The Stress ’ s State Analysis of Carbon Fibre Reinforced Concrete Elements Evaluating the Bond Influence

While strengthening structures with thin layered elements an important role is carried out by the bond between separate elements, but there are not any methods which would describe the distribution of shear stress in the contact zone when the concrete works plastically. In this article a single span reinforced concrete beam which is strengthened by CFRP (Carbon fiber reinforced polymer) is analysed and three methods which could describe shear stress through the length of the element are provided. The three compared methods are: finite elements method (FEM), theory of multiple rods and the proposed method of the authors. The results are provided as diagrams and the similarities and differences are discussed. The method that the authors propose is superior to other methods because it describes the plastic work of a concrete which is not possible to evaluate with the method of the multiple rods theory DOI: http://dx.doi.org/10.5755/j01.sace.23.2.20987


Introduction
While strengthening structures with thin layered elements an important role is carried out by the bond between separate elements, but there are not any methods which would describe the distribution of bond stress in the contact zone when the concrete behaviour is elastic and plastic.In this article a single span reinforced concrete beam which is strengthened by CFRP (Carbon fibre reinforced polymer) is analysed and three methods which could describe bond stress through the length of the element are provided.The three compared methods are: finite elements method (FEM), theory of multiple rods and the proposed method of the authors.The similarities and differences of results are shown by diagrams and discussed.The method that the authors propose is superior to other methods because of evaluation of the plastic behaviour of concrete which is not possible to evaluate it by the method of the theory of multiple rods.
Nowadays in modern world, when architecture is getting more complex and the structural requirements are growing higher, the strengthening of structures plays an important role.In this article the strengthening of reinforced concrete beam structures with thin layered materials is analysed.The main focus is given to analyse the contact zone between separate layers.Many authors analyse the contact zone between fibre and concrete, but these methods separately not evaluate the effect of concrete behaviour in reinforced concrete structures.In this article the analysis of reinforced flexural reinforced concrete structures is made, applying the finite elements method, the theory of multiple rods method and the method proposed by the authors.
The experiments show that the highest bond stress in the contact zone was at the front and the back end of the beam.However, when the load is increasing, the bond at the end of a beam is decreasing, so the highest bond stress shifts a little bit further from the end.It happens because at the end of the beam the tension strength of a concrete is exceeded.(Guo2005, Guo 2007) While analysing a single span beam which is strengthened by CFRP it was received that the shear contact stress increases evenly at the centre of a beam and increases greatly at the ends.(Lorenzis 2001) The examined composite beams are made of two elements so the interaction between the elements is analysed when there are different bonds in the elastic stage because of a hydrothermal load.To describe the bond between different elements the theory of multiple rods is being used.(Zabulionis 2005).

Methods
Fig. 2 Cross-section of beam Fig. 1 Test setup Table 1 Materials parameters E c E s E CFRP υ c υ s υ g υ CFRP 28 GPa 200 GPa 230 GPa 0,25 0,3 0,3 0,3 Where: E c , E s , E CFRP -concrete's, steel's or carbon fibber's elasticity module; υ c , υ s , υ g , υ CFRP -concrete's, steel's, glue's or carbon fiber's Poisson's ratio In this article a single span reinforced concrete beam reinforced by carbon fibre is analysed.The test setup and the cross-section of the beam are provided in the figures Fig. 1 and Fig. 2. Concrete, steel, carbon fibre and glue material parameters are provided in the Table 1.

Methods
In this article a single span reinforced concrete beam reinforced by carbon fibre is analysed.The test setup and the cross-section of the beam are provided in the figures Fig. 1 and Fig. 2. Concrete, steel, carbon fibre and glue material parameters are provided in the Table 1.Where: Ec, Es, ECFRP -concrete's, steel's or carbon fibber's elasticity module; υc, υs, υg, υCFRPconcrete's, steel's, glue's or carbon fiber's Poisson's ratio In this article three comparative calculation methods are used: finite elements method, theory of multiple rods method and a method that is created by the authors which is based on iteration method.
A three-dimensional calculation model from the provided test setup is made with the "ANSYS" program.Modelling the finite elements model the support point is chosen not at the end, but in the centre of the beam.This kind of scheme helps to avoid stress concentrators which distort the distribution of shear stress at the ends of the beam Fig. 3.When creating the test setup and describing the bond between carbon fibre and concrete a layer of 1 mm thickness of glue was chosen.When changing the glue layer elasticity modulus, the bond between the layers changes as well as the values of bond stress.In this article the following glue elasticity modulus was used:  � =0,2 GPa; 2,0 GPa; 28,0 GPa.The highest elasticity modulus (28 GPa), which equals to the concrete one, means that the bond between the layers is perfectly good.An elasticity modulus (2,0 GPa) represent the real glue elasticity modulus [Rabinovitch 2001, Adruini 1997].Smallest elasticity modulus (0,2 GPa) means that the bond between the layers is very poor.If the bond is even more reduced the stress in the concrete and the rebar does not change.In order to describe the concrete SOLID65 element was used.This element describes the nonlinear behaviour of a concrete.(ANSYS 2016) 2. Theory of multiple rods.
The theory of multiple rods was used to describe the behaviour of layered structures when there is a certain     Where: Ec, Es, ECFRP -concrete's, steel's or carbon fibber's elasticity module; υc, υs, υg, υCFRPconcrete's, steel's, glue's or carbon fiber's Poisson's ratio In this article three comparative calculation methods are used: finite elements method, theory of multiple rods method and a method that is created by the authors which is based on iteration method.
A three-dimensional calculation model from the provided test setup is made with the "ANSYS" program.Modelling the finite elements model the support point is chosen not at the end, but in the centre of the beam.This kind of scheme helps to avoid stress concentrators which distort the distribution of shear stress at the ends of the beam Fig. 3.When creating the test setup and describing the bond between carbon fibre and concrete a layer of 1 mm thickness of glue was chosen.When changing the glue layer elasticity modulus, the bond between the layers changes as well as the values of bond stress.In this article the following glue elasticity modulus was used:  � =0,2 GPa; 2,0 GPa; 28,0 GPa.The highest elasticity modulus (28 GPa), which equals to the concrete one, means that the bond between the layers is perfectly good.An elasticity modulus (2,0 GPa) represent the real glue elasticity modulus [Rabinovitch 2001, Adruini 1997].Smallest elasticity modulus (0,2 GPa) means that the bond between the layers is very poor.If the bond is even more reduced the stress in the concrete and the rebar does not change.In order to describe the concrete SOLID65 element was used.This element describes the nonlinear behaviour of a concrete.(ANSYS 2016) 2. Theory of multiple rods.
The theory of multiple rods was used to describe the  In this article three comparative calculation methods are used: finite elements method, theory of multiple rods method and a method that is created by the authors which is based on iteration method.

Finite elements method
A three-dimensional calculation model from the provided test setup is made with the "ANSYS" program.Modelling the finite elements model the support point is chosen not at the end, but in the centre of the beam.This kind of scheme helps to avoid stress concentrators which distort the distribution of shear stress at the ends of the beam Fig. 3.When creating the test setup and describing the bond between carbon fibre and concrete a layer of 1 mm thickness of glue was chosen.When changing the glue layer elasticity modulus, the bond between the layers changes as well as the values of bond stress.In this article the following glue elasticity modulus was used: Eg = 0,2 GPa; 2,0 GPa; 28,0 GPa.The highest elasticity modulus (28 GPa), which equals to the concrete one, means that the bond between the layers is perfectly good.An elasticity modulus  (2,0 GPa) represent the real glue elasticity modulus [Rabinovitch 2001, Adruini 1997].
Smallest elasticity modulus (0,2 GPa) means that the bond between the layers is very poor.If the bond is even more reduced the stress in the concrete and the rebar does not change.In order to describe the concrete SOLID65 element was used.This element describes the nonlinear behaviour of a concrete.(ANSYS 2016) 1 2

Theory of multiple rods
The theory of multiple rods was used to describe the behaviour of layered structures when there is a certain bond.The calculation method is used by (Ржаницын 1982, Ржаницын 1986).In Fig. 4 the certain geometrical parameters, bond stress and relative strains distribution in the contact zone are provided.The calculation formulas for the two layered structures are provided below.

Fig. 4
Shear stress and relative strains distribution in the contact zone  The bond stress of the contact zone through the element length of is calculated by the following equation: contact zone through the element length of is calculated by the following equation: le of layers;  �,� -cross-section area;  -distance between centres of layers;  �� -stiffness of layers joint alculation method.
posed calculation method consists of iteration (layers) 013).From the values received using this method the ress through the length of the element is drawn.Iteration n the cross-section of the element is divided into the finite . 5. Every separate layer has its own stiffness and strains.ten into the matrix form: s and elasticity module matrix: Fig. 5. Dividing of the cross-section in the layers ss of the contact zone through the element length of is calculated by the following equation: ty module of layers;  �,� -cross-section area;  -distance between centres of layers;  �� ment;  -stiffness of layers joint oposed calculation method.
this proposed calculation method consists of iteration (layers) lauskas 2013).From the values received using this method the f bond stress through the length of the element is drawn.Iteration ated when the cross-section of the element is divided into the finite yers Fig. 5. Every separate layer has its own stiffness and strains.are written into the matrix form: ns, forces and elasticity module matrix: Fig. 5. Dividing of the cross-section in the layers (2) ontact zone through the element length of is calculated by the following equation: of layers;  �,� -cross-section area;  -distance between centres of layers;  �� stiffness of layers joint lculation method.
sed calculation method consists of iteration (layers) 13).From the values received using this method the ss through the length of the element is drawn.Iteration the cross-section of the element is divided into the finite .Every separate layer has its own stiffness and strains.n into the matrix form: and elasticity module matrix: (3) t zone through the element length of is calculated by the following equation: ers;  �,� -cross-section area;  -distance between centres of layers;  �� ess of layers joint ion method.
alculation method consists of iteration (layers) From the values received using this method the ough the length of the element is drawn.Iteration oss-section of the element is divided into the finite ry separate layer has its own stiffness and strains.the matrix form: lasticity module matrix: he contact zone through the element length of is calculated by the following equation: dule of layers;  �,� -cross-section area;  -distance between centres of layers;  ��  -stiffness of layers joint d calculation method.
roposed calculation method consists of iteration (layers) s 2013).From the values received using this method the stress through the length of the element is drawn.Iteration hen the cross-section of the element is divided into the finite ig. 5. Every separate layer has its own stiffness and strains.ritten into the matrix form: ces and elasticity module matrix: (5)

Authors proposed calculation method
The base of this proposed calculation method consists of iteration (layers) method (Zadlauskas 2013).From the values received using this method the distribution of bond stress through the length of the element is drawn.Iteration method is created when the cross-section of the element Where: E 1,2 -elasticity module of layers; A 1,2 -cross-section area; ω -distance between centres of layers; M Ed -bending moment; ξ -stiffness of layers joint.
is divided into the finite numbers of layers Fig. 5. Every separate layer has its own stiffness and strains.All equations are written into the matrix form: ress of the contact zone through the element length of is calculated by the following equation: city module of layers;  �,� -cross-section area;  -distance between centres of layers;  �� oment;  -stiffness of layers joint proposed calculation method.
f this proposed calculation method consists of iteration (layers) dlauskas 2013).From the values received using this method the of bond stress through the length of the element is drawn.Iteration eated when the cross-section of the element is divided into the finite layers Fig. 5. Every separate layer has its own stiffness and strains.s are written into the matrix form: ins, forces and elasticity module matrix: Relative strains, forces and elasticity module matrix: he bond stress of the contact zone through the element length of is calculated by the following equation: here: �,� -elasticity module of layers;  �,� -cross-section area;  -distance between centres of layers;  �� bending moment;  -stiffness of layers joint .Authors proposed calculation method.
he base of this proposed calculation method consists of iteration (layers) ethod (Zadlauskas 2013).From the values received using this method the istribution of bond stress through the length of the element is drawn.Iteration ethod is created when the cross-section of the element is divided into the finite umbers of layers Fig. 5. Every separate layer has its own stiffness and strains.ll equations are written into the matrix form: elative strains, forces and elasticity module matrix: ess of the contact zone through the element length of is calculated by the following equation: ity module of layers;  �,� -cross-section area;  -distance between centres of layers;  �� oment;  -stiffness of layers joint roposed calculation method.
this proposed calculation method consists of iteration (layers) lauskas 2013).From the values received using this method the f bond stress through the length of the element is drawn.Iteration ated when the cross-section of the element is divided into the finite ayers Fig. 5. Every separate layer has its own stiffness and strains.are written into the matrix form: ins, forces and elasticity module matrix: Relative strains, forces and elasticity module matrix: where:  ���� -coefficient which evaluates the bond between carbon fibre and concrete,  � -crosssection area,  � -elasticity modulus of layer, ℎ � -height of layer,  � -relative strain,  -bending moment.
Fig. 5. Dividing of the cross-section in the layers All equations are written into the matrix form: Relative strains, forces and elasticity module matrix: where:  ���� -coefficient which evaluates the bond between carbon fibre and concrete,  � -crosssection area,  � -elasticity modulus of layer, ℎ � -height of layer,  � -relative strain,  -bending moment.
Fig. 5. Dividing of the cross-section in the layers where: k CFRP -coefficient which evaluates the bond between carbon fibre and concrete, A icross-section area, E i -elasticity modulus of layer, h i -height of layer, ε i -relative strain, Mbending moment.
When the first iteration is done the relative strains of every layer are recalculated.If the relative strains exceed the elastic limit of this layer, then the elasticity modulus of this layer is reducing.
If the relative strains exceed the ultimate strains, then the strain modulus of this layer is set to 0. Sufficiently precise strains are received after calculating a couple of iterations.According to the received strains of layers and the strain modulus it is possible to easily calculate the missing parameters.
The results of the analysis are shown in Fig. 6 -Fig.9.The maximum bonding coefficient according to the theory of multiple rods is ξ max = 4,69 • 10 11 N/m 2 .This coefficient is received by the method of approximation and accepted as maximal.
If this value would be increased the normal stress in the cross-section would remain as constant.According to the results in the diagrams it is seen that the maximum bond stress of the contact zone appears at the ends of the element and it distribute by parabola.Further from the end the bond stress decreases significantly and changes according to the line function.
Analysing the results according to FEM the uniqueness at the ends of the beam when the bond is small was noticed.Analysis of the shear stress obtained by FEM shows that the shear stress becomes decrease at the end of the beam.The maximum value of shear stress is reached not at the end of the structure, but a little bit before it.Such a distribution of stress is in-

Results
Fig. 6 Shear stress through the length of the element when the bond between the elements is small (E g =0,2 GPa)

Fig. 7
Shear stress through the length of the element when the bond between the elements is medium (E g =2,0 GPa) When the first iteration is done the relative strains of every layer are recalculated.If the relative strains exceed the elastic limit of this layer, then the elasticity modulus of this layer is reducing.If the relative strains exceed the ultimate strains, then the strain modulus of this layer is set to 0. Sufficiently precise strains are received after calculating a couple of iterations.According to the received strains of layers and the strain modulus it is possible to easily calculate the missing parameters.

Results
The results of the analysis are shown in Fig. 6 -Fig.9.The maximum bonding coefficient according to the theory of multiple rods is  ��� = 4,69 • 10 ��  � .This coefficient is received by the method of approximation and accepted as maximal.If this value would be increased the normal stress in the crosssection would remain as constant.According to the results in the diagrams it is seen that the maximum bond stress of the contact zone appears at the ends of the element and it distribute by parabola.Further from the end the bond stress decreases significantly and changes according to the line function.
Analysing the results according to FEM the uniqueness at the ends of the beam when the bond is small was noticed.Analysis of the shear stress obtained by FEM shows that the shear stress becomes decrease at the end of the beam.The maximum value of shear stress is reached not at the end of the structure, but a little bit before it.Such a distribution of stress is inherent when the concrete reaches the marginal shear stress (Gou 2005, Gou 2007).According to the proposed method there is not obtained the decrease of stress.
In the Fig. 9 the presented shear stress level is higher because of bigger bending moment that was assumed for the elastic plastic analysis.In Fig. 6-8 the normal stress in concrete are under elastic zone.n is done the relative strains of every layer are recalculated.If the relative strains t of this layer, then the elasticity modulus of this layer is reducing.If the relative mate strains, then the strain modulus of this layer is set to 0. Sufficiently precise ter calculating a couple of iterations.According to the received strains of layers it is possible to easily calculate the missing parameters.
ysis are shown in Fig. 6 -Fig.9.The maximum bonding coefficient according to rods is  ��� = 4,69 • 10 ��  � .This coefficient is received by the method of epted as maximal.If this value would be increased the normal stress in the crossas constant.According to the results in the diagrams it is seen that the maximum tact zone appears at the ends of the element and it distribute by parabola.Further stress decreases significantly and changes according to the line function.
ccording to FEM the uniqueness at the ends of the beam when the bond is small of the shear stress obtained by FEM shows that the shear stress becomes decrease .The maximum value of shear stress is reached not at the end of the structure, but ch a distribution of stress is inherent when the concrete reaches the marginal shear 2007).According to the proposed method there is not obtained the decrease of ed shear stress level is higher because of bigger bending moment that was assumed nalysis.In Fig. 6-8 the normal stress in concrete are under elastic zone.
through the length of the nd between the elements is Eg=0,2 GPa) Shear stress through the length of the element when the bond between the elements is perfect (E g =28 GPa) Fig. 9 Shear stress through the length of the element when the bond between the stress is medium (E g =2,0 GPa) (plastic behaviour of concrete is evaluated) Fig. 8. Shear stress through the length of the element when the bond between the elements is perfect (Eg=28 GPa) Fig. 9. Shear stress throug element when the bond be medium (Eg=2,0 GPa) (pl concrete is eva In Table 2 the maximum shear stress of a contact zone is provided.They were c methods evaluating various bonds.One case of calculation was chosen in order behaviour of a concrete.In order to reach the plastic behaviour of a concrete it w the bending moment of a beam  �� = 1,89 .The tension strength of a conc was accepted that the concrete strain modulus is varying linearly from the highest appears).This change is described in formulas (7, 8).Also, in this table the maxi of authors proposed method from FEM or the theory of multiple rods are pro difference of authors proposed method did not exceed 5,9 %.
Maximum relative strains when the concrete is in elastic stage: ��,�� = 0,4 •  ��  � Maximum relative strains before the crack appears: According to the results obtained by finite elements method and the theory of mu formulas to calculate the shear stress through the element length were derived.Coe bond between the concrete and the carbon fibre: Table 2 the maximum shear stress of a contact zone is provided.They were calculated by different thods evaluating various bonds.One case of calculation was chosen in order to evaluate the plastic aviour of a concrete.In order to reach the plastic behaviour of a concrete it was needed to increase bending moment of a beam  �� = 1,89 .The tension strength of a concrete  �� = 1,5 .It s accepted that the concrete strain modulus is varying linearly from the highest to zero (when a crack ears).This change is described in formulas (7, 8).Also, in this table the maximum deviation values authors proposed method from FEM or the theory of multiple rods are provided.The maximum ference of authors proposed method did not exceed 5,9 %.
ximum relative strains when the concrete is in elastic stage: ximum relative strains before the crack appears: herent when the concrete reaches the marginal shear stress (Gou 2005, Gou 2007).According to the proposed method there is not obtained the decrease of stress.
In the Fig. 9 the presented shear stress level is higher because of bigger bending moment that was assumed for the elastic plastic analysis.In Fig. 6-8 the normal stress in concrete are under elastic zone.
In Table 2 the maximum shear stress of a contact zone is provided.They were calculated by different methods evaluating various bonds.One case of calculation was chosen in order to evaluate the plastic behaviour of a concrete.In order to reach the plastic behaviour of a concrete it was needed to increase the bending moment of a beam.
The tension strength of a concrete .It was accepted that the concrete strain modulus is varying linearly from the highest to zero (when a crack appears).This change is described in formulas (7,8).Also, in this table the maximum deviation values of authors proposed method from FEM or the theory of multiple rods are provided.The maximum difference of authors proposed method did not exceed 5,9 %.
Maximum relative strains when the concrete is in elastic stage: ugh the length of the etween the elements is =28 GPa) Fig. 9. Shear stress through the length of the element when the bond between the stress is medium (Eg=2,0 GPa) (plastic behaviour of concrete is evaluated) shear stress of a contact zone is provided.They were calculated by different s bonds.One case of calculation was chosen in order to evaluate the plastic order to reach the plastic behaviour of a concrete it was needed to increase eam  �� = 1,89 .The tension strength of a concrete  �� = 1,5 .It rete strain modulus is varying linearly from the highest to zero (when a crack escribed in formulas (7, 8).Also, in this table the maximum deviation values od from FEM or the theory of multiple rods are provided.The maximum sed method did not exceed 5,9 %.
when the concrete is in elastic stage: before the crack appears: tained by finite elements method and the theory of multiple rods method, the ear stress through the element length were derived.Coefficient evaluating the and the carbon fibre: Maximum relative strains before the crack appears: ough the length of the between the elements is =28 GPa) Fig. 9. Shear stress through the length of the element when the bond between the stress is medium (Eg=2,0 GPa) (plastic behaviour of concrete is evaluated) shear stress of a contact zone is provided.They were calculated by different s bonds.One case of calculation was chosen in order to evaluate the plastic n order to reach the plastic behaviour of a concrete it was needed to increase beam  �� = 1,89 .The tension strength of a concrete  �� = 1,5 .It rete strain modulus is varying linearly from the highest to zero (when a crack escribed in formulas (7, 8).Also, in this table the maximum deviation values od from FEM or the theory of multiple rods are provided.The maximum osed method did not exceed 5,9 %.
when the concrete is in elastic stage: before the crack appears: According to the results obtained by finite elements method and the theory of multiple rods method, the formulas to calculate the shear stress through the element length were derived.Coefficient evaluating the bond between the concrete and the carbon fibre: he bond between the concrete and the carbon fibre it was noticed that the maximum shear ceived at the ends of structures and directly depends on normal stress of the fibre: at the ends of an element are distributed by a principle of parabola and could be calculated wing formulas: m the end of element where the parabola equation of shear stress is valid: s was made for simple supported beam 1 Fig. loaded by bending moment at the ends.For ding cases (if the form of moments diagram change) the additional analysis has to be made posed formulas have to be checked additionally.The plastic behaviour of concrete was the proposed method and the FEM but not by the theory of multiple rods s difference of values of shear stress using the proposed method did not exceed 5,9 % pared to FEM and the theory of multiple rods.imum shear stress of contact zone obtained at the ends of a beam or close to it.shear stress in contact zone at the ends of a beam, according to the proposed method, ibutes by a parabola function.ors proposed method is possible to be used for calculation of the reinforced concrete tures strengthened by carbon fibre at the elastic plastic stress stage of concrete.gment thank to Justinas Valeika for helping on English. (9) Analysing the bond between the concrete and the carbon fibre it was noticed that the maximum shear stress are received at the ends of structures and directly depends on normal stress of the fibre:] Analysing the bond between the concrete and the carbon fibre it was noticed that the maximum shear stress are received at the ends of structures and directly depends on normal stress of the fibre: Shear stress at the ends of an element are distributed by a principle of parabola and could be calculated by the following formulas: Distance from the end of element where the parabola equation of shear stress is valid:

Discussion
The analysis was made for simple supported beam 1 Fig. loaded by bending moment at the ends.For different loading cases (if the form of moments diagram change) the additional analysis has to be made and the proposed formulas have to be checked additionally.The plastic behaviour of concrete was evaluated by the proposed method and the FEM but not by the theory of multiple rods

Conclusions
1.The difference of values of shear stress using the proposed method did not exceed 5,9 % compared to FEM and the theory of multiple rods.2. Maximum shear stress of contact zone obtained at the ends of a beam or close to it.3. The shear stress in contact zone at the ends of a beam, according to the proposed method, distributes by a parabola function.4. Authors proposed method is possible to be used for calculation of the reinforced concrete structures strengthened by carbon fibre at the elastic plastic stress stage of concrete.
Shear stress at the ends of an element are distributed by a principle of parabola and could be calculated by the following formulas: he bond between the concrete and the carbon fibre it was noticed that the maximum shear ceived at the ends of structures and directly depends on normal stress of the fibre: at the ends of an element are distributed by a principle of parabola and could be calculated wing formulas: m the end of element where the parabola equation of shear stress is valid: s was made for simple supported beam 1 Fig. loaded by bending moment at the ends.For ding cases (if the form of moments diagram change) the additional analysis has to be made posed formulas have to be checked additionally.The plastic behaviour of concrete was y the proposed method and the FEM but not by the theory of multiple rods s difference of values of shear stress using the proposed method did not exceed 5,9 % pared to FEM and the theory of multiple rods.imum shear stress of contact zone obtained at the ends of a beam or close to it.shear stress in contact zone at the ends of a beam, according to the proposed method, ibutes by a parabola function.ors proposed method is possible to be used for calculation of the reinforced concrete tures strengthened by carbon fibre at the elastic plastic stress stage of concrete.
gment thank to Justinas Valeika for helping on English. ( he bond between the concrete and the carbon fibre it was noticed that the maximum shear ceived at the ends of structures and directly depends on normal stress of the fibre: at the ends of an element are distributed by a principle of parabola and could be calculated wing formulas: m the end of element where the parabola equation of shear stress is valid: s was made for simple supported beam 1 Fig. loaded by bending moment at the ends.For ding cases (if the form of moments diagram change) the additional analysis has to be made posed formulas have to be checked additionally.The plastic behaviour of concrete was the proposed method and the FEM but not by the theory of multiple rods s difference of values of shear stress using the proposed method did not exceed 5,9 % pared to FEM and the theory of multiple rods.imum shear stress of contact zone obtained at the ends of a beam or close to it.shear stress in contact zone at the ends of a beam, according to the proposed method, ibutes by a parabola function.ors proposed method is possible to be used for calculation of the reinforced concrete tures strengthened by carbon fibre at the elastic plastic stress stage of concrete.
gment thank to Justinas Valeika for helping on English. ( bond between the concrete and the carbon fibre it was noticed that the maximum shear ved at the ends of structures and directly depends on normal stress of the fibre: the ends of an element are distributed by a principle of parabola and could be calculated g formulas: the end of element where the parabola equation of shear stress is valid: as made for simple supported beam 1 Fig. loaded by bending moment at the ends.For ng cases (if the form of moments diagram change) the additional analysis has to be made sed formulas have to be checked additionally.The plastic behaviour of concrete was e proposed method and the FEM but not by the theory of multiple rods fference of values of shear stress using the proposed method did not exceed 5,9 % ed to FEM and the theory of multiple rods.um shear stress of contact zone obtained at the ends of a beam or close to it.ear stress in contact zone at the ends of a beam, according to the proposed method, tes by a parabola function.s proposed method is possible to be used for calculation of the reinforced concrete res strengthened by carbon fibre at the elastic plastic stress stage of concrete.ent ank to Justinas Valeika for helping on English. ) the additional analysis has to be made roposed formulas have to be checked additionally.The plastic behaviour of concrete was by the proposed method and the FEM but not by the theory of multiple rods ns e difference of values of shear stress using the proposed method did not exceed 5,9 % mpared to FEM and the theory of multiple rods.aximum shear stress of contact zone obtained at the ends of a beam or close to it.e shear stress in contact zone at the ends of a beam, according to the proposed method, tributes by a parabola function.thors proposed method is possible to be used for calculation of the reinforced concrete uctures strengthened by carbon fibre at the elastic plastic stress stage of concrete.dgment rs thank to Justinas Valeika for helping on English. ( Distance from the end of element where the parabola equation of shear stress is valid: alysing the bond between the concrete and the carbon fibre it was noticed that the maximum shear ess are received at the ends of structures and directly depends on normal stress of the fibre: The difference of values of shear stress using the proposed method did not exceed 5,9 % compared to FEM and the theory of multiple rods.2. Maximum shear stress of contact zone obtained at the ends of a beam or close to it.3. The shear stress in contact zone at the ends of a beam, according to the proposed method, distributes by a parabola function.4. Authors proposed method is possible to be used for calculation of the reinforced concrete structures strengthened by carbon fibre at the elastic plastic stress stage of concrete.knowledgment e authors thank to Justinas Valeika for helping on English. (15)

Conclusions
The analysis was made for simple supported beam Fig. 1 loaded by bending moment at the ends.
For different loading cases (if the form of moments diagram change) the additional analysis has to be made and the proposed formulas have to be checked additionally.The plastic behaviour of concrete was evaluated by the proposed method and the FEM but not by the theory of multiple rods 1 The difference of values of shear stress using the proposed method did not exceed 5,9 % compared to FEM and the theory of multiple rods.
2 Maximum shear stress of contact zone obtained at the ends of a beam or close to it.

Fig. 3 .
Fig. 3. Distribution of shear stress at the ends of the beam Fig. 3 Distribution of shear stress at the ends of the beam

Fig. 5
Fig. 5Dividing of the cross-section in the layers

Fig. 3 .Fig. 4 .
Fig. 3. Distribution of shear stress at the ends of the beam Fig. 5. Dividing of the cross-section in the layers

Fig
Fig. 5. Dividing of the cross-section in the layers Fig. 5. Dividing of the cross-section in the layers

Fig. 5 .
Fig. 5. Dividing of the cross-section in the layers

Fig. 5 .
Fig. 5. Dividing of the cross-section in the layers Fig. 5. Dividing of the cross-section in the layers

Fig. 6 .Fig. 7 .
Fig.6.Shear stress through the length of the element when the bond between the elements is small (Eg=0,2 GPa)

Fig. 7 .
Fig.7.Shear stress through the length of the element when the bond between the elements is medium (Eg=2,0 GPa)

Fig. 8 .Fig. 9 .
Fig.8.Shear stress through the length of the lement when the bond between the elements is perfect (Eg=28 GPa) stress   , Pa   , Pa   , Pa Max difference, % 4,89 • 10 � 5,07 • 10 � 5,00 • 10 � 2 the concrete and the carbon fibre it was noticed that the maximum shear received at the ends of structures and directly depends on normal stress of the fibre: ends of an element are distributed by a principle of parabola and could be calculated owing formulas: =  •  � +  •  (of elementwhere the parabola equation of shear stress is valid: for simple supported beam 1 Fig. loaded by bending moment at the ends.For oading cases (if the form of moments diagram change the ends of an element are distributed by a principle of parabola and could be calculated the following formulas: =  •  � +  • (end of element where the parabola equation of shear stress is valid: made for simple supported beam 1 Fig.loadedby bending moment at the ends.For ferent loading cases (if the form of moments diagram change) the additional analysis has to be made d the proposed formulas have to be checked additionally.The plastic behaviour of concrete was aluated by the proposed method and the FEM but not by the theory of multiple rods nclusions 1.

Table 1 .
Materials parameters

Table 1 .
Materials parameters

Table 2 .
Maximum shear stress