the denominator.Such not well on calculation tegrals.In this case more convenient the tensile stress essed by polynomials (Židonis 2007, 2009, 2010).Then tress of tensile concrete will be equal a -free members.
The free members corresponded the stress-strain ionship of the C30/37 class concrete is equal to: 105003697.6− , 2 34987.8 a = , 3 0.0127 a = .2 case.It is also convenient to apply the elastic plastic ficient λ to express the σ ε − relationship.If to me that this coefficient linear vary from the beginning astic plastic zone until to concrete fracture it could be essed as (Augonis 2013) (strain at the moment of failure) ( ) , f E ε ≈ .
Then the tensile stress could be expressed as 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.where 1 a , 2 a , 3 a -free members.
(5) ( ) 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 where a -free members.
The free members corresponded the stress-strain relationship of the C30/37 class concrete is equal to: 1 105003697.6a = − , 2 34987.8 a = , 3 0.0127 a = .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) where lim λ -ultimate value of the elastic plastic coefficient at the moment of failure, , f E ε ≈ .
Then the tensile stress could be expressed as 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.).1-when the elastic plastic coefficient is applied, 2-when the equation ( 5) is applied with the factor 0,7.

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.
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) When the ultimate concrete strain it is known the tensile stress could be expressed according to the adjusted EC2 equation: ( ) oncrete (class 40/50) obtained 4) is presented in fig. 2 where the the strain equal to 0.000165.
ion (4) is not so convenient to the tensile and compressed concrete moment also, because the strain ator.Such not well on calculation ore convenient the tensile stress (Židonis 2007, 2009, 2010).Then te will be equal embers.
corresponded the stress-strain 37 class concrete is equal to: 34987.8, 3 0.0127 a = .venient to apply the elastic plastic (strain at the moment of failure) ( ) , f E ε ≈ .
Then the tensile stress could be expressed as 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.).1-when the elastic plastic coefficient is applied, 2-when the equation ( 5) is applied with the factor 0,7. where 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 where a -free members.
The free members corresponded the stress-strain relationship of the C30/37 class concrete is equal to: 1 105003697.6a = − , 2 34987.8 a = , 3 0.0127 a = .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) where lim λ -ultimate value of the elastic plastic coefficient at the moment of failure, The stres the equation concrete is pr average value (fctk0,05).Fo values obtaine curves is diffe is differ also.( ) 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 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 34987.8 a = , 3 0.0127 a = .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) where lim λ -ultimate value of the elastic plastic coefficient at the moment of failure, , f E ε ≈ .
Then the tensile stress could be expressed as 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. is applied, 2-when the equation ( 5) is applied with the factor 0,7.

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.(2) 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.
Then the tensile stress could be expressed as 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 Fig. 3 The stress-strain relationship of tensile concrete ( ). 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 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.
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 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., Baikov 1991).Of course it is average value that could vary.But if to admit that value, the ultimate strain will be equal to 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 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 Then the resultant of tensile concrete zone would be equal: element reinforced by FRP reinforced by steel reinforcem moment calculated according to cracking moment calculated acc When the cross section are reinforcement area, the relations reinforcement ratio is presented the results obtained according to the results of EC2 and STR m increase of cracking momen reinforcement arises.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 ,lim ε 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 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 Then the resultant of tensile concrete zone would be equal: 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 ,lim ε ε = . According this the cracking moment could be expressed by the next equation (Augonis, 2013) ( ) where 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: , 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 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 . In case when the expression (7) The height of the compressed zone is calculated according to 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: 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) the elastic plastic coefficient the ultimate tensile strain is equal to 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 Then the resultant of tensile concrete zone would be equal: 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 where 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: , 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 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, 0001656 ct ε = . In case when the expression ( 7) 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 So, the cracking moment will be reached then the strain of the tensile concrete will reach the ultimate strain value 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: , 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 reinforcemen reinforcemen the results ob the results o increase of reinforcemen 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 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  5 where could be ccording to the equation ( 9) is d STR methods.In this figure ns the cracking moment of 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, 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."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 stressstrain 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, 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.
Analysing the influence of stressstrain relationship to the cracking moment it is important to evaluate the elastic modulus of reinforcement.So, the cracking moment will be reached then the strain of the tensile concrete will reach the ultimate strain value 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  So, the cracking moment will be reached then the strain of the tensile concrete will reach the ultimate strain value 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  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 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 six times.Also, cording such case the secant modulus of compressed ncrete at the peak stress () the tensile concrete.If to express the ultimate astic plastic coefficient at the stress peak it would obtained e concrete.If to assume the one d the same elastic plastic coefficient to the compressed d tensile concrete, the strain at the peak stress tensile concrete stress-strain relationship is investigation the concrete stress-strain were expressed in two cases.e. Analysing the elastic plastic behaviour of crete should be note that concrete but not describe the tensile to assume that ultimate strain of tensile concrete the ultimate tensile stress ( ) the form of concrete until the peak of stress, it is possible to EC2 standard expression of stress-strain .figures the font of text are changed in "times new in fig.1and fig. 2 are presented units of tensile .The tensile concrete stress-strain relationship C2 standard expression is used.Concrete class 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: bigger than to the tensile concrete.If to express the ultimate elastic plastic coefficient at the stress peak it would obtained the concrete.If to assume the one and the same elastic plastic coefficient to the compressed and tensile concrete, the strain at the peak stress Fig. 1.The tensile concrete stress-strain relationship when the EC2 standard expression is used.Concrete class C40/50.

(
The stress-strain relations the equation (5) and equatio concrete is presented in fig.3average values of stress ( ct f (fctk0,05).For this reason the f values obtained by equation (5) curves is differ but it should be is differ also.
bigger than to the tensile concrete.If to express the ultimate elastic plastic coefficient at the stress peak it would obtained the 0

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

Fig. 5 .
Fig. 5.The relationship of reinforcement coefficient when the constant and varies the height of cro

Fig. 5 .
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 .
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.
the cracking moment could be expressed by the next equation(Augonis, 2013)

Fig. 5 .
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 .
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. 4 .
Fig. 4. Strain and stress of element when compressive concrete behaviour is elastic.

Fig
Fig. 5. reinforcement constant and v

Fig. 5 .
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 .
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.
/37.When the and varies the height of cross of cracking moment and ted in fig.

Fig. 5 .
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 .
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 .
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. 4 .
Fig. 4. Strain and stress of element when compressive concrete behaviour is elastic.

Fig. 6 .Fig. 4 .
Fig. 6.The relationship of the cracking moment and the

Fig. 5 .
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 .
Fig. 6.The relationship of the cracking moment and the

Fig. 7
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)