Comparison of calculation models for flexural capacity of RC beams strengthened with TRC in China and Germany

The new innovative composite material textile reinforced concrete (TRC) has been intensively investigated in Germany since the end 1990s. It has become increasingly important in the construction industry. Compared with conventional steel reinforcement, TRC has advantages such as higher load‐bearing capacity, higher strength‐to‐weight ratio, better ductility, and non‐corrosive behavior. This made them a subject of extensive research and diverse applications both nationally and internationally. In 2004, Xu et al. started research on bond properties of TRC in China in cooperation with Hans‐Wolf Reinhardt et al. from the University of Stuttgart in Germany. Since then, there have been numerous researches on TRC in China. This article introduces a calculation method for the flexural capacity of reinforced concrete (RC) beams strengthened with TRC in China. For comparison, the dimensioning procedure in Germany is also presented. Subsequently, the two models are compared with each other in a case study. Both models in China and Germany have the same mathematical background and also provide similar results. However, they have some differences in definitions of material characteristics (e.g., design concrete compressive strength, strain, and stress distribution) and consideration of the damage resulting from the preloading stage.


| INTRODUCTION
Fiber reinforcement has been used for a long time in various forms: natural fibers, short fibers of glass, steel, fiber-reinforced polymer (FRP) composites, for example, carbon FRP (CFRP), glass FRP (GFRP), and so on. 1 The main application of fiber reinforcement can be found in textile-reinforced concrete (TRC), short fiber-reinforced concrete (FRC), and non-metallic reinforcing bars.In the composite material TRC, steel reinforcement is replaced by non-metallic, corrosion-resistant reinforcement. 2,3Alkali-resistant glass fibers, basalt fibers, aramid, or preferably continuous carbon fibers are mainly used.The threads, which are present as multifilament yarns, are combined with a polymer impregnation to form a highperformance composite reinforcement material.6][7] The fiber-based textile reinforcement represents a directional, high-performance reinforcement for concrete components and is different from the fibers in FRC, where short fiber bundles are added directly to the concrete and thus form non-directional reinforcement. 8In Germany, TRC has been deepened developed in mainly two collaborative research centers funded by the German Research Foundation (DFG) from 1999 to 2011: SFB 528 (textile reinforcement for structural reinforcement and repair) in Dresden and SFB 532 (textile-reinforced concretebasics for development of a new type of technology) in Aachen. 9A few years later, researchers in China started investigating TRC as well.In 2004, Xu et al. 10 investigated bond properties of textile reinforcement in mortar through a comparison between pull-out tests and numerical simulations.Furthermore, some authors, [11][12][13][14][15] have researched the flexural behavior of RC beams strengthened with TRC to gain a better understanding of their mechanical properties.
In this article, design methods of reinforced concrete components strengthened with TRC under bending are compared between China and Germany.

| CALCULATION MODELS FOR RC BEAMS STRENGTHENED WITH TRC UNDER BENDING IN CHINA AND GERMANY
2.1 | Calculation models for material properties in China and Germany

| Concrete
A parabola-rectangle diagram (Figure 1A) is used to describe the material behavior of concrete both in the Chinese standard GB50050-2010 16 and in Eurocode 2 (EC2). 17The parameters ε c2 (concrete strain by reaching the concrete strength f cd ), the exponent n for the stress-strain curve (see Figure 1A) and ε c2u (ultimate compressive strain of concrete) are dependent on concrete strength both in China and Germany.The concrete cube strength f ck,cube is defined as the characteristic compressive strength of concrete cubes with a size of 150 mm in both countries.Despite those similarities, there are some differences in calculation models for concrete in Chinese and German Standards.For concrete cube strength f ck,cube up to 50 N/mm 2 (concrete C40/50 in Germany or concrete C50 in China), ε c2 and n are identical in both countries.The parameter ε c2u , which is normally 3.3‰ in China and 3.5‰ in Germany, is only slightly different.For concrete with the cube strength ranging between 50 and 85 N/mm 2 , the difference of the parameters ε c2 , ε c2u , and n from the Chinese and German standards amounts to up to 10%.The values for those parameters both in China and Germany are given in Table 1.
Except the mentioned parabola-rectangle diagram, a rectangular stress block for concrete stress distribution is recommended for the calculation of flexural capacity of RC beams according to the Chinese standard GB50050-2010 (Figure 1B).Through the simplification of concrete stress distribution in compression zone, the depth of concrete compression zone x and consequently the flexural capacity could be easily calculated.Two coefficients α 1 and β 1 are given for the design concrete strength f cd and the depth of concrete compression zone x, respectively (see Figure 1B).Similarly, in EC2 a rectangular stress block for concrete stress distribution can also be found for cross-section dimensioning.The coefficients for f cd and x are defined as η and λ in EC2, respectively (see Figure 1B).The value pairs (α 1 , β 1 ) and (η, λ) from Chinese and German standards are also dependent on concrete strength and are in most cases similar.A comparison of those values is also presented in Table 1.

| Steel
The Chinese standard GB50010-2010 16 specifies an idealized bilinear stress-strain curve for reinforcing steel (Figure 2A).After the yield point the stress remains constant while the strain increases until fracture.In EC2 there are two typical models for the stress-strain curve of steel reinforcement (Figure 2B).Model 1 has a slight increase of the stress after the yield strength f yd has been reached.Model 2 is the same as the idealized bilinear curve in the Chinese standard.A typical value for the maximum strain of steel reinforcement ε su is 1% in China, while it is 2.5% for normal steel and 5% for high ductile steel in Germany.

| Textile reinforcement
In References 11-13,15,and 18 a linear stress-strain curve for textile reinforcement is used for the calculation of load-bearing capacity of TRC in China (Figure 3A).The strength of textile reinforcement f u,tex is determined by tensile experiments on textile rovings.In addition, some authors in China, 14,19,20 use a bilinear curve to describe the tensile behavior of the textile concrete layer instead of only textile rovings (Figure 3B).It is mainly used when the textile reinforcement is combined with high ductile concrete, such as engineered cementitious composite (ECC) and ultra-high toughness cementitious composite (UHTCC).These materials exhibit pseudo-ductile behavior with formatted multiple, fine cracks when subjected to tensile forces.After the concrete cracks, the tensile force is primarily transferred to fiber strands.Due to the high strain capacity of up to several percent, the contribution of the concrete to the load transfer is taken into account and thus the entire TRC layer is considered when determining the stress-strain curve.
In Germany, there are also two typical methods to determine the tensile strength of impregnated textile reinforcement. 21The roving test determines the tensile strength of a single roving (or multiple rovings).Alternatively, the tensile strength can be determined by testing the textile fabric embedded in concrete.The concrete used for the specimen may have an influence on the measured tensile strength.Thus, the tensile strength of textile fabric will be evaluated based on the number and width of concrete cracks. 22Figure 4 presents the stress-strain curve each for textile fabric and TRC specimen using the above-mentioned methods.Compared with the methods in China the tensile behavior of the TRC specimen is described in more detail in the second state after the first concrete crack appears.State IIa describes the tensile behavior of the TRC specimen during the concrete crack growth.State IIb represents the loading stage, where the concrete crack formation is closed.In Chinese literature, state IIa and IIb are simplified as one linear curve (Figure 3B).However, Schütze et al. 23 have also pointed out, that the three-state development of the tensile force of a TRC specimen could be far less pronounced particularly in the case of a small concrete cross-section and a high degree of reinforcement. 1. Plane section remains plane for the strengthened beam during the loading process and the strain distribution is linear over the height of the cross-section.

| Calculation model of RC beams strengthened with TRC under bending in China
2. The steel reinforcement in the compression zone and the normal force are not considered for a simplified calculation.
3. A rectangular stress block for concrete (Figure 1B), a bilinear stress-strain curve for reinforcing steel (Figure 2A), and a linear elastic model for textile reinforcement (Figure 3A) serve as the basis for the dimensioning.
4. The pre-loading before strengthening with TRC is not taken into account.
An RC beam of rectangular section strengthened with one layer of TRC is taken as an example.Figure 5 schematically shows the stress and strain distribution of the structural member based on F I G U R E 1 (A) Parabola-rectangle diagram for concrete under compression in both Chinese and German standards 16,17 ; (B) rectangular stress block for concrete in compressive zone. 16,17A B L E 1 Comparison of parameters for concrete with different strengths between China and Germany.the premise that the ultimate strain of textile reinforcement ε u,tex is much larger than the strain of steel reinforcement at yield point ε yd .From the strain distribution it can be seen that the failure state of the member can be distinguished among three different types (type 1-3 in Figure 5).And the concrete fracture occurs in compression zone before the ultimate strain of textile reinforcement is reached.However, after the yield point of steel reinforcement a rupture of textile reinforcement could also be critical instead of concrete crushing, which is presented here as type 3.It should be noticed that also other failure modes like interface debonding between TRC layer and RC concrete member could occur.
However, these modes are not discussed in detail in the article.
The depth of concrete compression zones x tex and x s (see Figure 5), which represent demarcations between different failure modes, are given in Equations ( 1) and ( 2), respectively.
F I G U R E 3 (A) A linear stress-strain curve for textile reinforcement used in China. 11(B) A bilinear stress-strain curve for textile concrete used in China. 14I G U R E 2 (A) Material property of reinforcing steel in Chinese standard. 16(B) Material property of reinforcing steel in German standard. 17I G U R E 4 Qualitative course of the stress-strain curve of test specimens with impregnated textile reinforcement and of the pure impregnated fiber strand. 22 where ε c2u , ε yd , ε u,tex , d tex , and d s are the ultimate compressive strain of concrete, the strain of steel reinforcement at yield point, the ultimate strain of textile reinforcement, the distance from the centroid of textile reinforcement to the top edge of the concrete compression zone, and the distance from the centroid of steel reinforcement to the top edge of the concrete compression zone, respectively.Based on the depth of concrete compression zone x, different failure mechanisms are discussed in the following.

| Type 1: x s < x
In this condition, the concrete crushing occurs whereas the tensile steel bars do not yet yield.The ultimate strain of textile reinforcement ε u,tex is not reached.Based on the equilibrium of forces Equation (3) can be obtained: where α 1 and β 1 are parameters for design concrete compressive strength f cd and depth of concrete compression zone x, respectively, when using the rectangular stress block (see Section 2.1.1);b is the width of the RC beam; σ s and A s are the stress and the cross-section area of tensile steel bars, respectively; σ tex and A tex are the stress and the cross-section area of textile reinforcement, respectively.Using Hooke's law and the assumption of a linear strain distribution the following equations can be obtained.
where ε s , ε tex , E s , and E tex are the strain of steel and textile reinforcement, the elastic modulus of steel and textile reinforcement, respectively.Substituting Equations ( 4)- (7) into Equation (3), the depth of concrete compression zone x can be deducted as follows: where the parameter F is defined as: The flexural capacity M u can be calculated as follows: where z tex and z are the distances from the centroid of concrete compressive force to the centroid of textile reinforcement and to the centroid of the tensile steel bars, respectively (see Figure 5).

| Type 2:
In this case, after the tensile steel bars yield, the concrete crushing occurs instead of the rupture of textile reinforcement.From F I G U R E 5 Strain, stress, and force distribution of a TRC reinforced RC beam with a rectangular cross-section. 13quations ( 3)-( 7) the depth of concrete compression zone x can be given in Equation ( 11): where the parameter F is defined as: Consequently, the flexural capacity M u can be obtained as follows: Different than type 2, the rupture of textile reinforcement occurs while the ultimate compressive strain of concrete is not yet reached.
Inserting the stress of tensile steel bars f yd and textile reinforcement f td,tex into Equation ( 3), the depth of concrete compression zone x can be obtained: The flexural capacity M u can be determined as follows: With an assumed value for the depth of concrete compression zone, the required cross-section area of textile reinforcement A tex can be calculated using the equilibrium of moments.Based on the calculated cross-section area the depth of compression zone x can be recalculated using the equilibrium of forces and compared with the assumed value.This process is repeated until the assumption is fulfilled.

| Calculation model for RC beams strengthened with TRC under bending in Germany
In Germany, an iterative method for calculating the flexural capacity of RC beams strengthened with TRC was developed and presented in detail in References 24,25.The textile tensile force, the resulting concrete compressive force, the steel tensile force, and the respective distances between the resulting compressive and tensile forces are determined on the basis of an assumed strain distribution.Then the equilibrium of forces and moments will be verified.After the strain distribution has been adjusted, the process is repeated until the balance of forces and moments is achieved. 24r a simple and practical handling of the iterative method, Frenzel and Curbach 25 have provided dimensioning tables in which a mechanical reinforcement ratio ω tex of textile reinforcement can be read from a related moment μ tex .
For an approximate hand calculation, Schladitz et al. 26 use a simplified method with a rectangular stress distribution in the concrete compression zone.Based on Schladitz et al., 26 Frenzel 27 has developed a design procedure that additionally takes into account the stress and strain conditions before strengthening with textile reinforcement.The fictitious strain ε tex0 of the textile layer before strengthening is added to the ultimate strain of textile reinforcement ε u,tex as a part of the total strain of the textile layer after strengthening.The Chinese model (see Section 2.2) ignores the strain resulting from the pre-loading stage before strengthening, which is a major difference between German and Chinese models.Analogous to the Chinese model, the German model is based on the following assumptions: 1. Plane section remains plane for the strengthened beam during the loading process and the strain distribution is linear over the height of the cross-section.
2. The steel reinforcement in the compression zone and the normal force are not considered for a simplified calculation.
3. A rectangular stress block for concrete (Figure 1B), a bilinear stress-strain curve for reinforcing steel (Figure 2B, model 2) and a linear elastic model for textile reinforcement (Figure 4, textile roving) serve as the basis for the dimensioning.
4. The pre-loading before strengthening with TRC is considered for the calculation.
Based on the rectangular stress block for concrete (Figure 1B) the following equations can be obtained according to the equilibrium of forces and moments at the pre-loading stage: where η and λ are parameters for the rectangular stress block (see Section 2.1.1),ε c0 is the strain of the concrete at the top edge of the compression zone before strengthening, ε s0 and σ s0 are the strain and stress of steel reinforcement before strengthening, x 0 is the depth of compression zone at the pre-loading stage under the moment M Ed0 (Figure 6).
The strain distribution before the application of the textile reinforcement can be determined iteratively by using the Equations ( 16)-( 19).On that basis, the depth of concrete compression zone x of the cross-section after strengthening with textile reinforcement can be calculated by Equation (21).According to the equilibrium of moments the required cross-section area of the textile reinforcement can be obtained and are shown in Equations ( 22)-( 24).
Since the calculation is based on the equilibrium of moments, the equilibrium of forces should also be verified by comparing the maximum concrete compressive stress σ c with the design concrete strength f cd :

| COMPARISON BETWEEN CHINESE AND GERMAN MODELS
In the following, a design example is used to compare the two models from the above-introduced Chinese and German literature.
Concrete with the same cube strength (25 N/mm 2 ) defined in the Chinese 16 and German 17 standards is each used for the Chinese and German model.Since the definition of design concrete strength f cd differs slightly in both countries, the values of f cd are also different even if the concrete cube strength is identical (see Table 1).The design compressive strength for each concrete and other initial values for the dimensioning are given in Table 2 on the basis of References 16,17,28,and 29.
3.1 | Model based on the Chinese literature [11][12][13][14][15] As described in Section 2.2 a certain value for the depth of concrete compression zone is assumed.Based on that value the required cross-section area for textile reinforcement is calculated.
Using the equilibrium of forces and moments the depth of concrete compression zone is recalculated and compared with the assumption.This process is repeated until the assumption is fulfilled or nearly fulfilled.
The iteration steps and the used equations for the calculation are listed in Table 3. From Table 3 it can be seen that after two proper iterations the difference between the calculated and assumed depth of concrete compression zone is already very small.A required crosssection area of 4.1 cm 2 for the textile reinforcement can be obtained for a flexural capacity of 70 kN m.
3.2 | Model based on the German literature 27,28 In the following example, the effect of pre-loading is considered according to Reference 27.The strain distribution of the cross-section before strengthening is determined iteratively from the given values in Table 2 by using the Equations ( 16)- (20).The iterative steps and the used equations are listed in Table 4.
The results in Table 4 show that in the pre-loading stage the steel reinforcement has reached a tensile strain of 2‰ and the fictitious strain of the textile layer ε tex0 comes to 2.4‰.Substituting ε tex0 into Equation ( 21) the depth of concrete compression zone x can be obtained: Strain, stress, and force distribution of a TRC reinforced RC beam with a rectangular cross-section. 27 ¼ Using Equations ( 22)-( 24) the required cross-section area of textile reinforcement can be calculated: T A B L E 2 Material and cross-section properties for the design example.
The maximum concrete compressive stress after strengthening can be calculated using Equation ( 25): It can be seen that the concrete compressive strength is not exceeded, which means that the assumption for the calculation is fulfilled.Therefore, the calculated cross-section area of textile reinforcement is valid.

| SUMMARY
The article presents a comparison of design models for RC beams strengthened with TRC under bending on the basis of Chinese and German literature.The similarities and differences can be summarized as follows: • Both German and Chinese models have the same mathematic background.
• Calculation models for material properties (e.g., stress-strain curve for concrete, steel, and textile reinforcement) are similar.
• Some definitions (e.g., the ultimate compressive strain of concrete ε c2u , the design concrete strength f cd , parameters for stress distribution) are slightly different (see Section 2.1).
• The Chinese model uses an iteration of the depth of concrete compression zone to determine the strain distribution, while the German model uses an iteration of concrete strain at the top edge of the compression zone.
With consideration of the pre-loading the German model provides a more conservative method compared with the Chinese model, which ignores the stress and strain from the pre-loading stage.

References 11 -
15 introduce a model for determining the flexural capacity of reinforced concrete (RC) beams strengthened with TRC based on the design for conventional reinforced concrete in China.The model is based on the following assumptions:

Type 1
represents a failure mode where concrete crushing occurs in compression zone while the yield strength of steel reinforcement is not yet reached.In type 2 the yield point of steel reinforcement is exceeded.
Comparing the two methods for calculation of flexural capacity of RC beams strengthened with TRC used in China and Germany, it can be stated that both have the same mathematical background and also provide similar results of the required amount of textile reinforcement.Beside this, the main difference between two models is that the German model additionally takes into account the stress and strain from the pre-loading stage while the Chinese model ignores it.The cross-section area of textile reinforcement is calculated using the equilibrium of moments considering steel and textile tensile forces.Therefore, the slightly different design concrete strengths in the Chinese (11.9 N/mm 2 ) and German (11.3 N/mm 2 ) model have no influence on the results.Thus, the comparatively bigger value using the German model (4.3 cm 2 ) than using the Chinese model (4.1 cm 2 ) can be attributed to the damage from the pre-loading stage.With the consideration of pre-loading effect, the German model provides a more conservative method.