Prediction of Deflection of Reinforced Concrete Beams Strengthened with Fiber Reinforced Polymer

The article analyses the calculation of the deflection of reinforced concrete beams strengthened with fiber reinforced polymer. This paper specifically focuses on estimating deflection when the yielding of reinforcement is reached. The article proposes a simple method for calculating deflection that was compared with the experimentally predicted deflection. The carried out comparison has showed that the proposed method is suitable not only for the strengthened beams but also for the reinforced concrete beams with a varying reinforcement ratio. The suggested calculation method is based on the effective moment of inertia, such as the one introduced in the ACI Committee 318 Building Code Requirement for Structural Concrete (ACI318). The development of deflection was divided into three stages, and equations for the effective moment of inertia were proposed considering separate stages. In addition, the put forward equations were modified attaching additional relative coefficients evaluating a change in the depth of the neutral axis.


Introduction
One of the greatest advantages that can provide strengthening with carbon fiber reinforced polymer (CFRP) is an increase in the flexibility of the beam.Failure in the reinforced concrete beam is related to steel yielding, concrete crashing, or shear failure.Short-term and long-term experiments have showed that strengthening RC beams with CFRP can delay steel yielding [1][2][3][4][5][6].Evenly, if steel yielding is reached or steel is rusted, the strengthened beams can serve until the rupture, delamination of the CFRP layer, steel fatigue fracture, or concrete crashing are achieved [7][8][9][10][11].Due to high strength and high elasticity, the tensioned layer of CFRP can intercept tensile forces (stresses) when the yielding of reinforcement is reached.That is why the deflection of the beam can develop, thus reaching the yielding of reinforcement at a later stage.However, there is a danger for premature debonding of CFRP layer.In order to prevent this, proper additional anchoring can delay this phenomenon [12].As well near surface mounted CFRP due to a larger perimeter-to-sectional-area-ratio can ensure better bond performance [13].
Various researches demonstrate that deflection development and reached yielding depend on the reinforcement (steel) ratio [14,15].This may be related to the exploitation of the compressed concrete.If the reinforcement ratio is low, the exploitation of the compressed concrete is also greatly reduced until the yielding of reinforcement is reached.Therefore, the deflection (when the yielding of reinforcement is reached) of the strengthened beams with a low reinforcement ratio is the biggest.This is due to the unexploited deformability of the compressed concrete.
The existing methods for calculating deflection can perform estimation until the yielding of reinforcement is reached.The most common and simplest methods are based on design guidelines ACI318 [16] and the Eurocode 2 [17].In addition, the multi-layer method can be used for calculating the deflection of the strengthened beams; however, this method is not that convenient for engineers, and therefore will not be discussed in this article.The calculation method based on ACI318 [16] evaluates the effective moment of inertia, and the method based on Eurocode 2 [17], usually evaluates the average curvature of the bending element.Both methods evaluate the moment of the inertia of the full cross-section and the moment of the inertia of the cross-section where the crack is opened.However, these methods evaluate stress strain state in the cross-section before yield stresses in reinforcement are reached.There are several methods [18][19][20][21] that can evaluate stress-strain state in the cross-section after yield stresses are reached but these methods are difficult to be applied by the designer.Several contributions based on the moment-curvature modeling are available [22,23].The accuracy of the proposed model [22,23] is impressive, however certain parameters like moment of inertia, depth of the neutral axis remains unknown.
The load carrying capacity of the strengthened beams can significantly increase such that the increased service load can locate in the range of the load-deflection curve where steel yielding is reached.The main objective of this article is to calculate the deflection of the strengthened beam when steel yielding is reached and when only the layer of CFRP intercepts tensile forces.

Analyzed Beams
RC strengthened beams with various reinforcement ratios were chosen to perform the calculation of deflection.The data about beams were collected from various research.The references and titles of the analyzed beams with a short description are presented in Table 1.The chosen beams are suitable for deflection analysis, because deflection develops when the yielding of reinforcement is reached.As mentioned above, a lower reinforcement ratio allows a higher increment in deflection when the yielding of reinforcement is reached.
The mechanical parameters of the material such as the modulus of elasticity and tensile strength are required in order to calculate the deflection of the beam.This and other mechanical parameters are presented in Table 2.

Calculation of Deflection
The development of the deflection of the strengthened and unstrengthened beams is divided into stages.At the first stage, deflection develops until vertical cracks open in the tensioned part of the cross-section.At the second stage, deflection develops when the vertical crack is opened until the yielding strength of the tensioned reinforcement is reached.At the third stage, deflection develops when the yielding strength of reinforcement is reached and only a layer of CFRP intercepts tensile force.Therefore, two deflection development stages exist for the unstrengthened beams and three stages for the strengthened ones (Figure 1).Bending moments M I and M I.S are shown in (Figure 1), which is the cracking moment of the unstrengthened and strengthened beam, respectively.Due to the CFRP layer, the contribution cracking moment of the strengthened beam is slightly bigger than that of the unstrengthened beam (M I.S > MI).Bending moments (M I.S and M I ) correspond to the end of the first stage.The maximal carrying bending moment of the unstrengthened beam (M R = M II ) is smaller than that of the bending moment of the strengthened beam (M II.S ) when the yielding of reinforcement is reached.These bending moments correspond to the end of the second stage.The maximum carrying bending moment of the strengthened beam is designated as M R.S = M III and corresponds to the end of the third stage.

Calculation of Deflection
The development of the deflection of the strengthened and unstrengthened beams is divided into stages.At the first stage, deflection develops until vertical cracks open in the tensioned part of the cross-section.At the second stage, deflection develops when the vertical crack is opened until the yielding strength of the tensioned reinforcement is reached.At the third stage, deflection develops when the yielding strength of reinforcement is reached and only a layer of CFRP intercepts tensile force.Therefore, two deflection development stages exist for the unstrengthened beams and three stages for the strengthened ones (Figure 1).Bending moments MI and MI.S are shown in (Figure 1), which is the cracking moment of the unstrengthened and strengthened beam, respectively.Due to the CFRP layer, the contribution cracking moment of the strengthened beam is slightly bigger than that of the unstrengthened beam (MI.S > MI).Bending moments (MI.S and MI) correspond to the end of the first stage.The maximal carrying bending moment of the unstrengthened beam (MR = MII) is smaller than that of the bending moment of the strengthened beam (MII.S) when the yielding of reinforcement is reached.These bending moments correspond to the end of the second stage.The maximum carrying bending moment of the strengthened beam is designated as MR.S = MIII and corresponds to the end of the third stage.The deflection of the beams at a certain stage is influenced by different flexural stiffness.Generally, bending stiffness E• I (the product of the modulus of elasticity and the moment of inertia) is influenced by the moment of inertia.The current methods for calculating deflection usually evaluate the modulus of elasticity like for an elastic material.Then, the development of deflection undergoes all stages, cracks in the tensioned part of the cross-section develop, therefore, the moment of the inertia is not constant.Thus, at a certain stage, the depth of the neutral axis and the moment of inertia are different.A change in the depth of the neutral axis of the strengthened and unstrengthened beams is presented in Figures 2 and 3. Thus, there are parts of the cross-section containing and having no cracks.Therefore, the effective moment of inertia should be evaluated.The prediction of the depth of the neutral axis at each stage confirms that the distribution of strains is linear.Stresses in the compressed part of the section are in the elastic range.In addition, a hypothesis about the plane section is valid.The strain of internal and external reinforcement is equal to the surrounded concrete strain (bond slip is not evaluated).The deflection of the beams at a certain stage is influenced by different flexural stiffness.Generally, bending stiffness E•I (the product of the modulus of elasticity and the moment of inertia) is influenced by the moment of inertia.The current methods for calculating deflection usually evaluate the modulus of elasticity like for an elastic material.Then, the development of deflection undergoes all stages, cracks in the tensioned part of the cross-section develop, therefore, the moment of the inertia is not constant.Thus, at a certain stage, the depth of the neutral axis and the moment of inertia are different.A change in the depth of the neutral axis of the strengthened and unstrengthened beams is presented in Figures 2  and 3. Thus, there are parts of the cross-section containing and having no cracks.Therefore, the effective moment of inertia should be evaluated.The prediction of the depth of the neutral axis at each stage confirms that the distribution of strains is linear.Stresses in the compressed part of the section are in the elastic range.In addition, a hypothesis about the plane section is valid.The strain of internal and external reinforcement is equal to the surrounded concrete strain (bond slip is not evaluated).The deflection of the strengthened beam at stage 1 up to the cracking of the tensioned part of the cross-section can be predicted by the equation: where l-the span length of the beam, a-distance from the support to the external load position, MI-acting moment, Ecm-the modulus of elasticity of concrete, II.red-the reduced moment of the inertia of the total cross-section according to the neutral axis of the cross-section.
At stage 1, the evaluated acting moment is 0 < MI ≤ MI.S, and the ultimate bending moment of stage 1 is the cracking moment: where fct-the tensile strength of concrete, yc.I-the centre of the gravity of the cross-section at stage 1.
The center of gravity can be predicted by the following equations: The deflection of the strengthened beam at stage 1 up to the cracking of the tensioned part of the cross-section can be predicted by the equation: where l-the span length of the beam, a-distance from the support to the external load position, MI-acting moment, Ecm-the modulus of elasticity of concrete, II.red-the reduced moment of the inertia of the total cross-section according to the neutral axis of the cross-section.
At stage 1, the evaluated acting moment is 0 < MI ≤ MI.S, and the ultimate bending moment of stage 1 is the cracking moment: where fct-the tensile strength of concrete, yc.I-the centre of the gravity of the cross-section at stage 1.
The center of gravity can be predicted by the following equations: The deflection of the strengthened beam at stage 1 up to the cracking of the tensioned part of the cross-section can be predicted by the equation: where l-the span length of the beam, a-distance from the support to the external load position, M I -acting moment, E cm -the modulus of elasticity of concrete, I I.red -the reduced moment of the inertia of the total cross-section according to the neutral axis of the cross-section.At stage 1, the evaluated acting moment is 0 < M I ≤ M I.S , and the ultimate bending moment of stage 1 is the cracking moment: where f ct -the tensile strength of concrete, y c.I -the centre of the gravity of the cross-section at stage 1.
The center of gravity can be predicted by the following equations: where A red -the reduced cross-section of the strengthened beam, A f -the cross section of carbon fibers, A s1 , A s2 -the cross-section of steel bars, S red -the static moment of the reduced cross-section of the strengthened beam, α f , α s1 , α s2 -coefficients of reduction, E f -the modulus of elasticity of fibers, E s1 , E s2 -the modulus of elasticity of the steel bars.
The reduced moment of the inertia of the cross-section can be predicted by the following equation: The deflection of the strengthened beam at stage 2, when the tensioned part of the cross-section is cracked and the yielding of the tensioned reinforcement is not reached, can be predicted by the equation: The acting bending moment at stage 2 is M II and the moment M I.S < M II ≤ M II.S .The moment when the yielding of reinforcement is reached is M II.S .The effective moment of inertia is evaluated using the Branson [45] equation for parameter I II : If change of the neutral axis is evaluated, then Equation ( 11) is modified like: where I II.red -the reduced moment of the inertia of the cross section where the vertical crack is opened.This moment of inertia can be predicted by the equation: Coefficients γ 1.c and γ 1.t evaluate a change in the neutral axis and can be predicted by equations: The depth of the neutral axis at stage 1 is predicted by the equation: The prediction of the depth of the neutral axis in the section having an opened crack is based on the previously mentioned assumptions.The hypothesis of plain sections is valid.The distribution of strains through the height of the section is linear (Figure 4b).Then, by the similarity of triangles, strains at each layer, in proportion with the strain of the compressed concrete layer, can be expressed, and the depth of the neutral axis should be expressed from the square equation.The depth of the neutral axis at stage 2 can be predicted by the equation: where coefficients A, B, and Materials 2019, 12, x FOR PEER REVIEW 14 of 41 The prediction of the depth of the neutral axis in the section having an opened crack is based on the previously mentioned assumptions.The hypothesis of plain sections is valid.The distribution of strains through the height of the section is linear (Figure 4b).Then, by the similarity of triangles, strains at each layer, in proportion with the strain of the compressed concrete layer, can be expressed, and the depth of the neutral axis should be expressed from the square equation.The depth of the neutral axis at stage 2 can be predicted by the equation: where coefficients A, B, and C: The deflection of the strengthened beam at stage 3, when the yielding strength of tensioned reinforcement is reached, can be predicted by the equation: The acting bending moment at stage 3 is MIII and the moment MII.u < MIII ≤ MIII.u.The ultimate bending moment at stage 3 is MIII.u.The new effective moment of inertia is evaluated in the equation for parameter IIII: If change of the neutral axis is evaluated, then Equation ( 22) is modified like: where IIII.red-the reduced moment of the inertia of the cross section where the vertical crack is opened.This moment of inertia can be predicted by the equation: The deflection of the strengthened beam at stage 3, when the yielding strength of tensioned reinforcement is reached, can be predicted by the equation: The acting bending moment at stage 3 is M III and the moment M II.u < M III ≤ M III.u .The ultimate bending moment at stage 3 is M III.u .The new effective moment of inertia is evaluated in the equation for parameter I III : where I III.red -the reduced moment of the inertia of the cross section where the vertical crack is opened.This moment of inertia can be predicted by the equation: Coefficients γ 2.c and γ 2.t : The depth of the neutral axis at stage 3 is also predicted from the similarity of triangles (Figure 5b).
Coefficients γ2.c and γ2.t: The depth of the neutral axis at stage 3 is also predicted from the similarity of triangles (Figure 5b).The depth of the neutral axis at stage 3 is predicted by the equation: Were coefficients A, B, and C: The deflection of the unstrengthened beams can be predicted by the same Equations ( 1) and (10).However, the parameters of the FRP layer in other equations should be ignored.If the beams are strengthened with the prestressed FRP, in this case it is necessary to calculate the additional curvature and the deflection from prestress force.The total deflection is obtained by summing up all the deflections.6-9) shows that the equation method is suitable for RC beams with various reinforcement ratios.Calculated deflections of all mentioned beams are presented in the Appendix A. In these figures, designation "Calc.I" is related to Equations ( 11) and (22).Designation "Calc.II" related with Equations ( 12) and (23).It is clear that the theoretical equation method gives brake points such as the cracking moment and steel yielding moment on the load deflection curve.The difference between the calculated and experimental deflection increases when the load level increases.This may happen because the theoretical method evaluates the elastic work of concrete and the constant depth of the neutral axis.Thus, the deflection curve curvature The depth of the neutral axis at stage 3 is predicted by the equation:

A comparison of deflections (Figures
Were coefficients A, B, and The deflection of the unstrengthened beams can be predicted by the same Equations ( 1) and (10).However, the parameters of the FRP layer in other equations should be ignored.If the beams are strengthened with the prestressed FRP, in this case it is necessary to calculate the additional curvature and the deflection from prestress force.The total deflection is obtained by summing up all the deflections.6-9) shows that the equation method is suitable for RC beams with various reinforcement ratios.Calculated deflections of all mentioned beams are presented in the Appendix A. In these figures, designation "Calc.I" is related to Equations ( 11) and (22).Designation "Calc.II" related with Equations ( 12) and (23).It is clear that the theoretical equation method gives brake points such as the cracking moment and steel yielding moment on the load deflection curve.The difference between the calculated and experimental deflection increases when the load level increases.This may happen because the theoretical method evaluates the elastic work of concrete and the constant depth of the neutral axis.Thus, the deflection curve curvature depends just from ratio of the bending moments.In order to increase the accuracy of the theoretical method, nonlinear stress-strain distribution across the height of the cross-section should be evaluated.The proposed method evaluates linear stress-strain distribution.The evaluation of nonlinear stress-strain distribution can be complex for designers, and thus triangular distribution is easier to assess.Furthermore, a comparison of the position of the center of the parabolic and triangular form gives little difference.The difference in results is also influenced by the accuracy of the experiment.In certain experiments, deflection at the cracking moment to big.The main drawback of the suggested method is the prediction of the bending moment when steel yielding is reached.It is difficult to predict the moment when the FRP layer is incorporated, because strains are not known in the compressed concrete and tensioned CFRP layer.In such a case, the problem must be solved by the iteration approach until the balance of internal forces is reached.This is also a complex task for designers.For this research values of cracking, yielding and ultimate moment were predicted from the deflection evolution plots.

A comparison of deflections (Figures
Experiments in which the deflection was measured from the frame mounted on a beam gives a more precise result.Calculated deflection (Calc.I) using the effective moment of inertia equation without any coefficients is suitable for this measurement system.Equation of the effective moment of inertia must be without coefficients-it is related with the neutral axis.Please note that the second stage does not have a horizontal straight line.The other experimental "deflection" results, which are more close to the "Calc.II" can be associated with the measured displacement.depends just from ratio of the bending moments.In order to increase the accuracy of the theoretical method, nonlinear stress-strain distribution across the height of the cross-section should be evaluated.The proposed method evaluates linear stress-strain distribution.The evaluation of nonlinear stress-strain distribution can be complex for designers, and thus triangular distribution is easier to assess.Furthermore, a comparison of the position of the center of the parabolic and triangular form gives little difference.The difference in results is also influenced by the accuracy of the experiment.In certain experiments, deflection at the cracking moment to big.The main drawback of the suggested method is the prediction of the bending moment when steel yielding is reached.It is difficult to predict the moment when the FRP layer is incorporated, because strains are not known in the compressed concrete and tensioned CFRP layer.In such a case, the problem must be solved by the iteration approach until the balance of internal forces is reached.This is also a complex task for designers.For this research values of cracking, yielding and ultimate moment were predicted from the deflection evolution plots.Experiments in which the deflection was measured from the frame mounted on a beam gives a more precise result.Calculated deflection (Calc.I) using the effective moment of inertia equation without any coefficients is suitable for this measurement system.Equation of the effective moment of inertia must be without coefficients-it is related with the neutral axis.Please note that the second stage does not have a horizontal straight line.The other experimental "deflection" results, which are more close to the "Calc.II" can be associated with the measured displacement.

Conclusions
According to the proposed method for calculating the deflection of the strengthened RC beam, it is possible to predict deflection when steel yielding is reached.When the deflection is calculated using the usual expression of an effective moment of inertia (Equations ( 11) and ( 22)), in some cases smaller deflections are obtained.This discrepancy may be due to an incorrectly determined experimental deflection, since in some experiments it is not clear whether the deflection is determined by compensating the lift of the neutral axis at the supports.In most cases, the most accurate calculation using the normal expression of an effective inertia moment (Equations ( 11) and ( 22)).Estimating the change in the neutral axis (Equations ( 12) and ( 23)) results in bigger deflections but are more precise when the deflections are lower with normal expression (Equations ( 11) and ( 22)).Another important criterion related to the accuracy of deflections is the coefficient of estimating the nature of the external load, since after the strengthening the evolution of cracks changes, the curvature development change too.In order to verify the accuracy of the experimental and computational results, further finite element analysis is required.

Conclusions
According to the proposed method for calculating the deflection of the strengthened RC beam, it is possible to predict deflection when steel yielding is reached.When the deflection is calculated using the usual expression of an effective moment of inertia (Equations ( 11) and ( 22)), in some cases smaller deflections are obtained.This discrepancy may be due to an incorrectly determined experimental deflection, since in some experiments it is not clear whether the deflection is determined by compensating the lift of the neutral axis at the supports.In most cases, the most accurate calculation using the normal expression of an effective inertia moment (Equations ( 11) and ( 22)).Estimating the change in the neutral axis (Equations ( 12) and ( 23)) results in bigger deflections but are more precise when the deflections are lower with normal expression (Equations ( 11) and ( 22)).Another important criterion related to the accuracy of deflections is the coefficient of estimating the nature of the external load, since after the strengthening the evolution of cracks changes, the curvature development change too.In order to verify the accuracy of the experimental and computational results, further finite element analysis is required.Acknowledgments: Thanks to all researchers who publish detailed theoretical and experimental information.

Conflicts of Interest:
The authors declare no conflict of interest.the coefficient evaluating the shape of stress distribution; l the span length of the beam; t f the thickness of the carbon fiber layer; x I , x II and x III the depth of the neutral axis at stages 1, 2 and 3; y c.I , y c.II , y c.III , the centre of the gravity of the beam cross-section at stages 1, 2 and 3; Funding: This research received no external funding.

Notation
Acknowledgments: Thanks to all researchers who publish detailed theoretical and experimental information.

*
a s = h-A s1 / ńs1 •b; ** evaluated individually; l-span length; b-total width of the beam; h-height of the beam; A s1 -cross-section of the tensioned steel bars; A s2 -cross-section of the compressed steel bars; d 1 -position of the tensioned steel bars; d 2 -position of the compressed steel bars; A f -cross-section of the tensioned fibers or FRP; ńs1-reinforcement ratio by A s1 .

Figure 1 .
Figure 1.The development of the deflection of the strengthened and unstrengthened beam.

Figure 1 .
Figure 1.The development of the deflection of the strengthened and unstrengthened beam.

Figure 2 .Figure 3 .
Figure 2. A change in the depth of the neutral axis of the RC strengthened beam: (a) Cross-section of the strengthened beam; (b) depth of the neutral axis before vertical cracks will open; (c) depth of the neutral axis when vertical cracks are opened; (d) depth of the neutral axis when steel yielding is reached.

Figure 2 .Figure 2 .Figure 3 .
Figure 2. A change in the depth of the neutral axis of the RC strengthened beam: (a) Cross-section of the strengthened beam; (b) depth of the neutral axis before vertical cracks will open; (c) depth of the neutral axis when vertical cracks are opened; (d) depth of the neutral axis when steel yielding is reached.

Figure 3 .
Figure 3.A change in the depth of the neutral axis of the RC beam: (a) Cross-section of the beam (b) depth of the neutral axis before vertical cracks will open; (c) depth of the neutral axis when vertical cracks are opened.

Figure 4 .
Figure 4. Stress-strain state in the strengthened RC beam until the yielding of reinforcement is reached: (a) Depth of the neutral axis; (b) distribution of strains; (c) distribution of stresses; (d) internal forces.

Figure 4 .
Figure 4. Stress-strain state in the strengthened RC beam until the yielding of reinforcement is reached: (a) Depth of the neutral axis; (b) distribution of strains; (c) distribution of stresses; (d) internal forces.

Figure 5 .
Figure 5. Stress-strain state in the strengthened RC beam when the yielding of reinforcement is reached: (a) Depth of the neutral axis; (b) distribution of strains; (c) distribution of stresses; (d) internal forces.

Figure 5 .
Figure 5. Stress-strain state in the strengthened RC beam when the yielding of reinforcement is reached: (a) Depth of the neutral axis; (b) distribution of strains; (c) distribution of stresses; (d) internal forces.
Notation A, B, C the designation of the equation for the depth of the neutral axis; Ac the cross section of the compressed concrete layer; Af the cross-section area of carbon fiber; Ared the transformed cross section of the beam; As1 and As2 the cross-section area of the tensioned and compressed reinforcement; Ecm the modulus of elasticity of concrete; Ef the modulus of elasticity of carbon fiber; Es1, Es2 the modulus of elasticity of steel bars; III, IIII the effective moment of inertia at stages 2 and 3; II.red,III.red, IIII.red the moment of the inertia of the transformed cross-section at stages 1, 2, and 3; MI and MI.S the cracking moment of the unstrengthened beam and strengthened beam respectively; MII.S the bending moment of the strengthened beam when the yielding of reinforcement is reached; MR, MII the maximum carrying bending moment of the unstrengthened beam; MR.S, MIII the maximum carrying bending moment of the strengthened beam; Sred the static moment of the transformed cross-section; a distance from the support to loading; b the width of the beam;d1 and d2 distance from the beam edge to the center of the tensioned and compressed reinforcement;
A, B, Cthe designation of the equation for the depth of the neutral axis; A c the cross section of the compressed concrete layer; A f the cross-section area of carbon fiber; A red the transformed cross section of the beam; A s1 and A s2 the cross-section area of the tensioned and compressed reinforcement; E cm the modulus of elasticity of concrete; E f the modulus of elasticity of carbon fiber; E s1 , E s2 the modulus of elasticity of steel bars; I II , I III the effective moment of inertia at stages 2 and 3; I I.red , I II.red , I III.red the moment of the inertia of the transformed cross-section at stages 1, 2, and 3; M I and M I.S the cracking moment of the unstrengthened beam and strengthened beam respectively;M II.S the bending moment of the strengthened beam when the yielding of reinforcement is reached; M R , M II the maximum carrying bending moment of the unstrengthened beam; M R.S , M III the maximum carrying bending moment of the strengthened beam; S red the static moment of the transformed cross-section; a distance from the support to loading; b the width of the beam; d 1 and d 2 distance from the beam edge to the center of the tensioned and compressed reinforcement; f c the compressive strength of concrete cylinders; f ct the tensile strength of concrete; h the height of the beam; k 1

Table 1 .
Characteristics of investigated experimental beams.

Table 2 .
Mechanical characteristics of investigated experimental beams materials.
γ 1.t relative coefficients evaluating a change in the depth of the neutral axis at stage 2;γ 2.c , γ 2.trelative coefficients evaluating a change in the depth of the neutral axis at stage 3; II the deflection of the control beam up to the end of stages I and II.ω I.S , ω II.S , ω III.S the deflection of the strengthened beam up to the end of stages I, II, and III.tf the thickness of the carbon fiber layer; xI, xII and xIII the depth of the neutral axis at stages 1, 2 and 3; yc.I, yc.II, yc.III, the centre of the gravity of the beam cross-section at stages 1, 2 and 3; the control beam up to the end of stages I and II.ωI.S, ωII.S, ωIII.S the deflection of the strengthened beam up to the end of stages I, II, and III.