Analysis of Flexural NSRC-HSRC Composite Members Cracking Behaviour and Concrete Properties

The application of high strength concrete to reinforced concrete structures is very popular nowadays. This paper presents two main topics. In the first part of this paper, it is presented elasto-plastic properties of normal and high strength compressive concrete. It is known that actual behaviour of concrete is nonlinear. Design standard EN 1992-1-1 provides nonlinear stress-strain relation of compressive concrete for structural analysis. Expression to calculate this relation is given in standard. In this article, there are suggested coefficients which evaluate elasto-plastic behaviour of normal and high strength compressive concrete. Good correlation can be observed between stress-strain relation of compressive concrete given in standard EC2 and calculated relation using suggested elasto-plastic coefficients. In the second part of the paper, it is presented an overview of cracking moment calculations of reinforced concrete composite flexural members made from normal and high strength concrete and normal cross-section beams. These calculations were performed using existing design standard, technical construction regulation, structural building code (EN 1992-1-1, STR 2.05.05:2005, ACI 318) and layer method. New approximate values of elasto-plastic coefficients were suggested for tension concrete.


Introduction
Journal of Sustainable Architecture and Civil Engineering 2014/3/8 84 According to STR and EC2, it is assumed that elastic zone of compressive concrete is approximately equal to 0.4ƒ cm .However, it is observed that for high strength concrete the zone of elastic stress-strain relation is higher than for normal strength concrete.It was determined by Iravani et al. (1994) that depending on the class of high strength concrete, the value of this coefficient can reach 0.85 or even higher.
Great attention is given to the applicability of high strength concrete to reinforced concrete structures.In order to optimize or improve behaviour of reinforced concrete flexural members, series of analysis are performed.Partial use of high strength concrete can increase stiffness of the reinforced concrete composite beams, ultimate capacity, initial cracking load and other parameters, when comparing with the beams which are made from normal strength concrete.It is observed that partial use of HSC (high strength concrete) can reduce crack width.Composite beams can be also named as hybrid beams.Various configurations of composite beams layers are possible.Strengthening by high strength concrete can be provided in compression zone, tension zone or in compression and tension zones together.Casting operation of composite NSRC-HSRC (two layers of normal and high strength concrete) beams can be performed in two cases.In the first case, both layers can be cast while the concrete is in fresh condition.This should insure full bond between different layers of the concrete.Also, it is observed that transverse reinforcement can increase bond between the layers.In the other case, the second layer of new concrete can be cast when the first layer of concrete reaches its design strength (after 28 days or even later).This case can be applied for the analysis of the structural members which have to be rehabilitated or repaired.However, long time between casting phases could require additional tools to insure full bond between two layers of the concrete.Application of steel fibre reinforced concrete is common in composite cross-section beams, but in this study only by steel bars reinforced concrete flexural members are analysed (Kheder et al., 2010;Lapko et al., 2005;Sadowska-Buraczewska et al., 2007).
There is no united design standard which provides the calculation method for such type of the flexural members.It should be noted that it is necessary to take into account nonlinear behaviour of tension and compression concrete.Therefore, this paper is intended for the analysis of the flexural NSRC-HSRC composite members cracking behaviour and concrete properties.

Estimation of coefficient
λ c,lim according to EC2 stress-strain relation Design standard EC2 provides stress-strain relation of normal and high strength compressive concrete for nonlinear structural analysis.It includes concrete classes from C12/15 to C90/105.
The relation between σ c and ε c can be calculated according to the expression: where: η = ε c /ε c1 ; ε c -compressive strain in the concrete; ε c1 -strain at peak stress; k = 1.05•E cm •|ε c1 |/f cm ; E cm -secant modulus of elasticity of the concrete; f cm -mean value of concrete cylinder compressive strength.
flexural NSRC-HSRC composite members cracking behaviour and concrete properties.

Estimation of coefficient λ c,lim according to EC2 stressstrain relation
Design standard EC2 provides stress-strain relation of normal and high strength compressive concrete for nonlinear structural analysis.It includes concrete classes from C12/15 to C90/105.
When the strain in the concrete reaches the limit elastic strain of compressive concrete, the behaviour of the concrete changes from elastic to elasto-plastic, and here secant modulus of elasticity of concrete has to be recalculated to the deformation modulus of concrete.This modulus can be calculated according to the expression: where: λ c -coefficient which evaluates elasto-plastic EC2 provides stress-strain relation until th ultimate strain ε cu1 .In this research crackin calculations are performed.The assumption is λ c,lim values are applied to obtain the limit strain concrete, due to this reason only the first part of s relation is analysed (until the limit strain of co concrete at the peak stress -ε c1 ).From Fig.
observed that stress-strain relation calculated ac expression (1), which is given in EC2, ha correlation with suggested stress-strain relation calculated using k el,c and λ c,lim coefficients.It a seen that with the increase of the concrete class part of stress-strain relation also increases.flexural NSRC-HSRC composite members cracking behaviour and concrete properties.

Estimation of coefficient λ c,lim according to EC2 stressstrain relation
Design standard EC2 provides stress-strain relation of normal and high strength compressive concrete for nonlinear structural analysis.It includes concrete classes from C12/15 to C90/105.
The relation between σ c and ε c can be calculated according to the expression: where: η = ε c /ε c1 ; ε c -compressive strain in the concrete; ε c1 -strain at peak stress; k = 1.05When the strain in the concrete reaches the limit elastic strain of compressive concrete, the behaviour of the concrete changes from elastic to elasto-plastic, and here secant modulus of elasticity of concrete has to be recalculated to the deformation modulus of concrete.This modulus can be calculated according to the expression: where: λ c -coefficient which evaluates elasto-plastic deformations of compressive concrete; The expression (3), given in Augonis et al. (2013) publication can be used to determine coefficient which evaluates elasto-plastic behaviour of compressive concrete.However, the limit value of this coefficient depends on the EC2 provides stress-strain relation until th ultimate strain ε cu1 .In this research crackin calculations are performed.The assumption is λ c,lim values are applied to obtain the limit strain concrete, due to this reason only the first part of s relation is analysed (until the limit strain of c concrete at the peak stress -ε c1 ).From Fig.
observed that stress-strain relation calculated ac expression (1), which is given in EC2, ha correlation with suggested stress-strain relation calculated using k el,c and λ c,lim coefficients.It a seen that with the increase of the concrete class part of stress-strain relation also increases.flexural NSRC-HSRC composite members cracking behaviour and concrete properties.

Estimation of coefficient λ c,lim according to EC2 stressstrain relation
Design standard EC2 provides stress-strain relation of normal and high strength compressive concrete for nonlinear structural analysis.It includes concrete classes from C12/15 to C90/105.
The relation between σ c and ε c can be calculated according to the expression: where: η = ε c /ε c1 ; ε c -compressive strain in the concrete; ε c1 -strain at peak stress; k = 1.05When the strain in the concrete reaches the limit elastic strain of compressive concrete, the behaviour of the concrete changes from elastic to elasto-plastic, and here secant modulus of elasticity of concrete has to be recalculated to the deformation modulus of concrete.This modulus can be calculated according to the expression: where: λ c -coefficient which evaluates elasto-plastic deformations of compressive concrete; The expression (3), given in Augonis et al. (2013) publication can be used to determine coefficient which evaluates elasto-plastic behaviour of compressive concrete.However, the limit value of this coefficient depends on the concrete class.
EC2 provides stressultimate strain ε cu1 .In calculations are performed λ c,lim values are applied to concrete, due to this reason relation is analysed (until concrete at the peak stre observed that stress-strain expression (1), which is correlation with suggested calculated using k el,c and seen that with the increase part of stress-strain relation flexural NSRC-HSRC composite members cracking behaviour and concrete properties.

Estimation of coefficient λ c,lim according to EC2 stressstrain relation
Design standard EC2 provides stress-strain relation of normal and high strength compressive concrete for nonlinear structural analysis.It includes concrete classes from C12/15 to C90/105.
The relation between σ c and ε c can be calculated according to the expression: where: η = ε c /ε c1 ; ε c -compressive strain in the concrete; ε c1 -strain at peak stress; k = 1.05When the strain in the concrete reaches the limit elastic strain of compressive concrete, the behaviour of the concrete changes from elastic to elasto-plastic, and here secant modulus of elasticity of concrete has to be recalculated to the deformation modulus of concrete.This modulus can be calculated according to the expression: where: λ -coefficient which evaluates elasto-plastic EC2 provides stress-strain relation until the nominal ultimate strain ε cu1 .In this research cracking moment calculations are performed.The assumption is made that λ c,lim values are applied to obtain the limit strain of tensile concrete, due to this reason only the first part of stress-strain relation is analysed (until the limit strain of compressive concrete at the peak stress -ε c1 ).From Fig. 1. can be observed that stress-strain relation calculated according to expression (1), which is given in EC2, has a good correlation with suggested stress-strain relation which is calculated using k el,c and λ c,lim coefficients.It also can be seen that with the increase of the concrete class, the linear part of stress-strain relation also increases.
When the strain in the concrete reaches the limit elastic strain of compressive concrete, the behaviour of the concrete changes from elastic to elasto-plastic, and here secant modulus of elasticity of concrete has to be recalculated to the deformation modulus of concrete.This modulus can be calculated according to the expression: where: λ c -coefficient which evaluates elasto-plastic deformations of compressive concrete; The expression (3), given in Augonis et al. (2013) publication can be used to determine coefficient which evaluates elasto-plastic behaviour of compressive concrete.However, the limit value of this coefficient depends on the concrete class.
where: λ c -coefficient which evaluates elasto-plastic deformations of compressive concrete; The expression (3), given in Augonis et al. (2013) publication can be used to determine coefficient which evaluates elasto-plastic behaviour of compressive concrete.However, the limit value of this coefficient depends on the concrete class.
where: k c = 1-λ c,lim ; λ c,lim -the limit value of coefficient which evaluates elasto-plastic deformations of compressive concrete; ε c,i -compressive strain in the concrete; ε c,el -limit elastic strain of compressive concrete; ε c,lim -the limit strain of compressive concrete at the peak stress.
The limit elastic strain of the compressive concrete can be calculated according to the expression: where: k el,c -coefficient which evaluates the limit of elastic strain of compressive concrete; The limit strain of compressive concrete at the peak stress can be calculated according to the expression: In order to calculate the limit elastic strain of the compressive concrete ε c,el , which approximately corresponds the shape of EC2 stress-strain curve, coefficient k el,c is suggested for different concrete classes.The values of this coeficient using mentioned method vary from 0.01 to 0.4 for concrete classes from C20/25 to C45/55, and respectively from 0.5 to 0.65 for the concrete classes from C50/60 to C90/105.It is a tendency that for ultra-high strength concrete the value of this coefficient could be even higher.However, in the cracking moment calculations not less than 0.4 is assumed to evaluate the elastic zone of concrete.
To obtain accurate values of k el,c coefficients, more experimental analysis is necessary.Summary of the specific publication can be used to determine coefficient which evaluates elasto-plastic behaviour of compressive concrete.However, the limit value of this coefficient depends on the concrete class.
where: k c = 1-λ c,lim ; λ c,lim -the limit value of coefficient which evaluates elasto-plastic deformations of compressive concrete; ε c,i -compressive strain in the concrete; ε c,el -limit elastic strain of compressive concrete; ε c,lim -the limit strain of compressive concrete at the peak stress.
The limit elastic strain of the compressive concrete can be calculated according to the expression: where: k el,c -coefficient which evaluates the limit of elastic strain of compressive concrete; The limit strain of compressive concrete at the peak stress can be calculated according to the expression: In order to calculate the limit elastic strain of the compressive concrete ε c,el , which approximately corresponds the shape of EC2 stress-strain curve, coefficient k el,c is suggested for different concrete classes.The values of this coeficient using mentioned method vary from 0.01 to 0.4 for concrete classes from C20/25 to C45/55, and respectively from 0.5 to 0.65 for the concrete classes from C50/60 to C90/105.It is a tendency that for ultra-high strength concrete the value of this coefficient could be even higher.However, in the cracking moment calculations not less than 0.4 is assumed to evaluate the elastic zone of concrete.
To obtain accurate values of k el,c coefficients, more experimental analysis is necessary.Summary of the specific However, the limit value of this coefficient depends on the concrete class.
where: k c = 1-λ c,lim ; λ c,lim -the limit value of coefficient which evaluates elasto-plastic deformations of compressive concrete; ε c,i -compressive strain in the concrete; ε c,el -limit elastic strain of compressive concrete; ε c,lim -the limit strain of compressive concrete at the peak stress.
The limit elastic strain of the compressive concrete can be calculated according to the expression: where: k el,c -coefficient which evaluates the limit of elastic strain of compressive concrete; The limit strain of compressive concrete at the peak stress can be calculated according to the expression: In order to calculate the limit elastic strain of the compressive concrete ε c,el , which approximately corresponds the shape of EC2 stress-strain curve, coefficient k el,c is suggested for different concrete classes.The values of this coeficient using mentioned method vary from 0.01 to 0.4 for concrete classes from C20/25 to C45/55, and respectively from 0.5 to 0.65 for the concrete classes from C50/60 to C90/105.It is a tendency that for ultra-high strength concrete the value of this coefficient could be even higher.However, in the cracking moment calculations not less than 0.4 is assumed to evaluate the elastic zone of concrete.
To obtain accurate values of k el,c coefficients, more experimental analysis is necessary.Summary of the specific flexural NSRC-HSRC composite members cracking behaviour and concrete properties.

Estimation of coefficient λ c,lim according to EC2 stressstrain relation
Design standard EC2 provides stress-strain relation of normal and high strength compressive concrete for nonlinear structural analysis.It includes concrete classes from C12/15 to C90/105.
The relation between σ c and ε c can be calculated according to the expression: where: Here ε cu1 is the nominal ultimate strain.
When the strain in the concrete reaches the limit elastic strain of compressive concrete, the behaviour of the concrete changes from elastic to elasto-plastic, and here secant modulus of elasticity of concrete has to be recalculated to the deformation modulus of concrete.This modulus can be calculated according to the expression: where: λ c -coefficient which evaluates elasto-plastic deformations of compressive concrete; The expression (3), given in Augonis et al. (2013) publication can be used to determine coefficient which evaluates elasto-plastic behaviour of compressive concrete.However, the limit value of this coefficient depends on the concrete class.
where: k c = 1-λ c,lim ; λ c,lim -the limit value of coefficient which evaluates elasto-plastic deformations of compressive concrete; ε c,i -compressive strain in the concrete; ε c,el -limit elastic strain of compressive concrete; ε c,lim -the limit strain of compressive concrete at the peak stress.
The limit elastic strain of the compressive concrete can be calculated according to the expression: where: k el,c -coefficient which evaluates the limit of elastic strain of compressive concrete; The limit strain of compressive concrete at the peak stress can be calculated according to the expression: EC2 provides stress-strain relation until the nominal ultimate strain ε cu1 .In this research cracking moment calculations are performed.The assumption is made that λ c,lim values are applied to obtain the limit strain of tensile concrete, due to this reason only the first part of stress-strain relation is analysed (until the limit strain of compressive concrete at the peak stress -ε c1 ).From Fig. 1. can be observed that stress-strain relation calculated according to expression (1), which is given in EC2, has a good correlation with suggested stress-strain relation which is calculated using k el,c and λ c,lim coefficients.It also can be seen that with the increase of the concrete class, the linear part of stress-strain relation also increases.In order to calculate the limit elastic strain of the compressive concrete ε c,el , which approximately corresponds the shape of EC2 stress-strain curve, coefficient k el,c is suggested for different concrete classes.The values of this coeficient using mentioned method vary from 0.01 to 0.4 for concrete classes from C20/25 to C45/55, and respectively from 0.5 to 0.65 for the concrete classes from C50/60 to C90/105.It is a tendency that for ultra-high strength concrete the value of this coefficient could be even higher.However, in the cracking moment calculations not less than 0.4 is assumed to evaluate the elastic zone of concrete.
To obtain accurate values of k el,c coefficients, more experimental analysis is necessary.Summary of the specific lasto-plastic behaviour of compressive concrete.he limit value of this coefficient depends on the ss.
= 1-λ c,lim ; λ c,lim -the limit value of coefficient uates elasto-plastic deformations of compressive ,i -compressive strain in the concrete; ε c,el -limit n of compressive concrete; ε c,lim -the limit strain sive concrete at the peak stress.mit elastic strain of the compressive concrete can d according to the expression: -coefficient which evaluates the limit of elastic mpressive concrete; imit strain of compressive concrete at the peak e calculated according to the expression: In order to calculate the limit elastic strain of the compressive concrete ε c,el , which approximately corresponds the shape of EC2 stress-strain curve, coefficient k el,c is suggested for different concrete classes.The values of this coeficient using mentioned method vary from 0.01 to 0.4 for concrete classes from C20/25 to C45/55, and respectively from 0.5 to 0.65 for the concrete classes from C50/60 to C90/105.It is a tendency that for ultra-high strength concrete the value of this coefficient could be even higher.However, in the cracking moment calculations not less than 0.4 is assumed to evaluate the elastic zone of concrete.
To obtain accurate values of k el,c coefficients, more experimental analysis is necessary.Summary of the specific xpression (3), given in Augonis et al. (2013) can be used to determine coefficient which lasto-plastic behaviour of compressive concrete.he limit value of this coefficient depends on the ss.
= 1-λ c,lim ; λ c,lim -the limit value of coefficient uates elasto-plastic deformations of compressive ,i -compressive strain in the concrete; ε c,el -limit n of compressive concrete; ε c,lim -the limit strain sive concrete at the peak stress.mit elastic strain of the compressive concrete can d according to the expression: -coefficient which evaluates the limit of elastic mpressive concrete; imit strain of compressive concrete at the peak e calculated according to the expression: In order to calculate the limit elastic strain of the compressive concrete ε c,el , which approximately corresponds the shape of EC2 stress-strain curve, coefficient k el,c is suggested for different concrete classes.The values of this coeficient using mentioned method vary from 0.01 to 0.4 for concrete classes from C20/25 to C45/55, and respectively from 0.5 to 0.65 for the concrete classes from C50/60 to C90/105.It is a tendency that for ultra-high strength concrete the value of this coefficient could be even higher.However, in the cracking moment calculations not less than 0.4 is assumed to evaluate the elastic zone of concrete.
To obtain accurate values of k el,c coefficients, more experimental analysis is necessary.Summary of the specific e: λ c -coefficient which evaluates elasto-plastic ns of compressive concrete; expression (3), given in Augonis et al. (2013) n can be used to determine coefficient which elasto-plastic behaviour of compressive concrete.the limit value of this coefficient depends on the lass.
= 1-λ c,lim ; λ c,lim -the limit value of coefficient luates elasto-plastic deformations of compressive c,i -compressive strain in the concrete; ε c,el -limit in of compressive concrete; ε c,lim -the limit strain ssive concrete at the peak stress.imit elastic strain of the compressive concrete can ted according to the expression: c -coefficient which evaluates the limit of elastic ompressive concrete; limit strain of compressive concrete at the peak be calculated according to the expression: In order to calculate the limit elastic strain of the compressive concrete ε c,el , which approximately corresponds the shape of EC2 stress-strain curve, coefficient k el,c is suggested for different concrete classes.The values of this coeficient using mentioned method vary from 0.01 to 0.4 for concrete classes from C20/25 to C45/55, and respectively from 0.5 to 0.65 for the concrete classes from C50/60 to C90/105.It is a tendency that for ultra-high strength concrete the value of this coefficient could be even higher.However, in the cracking moment calculations not less than 0.4 is assumed to evaluate the elastic zone of concrete.
To obtain accurate values of k el,c coefficients, more experimental analysis is necessary.Summary of the specific where: k c = 1-λ c,lim ; λ c,lim -the limit value of coefficient which evaluates elasto-plastic deformations of compressive concrete; ε c,i -compressive strain in the concrete; ε c,el -limit elastic strain of compressive The limit elastic strain of the compressive concrete can be calculated according to the expression: The limit strain of compressive concrete at the peak stress can be calculated according to the expression: where: k el,c -coefficient which evaluates the limit of elastic strain of compressive concrete; EC2 provides stress-strain relation until the nominal ultimate strain ε cu1 .In this research cracking moment calculations are performed.The assumption is made that λ c,lim values are applied to obtain the limit strain of tensile concrete, due to this reason only the first part of stressstrain relation is analysed (until the limit strain of compressive concrete at the peak stress -ε c1 ).From Fig. 1. can be observed that stress-strain relation calculated according to expression (1), which is given in EC2, has a good correlation with suggested stress-strain relation which is calculated using k el,c and λ c,lim coefficients.It also can be seen that with the increase of the concrete class, the linear part of stress-strain relation also increases.In order to calculate the limit elastic strain of the compressive concrete ε c,el , which approximately corresponds the shape of EC2 stress-strain curve, coeffi-Fig. 1 Comparison of stressstrain relation of compressive concrete calculated according to EN 1992-1-1 and stress-strain relation (*) calculated using coefficients k el,c and λ c .cient k el,c is suggested for different concrete classes.The values of this coeficient using mentioned method vary from 0.01 to 0.4 for concrete classes from C20/25 to C45/55, and respectively from 0.5 to 0.65 for the concrete classes from C50/60 to C90/105.It is a tendency that for ultra-high strength concrete the value of this coefficient could be even higher.However, in the cracking moment calculations not less than 0.4 is assumed to evaluate the elastic zone of concrete.
To obtain accurate values of k el,c coefficients, more experimental analysis is necessary.Summary of the specific values of approximate coefficients are given in Table 1 and Table 2. To determine the limit strain of the compressive concrete at the peak stress, λ c,lim coefficient is concrete; ε c,lim -the limit strain of compressive concrete at the peak stress.
used.The approximate values of this coefficient vary from 0.467 to 0.613 for concrete classes from C20/25 to C45/55, and respectively from 0.640 to 0.795 for the concrete classes from C50/60 to C90/105.The same tencency as for coeffient k el,c remains that for ultra-high strength concrete the value of this coefficient could be even higher.Summary of the specific values of approximate coefficients are given in Table 1 and Table 2.In Fig. 2. it is presented the variation of λ c coefficient, depending on the class of the concrete.The similar values of these coefficients are calculated by Židonis cients are given in Table 1 and it strain of the compressive λ c,lim coefficient is used.The coeficient vary from 0.467 to from C20/25 to C45/55, and 0.795 for the concrete classes same tencency as for coeffient h strength concrete the value of ven higher.Summary of the ate coefficients are given in the variation of λ c coefficient, concrete.The similar values of ted by Židonis (2013).It can be deformations for high strength er strain level than for normal crete behaviour is elastic, it is n modulus of the concrete is lasticity, and then the value of  W pl -plastic moment of resistance of RC member effective cross-section.

Design standard EN 1992-1-1
Design standard EC2 provides elastic stress-strain relation in the calculations of the cracking moment.Here, mean value of axial tensile strength of the concrete and elastic moment of resistance are used to determine M crc : where: f ctm -mean value of axial tensile strength of concrete; W el -elastic moment of resistance of RC member effective cross-section.

ACI 318
Structural building code ACI 318 also uses elastic moment of resistance of RC member effective cross-section.However, modulus of rupture of the concrete is used instead of the axial tensile strength.M crc can be calculated by expression: (2013).It can be observed that elasto-plastic deformations for high strength concrete occurs at much higher strain level than for normal strength concrete.When concrete behaviour is elastic, it is assumed that the deformation modulus of the concrete is equal to secant modulus of elasticity, and then the value of coefficient λ c is equal to 1.

Variation of coefficient λ c for different concrete classes
Table 1 Summary of results for concrete classes from C20/25 to C45/55

Calculation of cracking moment according to STR, EC2, ACI 318 and LM
A short comparison of the different cracking moment calculation methods was carried out.Calculations of the cracking moments (M crc ) were performed according to STR 2.05.05:2005,EC2, ACI 318 and Layer method (LM).When the calculations were performed according to STR, EC2 and ACI 318, in all cases was assumed the transformed crosssection of the reinforced concrete beam.

Technical construction regulation STR 2.05.05:2005
In the calculations according to STR, it is assumed to use characteristic axial tensile strength of the concrete and plastic moment of resistance of reinforced concrete member effective cross-section.Cracking moment can be calculated by expression: where: f ctk -characteristic axial tensile strength of concrete; W pl -plastic moment of resistance of RC member ef cross-section.

Design standard EN 1992-1-1
Design standard EC2 provides elastic stress relation in the calculations of the cracking moment mean value of axial tensile strength of the concre elastic moment of resistance are used to determine M c where: f ctm -mean value of axial tensile strength of co W el -elastic moment of resistance of RC member ef cross-section.

ACI 318
Structural building code ACI 318 also uses moment of resistance of RC member effective cross-s However, modulus of rupture of the concrete is used of the axial tensile strength.M crc can be calcula expression: W pl -plastic moment of resistance of RC member effective cross-section.

Design standard EN 1992-1-1
Design standard EC2 provides elastic stress-strain relation in the calculations of the cracking moment.Here, mean value of axial tensile strength of the concrete and elastic moment of resistance are used to determine M crc : where: f ctm -mean value of axial tensile strength of concrete; W el -elastic moment of resistance of RC member effective cross-section.

ACI 318
Structural building code ACI 318 also uses elastic moment of resistance of RC member effective cross-section.However, modulus of rupture of the concrete is used instead of the axial tensile strength.M crc can be calculated by expression: W pl -plastic moment of resistance of RC member effective cross-section.

Design standard EN 1992-1-1
Design standard EC2 provides elastic stress-strain relation in the calculations of the cracking moment.Here, mean value of axial tensile strength of the concrete and elastic moment of resistance are used to determine M crc : where: f ctm -mean value of axial tensile strength of concrete; W el -elastic moment of resistance of RC member effective cross-section.

ACI 318
Structural building code ACI 318 also uses elastic moment of resistance of RC member effective cross-section.However, modulus of rupture of the concrete is used instead of the axial tensile strength.M crc can be calculated by expression: short comparison of the different cracking moment tion methods was carried out.Calculations of the g moments (M crc ) were performed according to STR 5:2005, EC2, ACI 318 and Layer method (LM).When culations were performed according to STR, EC2 and 18, in all cases was assumed the transformed crossof the reinforced concrete beam.
chnical construction regulation STR 2.05.05:2005 the calculations according to STR, it is assumed to aracteristic axial tensile strength of the concrete and moment of resistance of reinforced concrete member ve cross-section.Cracking moment can be calculated ression: f ctk -characteristic axial tensile strength of concrete; W pl -plastic moment of resistance of RC member effective cross-section.

Design standard EN 1992-1-1
Design standard EC2 provides elastic stress-strain relation in the calculations of the cracking moment.Here, mean value of axial tensile strength of the concrete and elastic moment of resistance are used to determine M crc : where: f ctm -mean value of axial tensile strength of concrete; W el -elastic moment of resistance of RC member effective cross-section.

ACI 318
Structural building code ACI 318 also uses elastic moment of resistance of RC member effective cross-section.However, modulus of rupture of the concrete is used instead of the axial tensile strength.M crc can be calculated by expression:

Calculation of cracking moment according to STR, EC2, ACI 318 and LM
A short comparison of the different cracking moment calculation methods was carried out.Calculations of the cracking moments (M crc ) were performed according to STR 2.05.05:2005,EC2, ACI 318 and Layer method (LM).When the calculations were performed according to STR, EC2 and ACI 318, in all cases was assumed the transformed cross-section of the reinforced concrete beam.
Technical construction regulation STR 2. 05.05:2005In the calculations according to STR, it is assumed to use characteristic axial tensile strength of the concrete and plastic moment of resistance of reinforced concrete member effective cross-section.Cracking moment can be calculated by expression: where: f ctk -characteristic axial tensile strength of concrete; W pl -plastic moment of resistance of RC member effective cross-section.

Design standard EN 1992-1-1
Design standard EC2 provides elastic stress-strain relation in the calculations of the cracking moment.Here, mean value of axial tensile strength of the concrete and elastic moment of resistance are used to determine M crc : where: f ctm -mean value of axial tensile strength of concrete; W el -elastic moment of resistance of RC member effective cross-section.

ACI 318
Structural building code ACI 318 also uses elastic moment of resistance of RC member effective cross-section.However, modulus of rupture of the concrete is used instead of the axial tensile strength.M crc can be calculated by expression: where: f r -modulus of rupture of concrete; Modulus of rupture of concrete can be expressed: where: f r -modulus of rupture of concrete; Modulus of rupture of concrete can be expressed: where: f ' -specified compressive strength of concrete.

Layer method
Layer method was used to calculate the cracking force and moment equations of equilibrium: where: E ci -deformation modulus of concrete of i layer; A ci -area of concrete of i layer, ε -strain of concrete in i layer Modulus of rupture of concrete can be expressed: where: f ' -specified compressive strength of concrete.

Layer method
Layer method was used to calculate the cracking moment evaluating that stress-strain relation of tension where: E ci -deformation modulus of concrete of i layer; A ci scientists' experimental research.Cross beams are given in Fig. 3.
Three different types of the beams we where: E ci -deformation modulus o -area of concrete of i layer, ε ci -str ("+" in compression zone, "-" in ten force acting in concrete i layer (dist c c where: f r -modulus of rupture of concrete; Modulus of rupture of concrete can be expressed: where: f ' -specified compressive strength of concrete.

Layer method
Layer method was used to calculate the cracking moment evaluating that stress-strain relation of tension where: E ci -deformation modulus of concrete of i layer; A ci -area of concrete of i layer, ε ci -strain of concrete in i layer ("+" in compression zone, "-" in tension zone); z ci -arm of force acting in concrete i layer (distance from the center of scientists' experimental resea beams are given in Fig. 3  where: f r -modulus of rupture of concrete; Modulus of rupture of concrete can be expressed: where: f ' -specified compressive strength of concrete.
divided into small layers.Strain, stress, and other characteristics were calculated in each layer.Calculations are performed by iterations.LM can be expressed from force and moment equations of equilibrium: scientists' experimental research.Cross-sections of the beams are given in Fig. 3.

Layer method
Layer method (Augonis et al. 2013) was used to calculate the cracking moment evaluating that stress-strain relation of tension concrete is elasto-plastic.The cross-section of the beam was divided into small layers.Strain, stress, and other characteristics were calculated in each layer.Calculations are performed by iterations.LM can be expressed from force and moment equations of equilibrium: where: E ci -deformation modulus of concrete of i layer; A ci -area of concrete of i layer, ε ci -strain of concrete in i layer ("+" in compression zone, "-" in tension zone); z ci -arm of force acting in concrete i layer (distance from the center of the most tension layer to the center of i layer); E s1 , E s2 -design value of modulus of elasticity of reinforcing steel; ε s1 , ε s2 -strain in tension ("-") and compression ("+") reinforcement; z s1 , z s2 -arm of force acting in reinforcement (distance from the center of the most tension layer to the center of reinforcement); M -external bending moment.
Expression (3) is given to calculate coefficient which evaluates elasto-plastic behaviour of compressive concrete.Good correlation was obtained between calculated stress-strain curve and EC2 stress-strain relation.However, to evaluate elasto-plastic behaviour of tension concrete in cracking moment calculations, more accurate expression was suggested.Coefficients λ ct can be calculated according to the experssion: where: E ci -deformation modulus of concrete of i layer; A ci -area of concrete of i layer, ε ci -strain of concrete in i layer ("+" in compression zone, "-" in tension zone); z ci -arm of force acting in concrete i layer (distance from the center of the most tension layer to the center of i layer); E s1 , E s2design value of modulus of elasticity of reinforcing steel; ε s1 , ε s2 -strain in tension ("-") and compression ("+") reinforcement; z s1 , z s2 -arm of force acting in reinforcement (distance from the center of the most tension layer to the center of reinforcement); M -external bending moment.Expression (3) is given to calculate coefficient which evaluates elasto-plastic behaviour of compressive concrete.Good correlation was obtained between calculated stressstrain curve and EC2 stress-strain relation.However, to evaluate elasto-plastic behaviour of tension concrete in cracking moment calculations, more accurate expression was suggested.Coefficients λ ct can be calculated according to the experssion: (11) where: λ ct,lim -the limit value of coefficient which evaluates elasto-plastic deformations of tensile concrete, f ctkcharacteristic axial tensile strength of concrete; σ ct,el -the limit elastic stress of tensile concrete; ε ct,i -strain of tensile concrete in i layer; E c -modulus of elasticity of concrete.
The parameters of the beams cross-sections are given below.Such parameters were used in order to compare calculated values of M crc with experimental values.Experimental values of the cracking moments were calculated according to the information (initial cracking load, loading scheme, etc.) given in Kheder et al. (2010) publication.Therefore, some data were used from these

Fig. 3. Parameters of beams cros
Cylindrical compressive elasticity of tested beams w (2010) experiment.Tensile s calculated according to EC2 e in Table 3.The beams were double reinforced.Four types of reinforcement were used for beams.

Fig. 3. Parameters of beams cross-sections
Cylindrical compressive strength and modulus of elasticity of tested beams were used from Kheder et al. (2010) experiment.Tensile strength of the concrete was calculated according to EC2 expressions and it is presented in Table 3. where: E ci -deformation modulus of concrete of i layer; A ci -area of concrete of i layer, ε ci -strain of concrete in i layer ("+" in compression zone, "-" in tension zone); z ci -arm of force acting in concrete i layer (distance from the center of the most tension layer to the center of i layer); E s1 , E s2design value of modulus of elasticity of reinforcing steel; ε s1 , ε s2 -strain in tension ("-") and compression ("+") reinforcement; z s1 , z s2 -arm of force acting in reinforcement (distance from the center of the most tension layer to the center of reinforcement); M -external bending moment.Expression (3) is given to calculate coefficient which evaluates elasto-plastic behaviour of compressive concrete.Good correlation was obtained between calculated stressstrain curve and EC2 stress-strain relation.However, to evaluate elasto-plastic behaviour of tension concrete in cracking moment calculations, more accurate expression was suggested.Coefficients λ ct can be calculated according to the experssion: where: λ ct,lim -the limit value of coefficient which evaluates elasto-plastic deformations of tensile concrete, f ctkcharacteristic axial tensile strength of concrete; σ ct,el -the limit elastic stress of tensile concrete; ε ct,i -strain of tensile concrete in i layer; E c -modulus of elasticity of concrete.
The parameters of the beams cross-sections are given below.Such parameters were used in order to compare calculated values of M crc with experimental values.Experimental values of the cracking moments were calculated according to the information (initial cracking load, loading scheme, etc.) given in Kheder et al. (2010) publication.Therefore, some data were used from these

Fig. 3. Parameters of beams cross-sections
Cylindrical compressive strength and elasticity of tested beams were used from K (2010) experiment.Tensile strength of the c calculated according to EC2 expressions and it in Table 3. where: xxxx/y: xxxx -strength of concrete H -high, NH -composite beam from norm strength of concrete); R -reinforced; y -are reinforcement (mm 2 ); f ctk -characteristic strength of concrete calculated according to characteristic axial compressive strength of co modulus of elasticity of concrete where: λ ct,lim -the limit value of coefficient which evaluates elasto-plastic deformations of tensile concrete, f ctkcharacteristic axial tensile strength of concrete; σ ct,el -the limit elastic stress of tensile concrete; ε ct,i -strain of tensile concrete in i layer; E c -modulus of elasticity of concrete.
The parameters of the beams cross-sections are given below.Such parameters were used in order to compare calculated values of M crc with experimental values.Experimental values of the cracking moments were calculated according to the information (initial cracking load, loading scheme, etc.) given in Kheder et al. (2010) publication.Therefore, some data were used from these scientists' experimental research.Cross-sections of the beams are given in Fig. 3.The beams were double reinforced.Four types of reinforcement were used for beams.
Cylindrical compressive strength and modulus of elasticity of tested beams were used from Kheder et al. (2010) experiment.Tensile strength of the concrete was calculated according to EC2 expressions and it is presented in Table 3.

Results
Analysis of the cracking moment calculations was performed according to STR, EC2, ACI 318 and LM.All results were compared with the experimental research results (Kheder et al. 2010).Due to this reason, the mean values of concrete properties were used in the calculation methods given in STR, EC2 and ACI 318.When the calculations were performed according to the available regulations or other standards, it can be observed from the results presented in Two cases of the calculations were performed using layer method.In the first case, calculations were performed assuming that the values of the coefficient λ ct,lim are equal to λ c,lim (this coefficient was obtained in Section 2, see Table 1 and Table 2, the interpolated values of λ c , lim were used depending on f cm ).In the second case, λ ct,lim was expressed from the experimental research results.
CASE 1.The results of the calculations are given in Table 4.The assumptions were made that coefficient λ ct,lim is equal to λ c,lim , k el,t and k el,c are equal to 0.4 for NSC and 0.6 for HSC.The values of the calculated cracking moments were obtained smaller than experimental: NSRC (7 -21%), HSRC (23 -28%), NHSRC (6 -30%).The largest errors were obtained for HSRC and NHSRC beams.
CASE 2. In this case, new approximate values of coefficient λ ct,lim were expressed from experimental data (NSRC and HSRC beams) for normal strength concrete (f cm ~21-23.6MPa -λ ct,lim vary from 0.281 to 0.385) and for high strength concrete (f cm ~65.4-74.9MPa -λ ct ,lim vary from 0.351 to 0.457).Assumption was made that coefficients k el,t and k el,c are equal to 0.4 for normal strength concrete and 0.6 for high strength concrete.λ c,lim was used the same as in the case 1.It was assumed that for NSRC and HSRC beams M crc (calculated) is equal to M crc (experimental).Expressed coefficients were used to calculate the cracking moments of composite beams.Calculated values of M crc of NHSRC beams were obtained close to the experimentally determined values.The difference between calculated and experimental values of NHSRC beams varies from 2 to 13%.In the calculations of the flexural members cracking moment, it is very important to know the real limit elastic strain of the tensile concrete and the limit strain of tensile concrete at the peak stress, however design standards provide these approximate values only for compressive concrete.Due to these reasons, coefficients k el,t and λ ct,lim can be corrected after more analysis.
According to the performed calculations, it can be seen that until the cracking moment, compression zone of the concrete beam behaves elastically (λ c =1).However, the behaviour of tension concrete is quite mysterious, and it is difficult to estimate its real behaviour, therefore it should be noted that suggested coefficients are approximate.

Fig. 4
Comparison of cracking moments Conclusions 1 Stress-strain curve calculated according to the expression (3) using k el,c and λ c,lim coefficients has a good correlation with stress-strain relation given in design standard EC2.However, for cracking moment calculation more accurate expression (11) was suggested.
2 The values of the cracking moments which were calculated according to STR and ACI 318 are closer to the experimental values as compared to the calculations according to EC2.
3 λ ct,lim coefficients, expressed from the experimentally determined cracking moments of NSC and HSC beams are approximate and can be corrected after more experimental analysis.It is very important to know the real limit elastic strain of the tensile concrete and the limit strain at the peak tensile stress, which are not defined by design standards for the tensile concrete.

Fig. 1 .
Fig. 1.Comparison of stress-strain relation of compressive concrete calculated according to EN 1992-1-1 and stress-strain relation (*) calculated using coefficients k el,c and λ c .

Fig. 1 .
Fig. 1.Comparison of stress-strain relation of compressive concrete calculated according to EN 1992-1-1 and stress-strain relation (*) calculated using coefficients k el,c and λ c .

Fig. 1 .
Fig. 1.Comparison of stress-strain relation of compressive concrete calculated according to EN 1992-1-1 and stress-strain relation (*) calculated using coefficients k el,c and λ c .

Fig. 1 .
Fig. 1.Comparison of stress-strain relation of compressive concrete calculated according to EN 1992-1-1 and stress-strain relation (*) calculated using coefficients k el,c and λ c .

Fig. 1 .
Fig. 1.Comparison of stress-strain relation of compressive concrete calculated according to EN 1992-1-1 and stress-strain relation (*) calculated using coefficients k el,c and λ c .

Fig. 1 .
Fig. 1.Comparison of stress-strain relation of compressive concrete calculated according to EN 1992-1-1 and stress-strain relation (*) calculated using coefficients k el,c and λ c .

Fig. 1 .
Fig. 1.Comparison of stress-strain relation of compressive concrete calculated according to EN 1992-1-1 and stress-strain relation (*) calculated using coefficients k el,c and λ c .

Fig. 2 .
Fig. 2. Variation of coefficient λ c for different concrete classes

Fig. 2
Fig. 2 r -modulus of rupture of concrete; . Three different types of th • normal strength concrete • high strength concrete b • composite beams from concrete (90(h 2 )/185(h strength of concrete.

Fig. 3 .
Fig. 3. Parameters of beams cross-sections Cylindrical compressive strength and modulus of elasticity of tested beams were used from Kheder et al.

Table 3 .
Properties of tested conc

Table 3 .
Properties of tested concrete(Kheder et al.

Table 4
where: xxxx/y: xxxx -strength of concrete (N -normal, H -high, NH -composite beam from normal and high strength of concrete); R -reinforced; y -area of tension reinforcement (mm 2 ); E c -modulus of elasticity of concrete.
Table4and Fig.4that for all types of the beams, the closest to experimental results cracking moments were obtained according to STR and ACI 318.According to EC2 standard, larger errors were obtained.Calculated results for all types of the beams are smaller than experimental values.