Experimental and theoretical investigations on crack spacing and stiffness of textile-reinforced concrete

Abstract

To improve the theory of the bending cracks spacing and the deformation of the reinforced concrete beam strengthened with textile-reinforced concrete, the experimental research was conducted first on the cracks spacing and bending performance of reinforced concrete beams with reinforced concrete through four-point bending test. Then, two theoretical derivations were conducted, analyzing the average crack spacing and the stiffness of the strengthened beam through the transformed-section method and the effective moment of inertia approach, respectively. Experimental results showed that no matter using the single-sided or U-type strengthening form, the flexural bearing capacity and crack forms of reinforced concrete beams were improved. Distribution characteristics of cracks in the strengthened beam were presented “multiple and dense” at the bottom and “few and sparse” on the top. The stiffness development of the strengthened beam could be divided into three phases: before-cracking phase, after-cracking phase, and after-yield phase, and the stiffness could be considered as unchanged in every phase. Results of average crack spacing calculated by transformed-section method corresponded with experimental results. By the effective moment of inertia approach, the calculated stiffness of the strengthened beam was slightly bigger than experimental values, which suggested that these calculating solutions may be applied into practical engineering design.

Keywords

analytical model bearing capacity bending stiffness crack control mechanical properties textile-reinforced concrete

Introduction

Textile-reinforced mortar (TRM), textile-reinforced concrete (TRC), fiber-reinforced cementitious mortar (FRCM), and so on all prove the effectiveness of reinforced concrete (RC) structures strengthened with cement-based fiber composites (Al-Salloum et al., 2011; Babaeidarabad et al., 2014; Elsanadedy et al., 2013; Mechtcherine, 2013). As one sort of cement-based fiber composites, TRC, which is composed of multi-axial textile and high-performance fine-grain concrete, has favorable bearing capacity, crack control, impermeability, and corrosion resistance as well as the capability of multi-cracking which can be self-healing (Mechtcherine, 2013; Pourasee et al., 2011). Advantages of methods of fiber-reinforced polymer (FRP) and reinforced shotcrete can be fully combined in construction sites (Weiland et al., 2007). When TRC is used as repairing and strengthening layer, flexural capacity (Brückner et al., 2006; Gopinath et al., 2015; Schladitz et al., 2012; Weiland et al., 2007; Yin et al., 2014), shear capacity (Si Larbi et al., 2010), and earthquake resistance (Harajli et al., 2011) can be obviously enhanced.

Currently, many scholars have researched on concrete structures strengthened with TRC materials about their rules of flexural behavior and crack development. Schladitz et al. (2012) investigated the flexural bearing capacity of the large-span plate strengthened with TRC, and their study showed that TRC significantly increases the bearing capacity of RC plate, deflection of which decreases with the increase in the textile reinforcement ratio. Besides, results computed by finite element analysis software were nearly same as experimental results. Elsanadedy et al. (2013) found that the flexural bearing capacity of beams strengthened with carbon fiber–reinforced polymer (CFRP) slightly exceeds that of beams strengthened with TRM, but as for ductility, TRM has an obvious advantage; results from numerical simulation indicated that adding U-type anchorages into TRC-strengthening components can effectively control debonding failure. Research conducted by Yin et al. (2014) on flexural bearing capacity of RC beams strengthened with TRC showed that the flexural bearing capacity of TRC-strengthened beam increased, crack patterns of the beam were improved, and the development of cracks was delayed, which presented “multiple and dense”; the increasing textile ratio in TRC can improve flexural bearing capacity of the beam obviously and reduce crack width and spacing. D’Ambrisi and Focacci (2011) used different kinds of fibers, different specifications of textile, and number of textile layers as influential parameters in order to study flexural behavior of RC beam with TRC and analyze reasons for delamination of TRC composites due to bending; in addition, two different theoretical models for the calculations of textile strain were put forward, both of which prove accordant to results from the experiment.

Previous studies have proven that TRC has conspicuous influences on strengthening the flexural behavior of RC slabs and beams. However, considering crack development and deformation theory of flexural beams with TRC, these investigations are still not comprehensive enough. In order to improve the reinforcement theory of the RC beam strengthened with TRC and then provide the theoretical basis of TRC when it is applied in practical engineering, this article first conducted experimental research on the crack spacing and bending performance of the beam strengthened with TRC considering the different strengthening methods and different reinforcement ratios. Then, transformed-section method and effective moment of inertia were put to use, respectively, to infer the average crack spacing and stiffness of TRC-strengthened beam, which were all under check process.

Experimental program

TRC used in test

Textile

In this investigation, the hybrid textile made up of carbon and E-glass yarns was used (Figure 1(a)). The carbon yarns were used to bear loads. The E-glass yarn cannot withstand alkaline conditions in concrete for a longer time; thus, its load-carrying contribution was not considered in this study. It was used merely to fix the carbon yarn. In addition, to improve the mechanical behavior of yarns and interface properties between yarns and fine-grain concrete, textile, according to the advice given by Yin et al. (2015), was coated with epoxy resin (Figure 1(b)) and stuck on sand before it was embedded into concrete, and thus, the alkali-resistance of the E-glass yarn could be improved. The mesh size of the textile was 10 mm × 10 mm; details of its mechanical properties and geometric parameters were provided in the literature (Yin et al., 2015), which can be found in Tables 1 and 2.

Figure 1.

Hybrid textiles made up of carbon and E-glass yarns: (a) not impregnated and (b) impregnated by epoxy resin and covered with sand.

Table 1.

Mechanical properties and geometric parameters of fiber yarns of textile.

Fiber type	Number of filaments per yarn	Filament tensile strength (MPa)	Filament elasticity model (GPa)	Filament break extension (%)	Yarn textile (g/km)	Yarn density (g/cm³)
T700S	12k	4660	231	2.0	801	1.78
E-glass	4k	3200	65	4.5	600	2.58

Table 2.

Mechanical properties of weft carbon yarn.

Fiber type	Number of filaments per yarn	Tensile strength (MPa)	Elasticity model (GPa)	Ultimate strain	Theoretical area (mm²)	Yarn textile (g/km)	Yarn density (g/cm³)
Toray carbon filament	12k	4100	180	0.023	0.45	801	1.78

Fine-grain concrete

The mix proportion of cement: fly ash: silica fume:water:fine sand:coarse sand:water reducer, for fine-grain concrete used, was 475:168:35:262:460:920:9.1. The materials included 52.5R Portland cement, first-class fly ash (FA I), a high-performance water reducer of a polycarboxylic series, 32-mesh to 64-mesh (diameter from 0.23 to 0.5 mm) common quartz sand (i.e. fine sand), and 26-mesh to 32-mesh (diameter from 0.5 to 0.65 mm) common quartz sand (i.e. coarse sand). The real compressive strength of the fine-grain concrete when formed into cubes with dimensions of 70.7 mm × 70.7 mm × 70.7 mm was measured to be 53 MPa at 28 days.

Concrete used in test

The concrete in each specimen was C40, with cement: water:sand:gravel mix proportion of 415:196:643:1181, respectively. The cement was PO 42.5, which was produced by China United Cement Corporation. The sand distribution was medium, and the maximum diameter of gravel was 10 mm. The actual compressive strength of concrete on the cubes with a dimension of 150 mm × 150 mm × 150 mm was measured to be 44.8 MPa at 28 days.

Steel bar

As shown in Figure 2, the HRB400 grade steel bar was used as the longitudinal tensile reinforcement, the diameter of which was 12, 14, or 16 mm (yield strength was 545, 500, and 415 MPa, respectively). Each beam had two longitudinal tensile reinforcements and two erection bars. The stirrup grade was HPB300 in the diameter of 6.5 mm. Stirrup spacing was s = 100 mm, and in pure bending area, the spacing was 200 mm.

Figure 2.

Cross section of strengthened beam (unit: mm):(a) single-sided and (b) U-type.

Design and making process of specimen

A total of seven RC beams were used. The length of all beams was 2400 mm, and the calculation span of which was 2200 mm. For beams without TRC, the cross-sectional dimension was 120 mm × 240 mm. For beams with TRC, the cross-sectional dimension was 120 mm × 230 mm before strengthening and 120 mm × 240 mm after strengthening. The control specimens had a normal concrete cover of 25 mm, and other specimens were designed to study the effect of using TRC as a strengthening layer. The thickness of the TRC in the test was 10 mm. The specific parameters of beams can be seen in Table 3. Reinforcement ratio in Table 3 is the ratio of the area of steel bars to effective cross-sectional area of beam. The effective cross section is equal to the width of the beam multiplying the effective height which is the distance from the point of the force action of steel bars to the edge of the compression zone of the beam.

Table 3.

Basic information of test beams.

No.	Reinforcement ratio (%)	Longitudinal steel	Method of reinforcement	Layer of reinforcement
H1	0.90	12	Nothing	Nothing
H2	1.23	14	Nothing	Nothing
H3	1.62	16	Nothing	Nothing
H4	0.90	12	Single-sided	Two layers of textile
H5	1.23	14	Single-sided	Two layers of textile
H6	1.62	16	Single-sided	Two layers of textile
H7	0.90	12	U-type	Two layers of textile (including one layer aside)

Single-sided strengthening and U-type strengthening forms (Yin et al., 2016) were used in this test (Figure 2). Meanwhile, U-type strengthening form means that the beam is strengthened at both the bottom and side surfaces: for the arrangement of two layers of textile, one layer was spread on the bottom of the beam and the other layer was spread on two sides whose width was half of the textile on the bottom.

The making process of specimens was as follows: (1) first step was to pour RC beam. For the strengthened specimen, its height was 230 mm and its cover thickness was 15 mm. (2) After 28 days of maintenance, artificial chiseling was adopted to rough the interfaces of aged concrete, then dusts on surfaces were cleaned up. (3) Before strengthening, the rough surface of the concrete was kept moist. (4) A thin layer of fine-grain concrete was poured on the rough surfaces and then a layer of textile was placed on the surface of fine-grain concrete, and then, the textile was fixed on both sides with battens to control the thickness of the fine-grain concrete. (5) The step (4) was repeated sequentially in terms of different layers. Finally, fine-grain concrete was poured, and the total thickness of TRC was 10 mm or so.

Load way and experimental content

As shown in Figure 3, four-point bending test was applied. Distance of two loading points (space of pure bending) was 800 mm. Under control of step loading, each step of loading remained for 3–5 min. The resistance stain gauges which were 100-mm length were set on both top surface and side of beams to measure concrete strain, while a micrometer gauge was set at the site of TRC layer to detect the strain of textile. Load sensor was used to record load. Mid-span deflection and deformation of support were measured by displacement gauge. A static data collector was used as the collecting devices. The crack width was obtained using the DJCK-2 Crevice Width Finder with an accuracy of 0.02 mm.

Figure 3.

Loading and layout schematic (unit: mm).

Experimental results and discussion

Each beam’s experimental results of crack load, yield load, and ultimate load can be seen in Table 4.

Table 4.

Test results.

No.	P_cr (kN)	ω_cr (mm)	P_y (kN)	ω_y (mm)	P_u (kN)	ω_u (mm)	Damage form	Extent of increase in P_cr (%)	Extent of increase in P_y (%)	Extend of increase in P_u (%)
H1	7.12	0.50	63.10	15.30	69.78	48.12	Under-reinforced beam failure	−	−	−
H2	7.26	0.48	80.12	15.23	85.00	53.87	Under-reinforced beam failure	−	−	−
H3	7.23	0.48	85.60	14.98	96.45	57.35	Under-reinforced beam failure	−	−	−
H4	8.16	0.33	68.32	15.64	82.34	26.83	Textile rupture	14.61	8.27	18.00
H5	8.20	0.69	82.30	15.38	95.00	24.72	Textile rupture	12.95	2.72	11.76
H6	8.28	0.63	88.85	15.46	103.23	40.12	Textile rupture	16.04	3.80	7.03
H7	9.98	0.50	67.48	12.92	75.00	30.92	Textile rupture	40.17	6.94	7.47

P_cr is the crack load; P_y is the yield load; P_u is the ultimate load.

Analysis on damage features and bearing capacity

Damage features

The ultimate damage patterns of beams from H4 to H7 can be seen in Figure 4. H4 to H6 beams were single-sided strengthened beams. Although steel bars in H5 beam yielded in damage process, one layer of textile broke away from the beam, so the beam H5 was not in normal damage pattern. The ultimate damage patterns of single-sided strengthened beams were different from ordinary RC beams. Single-sided strengthened beams had similar damage processes and patterns: with the increase in the load, steel bars yielded at first; as the load continued increasing, the mid-span deflection grew rapidly, and loud sounds of fracture sent out when textiles had tensile failures. At the place of rupture, longitudinal cracks were generated along steel bars. The study of Yin et al. (2014) also found this pattern. The reasons for this phenomenon above are as follows: the insufficient concrete cover of the steel rebar in combination with an insufficient confinement by the steel stirrups which are not quite dense in the middle of the beam that lead to the longitudinal cracks. In the case of the U-shaped TRC, the stirrups became also strengthened and therefore increased the confinement of the longitudinal rebar, which prevents the concrete cover from longitudinal splitting.

Figure 4.

Ultimate failure of beams (a) H4, (b) H5, (c) H6, and (d) H7.

Beam H7 was a sort of U-type reinforcement, damage process and form of which were not totally as same as that of single-sided strengthened beams, as shown in Figure 4(d). TRC layer of U-type did not break away from aged concrete after damage. The reason is that the lateral textile of U-type shared some load which was generated by the rupture of bottom textile. The integrity of beam H7 excelled beam H4 and H6 after damage.

Analysis on bearing capacity

As indicated in Table 4, no matter single-sided or U-type strengthened beams, compared with non-strengthened beams that had the same reinforcement ratio of steel bars, both of them had significant enhancement in bearing capacity. For crack load, yield load, and ultimate load, beam H4 increased by 14.61%, 8.27%, 18.00%, respectively, and beam H5 increased by 12.95%, 2.72%, 11.76%, respectively, and beam H6 increased by 16.04%, 3.80%, 7.03%, respectively, and the U-type strengthened beam H7 increased by 40.17%, 6.94%, 7.47%, respectively. Considering improvement of crack load, the U-type strengthening form was better than the single-sided one because U-type one increased its own cross-sectional area, which at the same time, improved the stiffness of beams, and thus, it controlled well on the appearance of initial cracks on beam. In terms of improvement of ultimate load, the single-sided strengthening form were better than the U-type one, for the reason that the strain of the lateral textile of the U-type strengthened beam diminished along with the beam height, causing the strength of the textile could not be made the most, so that the bearing capacity of the U-type strengthened beam was inferior to that of single-sided one. Comparing crack load, yield load, and ultimate load of beams H4, H5, and H6, it is demonstrated that in terms of improving the ultimate load of the beam, strengthening the beam with lower reinforcement ratio had higher increased range than strengthening ones with higher reinforcement ratio, and the increased range could reduce with increase in reinforcement ratio.

Number of cracks and distribution

As shown in statistics (Table 5), in pure bending area, the number of cracks at the bottom and on the flank of the strengthened beam was different. Taking the situation of longitudinal steels as the dividing line, the crack number of strengthened beams presented a feature of “multiple and dense” at the bottom of the line and “few and sparse” on the top of the line, as shown in Figure 5. Besides, comparing with ordinary RC beams, the number of cracks of single-sided strengthened beams was more and the spacing and width of cracks were smaller. These characteristics showed TRC had excellent restricted and dispersing capability on concrete cracks. The fiber yarns of the textile used by TRC had small diameter, and the number of fiber yarns was relatively large. It is known that when the number of steel bar becomes more and diameters of steel bars are less, the cracks will be denser (Yin et al., 2014). And, textile and steel bar in controlling cracks have similar characteristics.

Table 5.

Number and space of cracks in pure bending.

Number	Number of cracks at the bottom of the beam	Number of cracks on the flank of the beam	Average crack spacing at the bottom of the beam (mm)	Average crack spacing at the flank of the beam (mm)
H4	13	10	66.67	88.89
H5	9	9	100	100
H6	15	10	57.14	88.89
H7	12	9	72.73	100

Average spacing of cracks = length of pure bending/(number of cracks: 1).

Figure 5.

Cracks figure of beams (a) H4 and (b) H7.

The number of cracks of beam H4 was slightly more than beam H7. In addition, because beam H7 had fewer cracks than Beam H4, the average spacing of cracks at the bottom of the beam H7 was more than that of beam H4, which corresponded to laws of crack spacing calculated by later formula (2). This is because when U-type reinforcement was used, the bottom of the beam was only set with one layer of textile, whose restricted capability on concrete was less than that of single-sided reinforcement. Although the flank of U-type one was also set with the textile, the textile at this site could not be made full use of, and thus, its constraint capability was inferior to the textile at the bottom of the beam. But, from Figure 5, the number of inclined cracks on U-type one was less than single-sided one. This is because TRC was also assigned at the flank of beams when U-type reinforcement was used, that means, adding the stirrups in beams and increasing the stirrup ratio.

From Table 5, the number of cracks at the bottom of the beam H6 was more than that of beam H4, illustrating that crack spacing of the beam had a decreasing trend as the reinforcement ratio increased, which also corresponded to laws of crack spacing calculated by later formula (2). However, due to the restriction role of TRC in beam, H5 was not full use of, as shown in “Damage features,” the number of cracks at the bottom of the beam H5 was less than that of beam H4.

Analysis on load-deflection

From Figure 6, load-mid-span deflection diagram of the strengthened beam can be divided into three phases: before-cracking phase of concrete, post-cracking phase of concrete, and post-yield phase of steel bars. Each period’s load-deflection diagram presented mainly a linear relationship and slopes of curves decreased. The statement above indicated that the stiffness of the strengthened beam changed twice in the whole force process, and the value decreased after each time of change.

Figure 6.

Load-mid-span deflection curves.

As shown in Figure 6, during the period between the initial stage of loading and the yield of steel bars, the deflection of U-type strengthened beam was slightly smaller than that of single-sided strengthened beam at the same load. After steel bars yield, the deflection of U-type strengthened beam grew faster than that of single-sided one at the same level of load. Put what is stated above into analysis, before steel bars reached the point of yield, the crack wide in concrete was still small due to the capability of TRC-strengthening layer delaying the development of cracks, and thus, the cracks had no significant influence on the stiffness of the beam. Compared with the single-sided strengthened beam, the TRC layer of U-type one had larger cross-sectional area, thus U-type one had larger stiffness and smaller deformation; after steel bars yield, the tensile strain of the textile in TRC layer would increase rapidly, but the strain of the lateral textile did not coincide with that of textile at the bottom, leading to the textile not fully utilized, and thus, the ability of the TRC to control crack decreased. However, the single-sided beam with two layers of textile which were all set at the bottom had a relatively consistent deformation of the textile, and thus, the ability of the TRC to control crack played better, so single-sided strengthened beam had higher stiffness than U-type one, and therefore, its deformation was smaller.

Besides, as also demonstrated in Figure 6, before cracking of concrete, curves of beams H4, H5, and H6 almost coincided. After cracking of concrete, the slope of H4 curve decreased most, and the slope of H6 curve was slightly larger than that of H5. At the same level of load, mid-span deflection of the beam with higher reinforcement ratio was smaller than that of the beam with lower reinforcement ratio. Laws stated above indicate the reinforcement ratio was one of the chief factor that influenced the bending stiffness of beams.

Theoretical analysis

Average spacing of TRC-strengthened beams

Comparing European code with China code

EN 1992-1-1 (2004) shows how to calculate the maximum crack spacing of RC beam. GB50010-2010 (2010) shows how to calculate the average crack spacing of RC beam.

From EN 1992-1-1 (2004), the formula of maximum crack spacing is

s_{r, \max} = k_{3} c + \frac{k_{1} k_{2} k_{4} ϕ}{ρ_{p, eff}}

(1)

where ϕ is the bar diameter. When a mixture of bar diameters is used in a section, an equivalent diameter φ_eq should be used. For a section with n₁ bars of diameter φ₁ and n₂ bars of diameter φ₂, the following expression should be used

φ_{eq} = \frac{n_{1} φ_{1}^{2} + n_{2} φ_{2}^{2}}{n_{1} φ_{1} + n_{2} φ_{2}}

c is the cover to the longitudinal reinforcement; k₁ is a coefficient which takes account of the bond properties of the bonded reinforcement (=0.8 for high bond bars and =1.6 for bars with an effectively plain surface (e.g. pre-stressing tendons)); k₂ is a coefficient which takes account of the distribution of strain (=0.5 for bending and =1.0 for pure tension); and ρ_p,eff is the effective reinforcement ratio.

The values of k₃ and k₄ for use in a Country may be found in its National Annex. The recommended values are 3.4 and 0.425, respectively.

From GB50010-2010 (2010), the formula of average crack spacing is

l_{cr} = β (1.9 c_{s} + 0.08 \frac{d_{eq}}{ρ_{te}})

(2)

Comparing formula (1) with formula (2), the same form of formula calculating the crack spacing can be found, which shows that the cover thickness and bond length are main factors affecting crack spacing. According to the bond-slip and no slip theory, the difference between the formula of the average crack spacing and the formula of the maximum crack spacing is only a coefficient. Therefore, European code and China code take the same way to calculate crack spacing. The formula of calculating crack spacing from China code was used in this article.

Basic assumption

According to the data processing and analyzing process in this test, the following assumptions will be used: (1) number and position arrangements of equivalent steel bars from the textile are the same as that of the original carbon fiber yarns; (2) the stress–strain relationship of carbon fibers is σ_f = E_fε_f; (3) the stress–strain relationship before the yield of steel bars is σ_s = E_sε_s, and after the yield of steel bars is σ_s = E_p(ε_s − ε_y)+σ_y.

Calculations of crack spacing

Strengthened beams at their serviceability limit stage, and steel bars and carbon fiber yarns are still at the elastic period. Thus, calculative method of homogeneous material is employed, that is, through the conversion of the ratio of steel bars’ elasticity modulus and carbon fibers’, the conversion area of the steel bar can be achieved

A_{s 1} = \frac{E_{f}}{E_{s}} A_{fl}

(3)

The number of equivalent steel bars and arranged position is the same as that of original carbon fiber yarns, so

ρ_{te} = \frac{λ A_{s 1} + A_{s}}{A_{te}}

(4)

A_{te} = 0.5 bh

(5)

Use the diameter of equivalent steel bars to calculate $d_{eq}$ .

Comparison between experimental and calculative results

According to the test, values from which are as follows: E_s = 2.0 × 10⁵ N/mm², E_c = 3.25 × 10⁴ N/mm², E_p = 3.33 × 10⁴ N/mm², E_f = 1.8×10⁵ N/mm², n = E_s/E_c = 6.15, and m = E_f/E_c = 5.54.

As demonstrated in Figure 4, the size of grids on beams was 40 mm × 40 mm and the crack spacing around the middle of the height of the beam was about 80 mm. While cracks on the TRC layer were denser, and the spacing of which was smaller, outside the TRC layer, these cracks joined together toward the top of the beam, forming one thick crack (see Figures 4 and 5).

Table 6 presents the comparative results between calculative values and experimental values. From Table 6, calculative values and experimental values are very close, indicating that using this method to calculate the crack spacing of beams strengthened with TRC may be feasible.

Table 6.

Space of strengthened beams.

Number	l_cr (mm)	l_cr,t (mm)	l_cr/l_cr,t
H4	66.67	76.22	0.87
H5	100	72.62	1.31
H6	57.14	69.63	0.82
H7	72.73	76.22	0.95

l_cr is experimental average spacing of cracks; l_cr,t is calculative average spacing of cracks.

Calculations of stiffness of TRC-reinforced beam

Guo (2014) introduced the method of transformed inertia moment of section to calculate the stiffness of RC beam. The main principle of this method to calculate the stiffness is that it converts the area of steel bars to corresponding concrete area by their ratio of elastic modulus to get equivalent uniform material conversion section. Then, the corresponding calculation formula is derived and established. This article used this principle to derive the formula about calculation of the stiffness of TRC-strengthened beam.

Material constitutive model and stiffness model

According to the data processing and analyzing process in this test, several assumptions were proposed: (1) the stress–strain relationship of carbon fibers is σ_f = E_fε_f; (2) the stress–strain relationship before the yield of steel bars is σ_s = E_sε_s, after the yield of steel bars is σ_s = E_p(ε_s − ε_y) + σ_y, taking $E_{p} = (1 / 6) E_{s}$ ; (3) load–displacement curves of the beam obey the tri-linear constitutive equation, that means, the stiffness of strengthened beam in its whole life has three different values: B₀, B_cr, and B_y.

Calculation of stiffness of single-sided strengthened beam

Conversion cross-sectional moment of inertia before cracking.

Before the beam cracking, the whole cross section is subjected to force. The area of the tensile reinforcement is A_s, and the area of carbon fiber yarns at the beam bottom is A_f, and the equivalent cross sections are nA_s and mA_f, respectively, where n = E_s/E₀ and m = E_f/E₀. Subtracting the original area of steel bars and carbon fiber yarns, it should add the additional area (n − 1)A_s and (m − 1)A_f, respectively, at the same height of section, for example, Figure 7(b).

Figure 7.

Transformed cross section of single-sided strengthened beam: (a) original section, (b) before cracking, and (c) after cracking.

The total area of the conversion section can be gotten from formula (6)

A_{0} = bh + (n - 1) A_{s} + (m - 1) A_{f}

(6)

The height of the compression zone x₀ which is confirmed by the equal area moment of tension, and compression zone can be obtained from formula (7)

\frac{1}{2} b x_{0}^{2} = \frac{1}{2} b {(h - x_{0})}^{2} + (n - 1) A_{s} (h_{0} - x_{0}) + (m - 1) A_{f} (h_{f} - x_{0})

(7)

The conversion section moment of inertia is that

I_{0} = \frac{b}{3} [x_{0}^{3} + {(h - x_{0})}^{3}] + (n - 1) A_{s} {(h_{0} - x_{0})}^{2} + (m - 1) A_{f} {(h_{f} - x_{0})}^{2}

(8)

Therefore, the section stiffness of the strengthened beam before cracking can be get from formula (9)

B_{0} = E_{0} I_{0}

(9)

2. Conversion cross-sectional moment of inertia after cracking.

Assume only reinforcement and carbon fiber yarns bear the tension force when the beam cracks. The same method is used to determine the height of the compression zone of the crack section. The compression zone x_cr can be calculated from formula (10)

\frac{1}{2} {bx}_{cr}^{2} = n A_{s} (h_{0} - x_{cr}) + {mA}_{f} (h_{f} - x_{cr})

(10)

The conversion moment of inertia at cracking section and section stiffness can be calculated from formulae (11) and (12), respectively

I_{cr} = \frac{b}{3} x_{cr}^{3} + n A_{s} {(h_{0} - x_{cr})}^{2} + m A_{f} {(h_{f} - x_{cr})}^{2}

(11)

B_{cr} = E_{0} I_{cr}

(12)

After the steel bars have yielded, using E_p as elastic modulus when calculating inertia moment of section of the component, other parameters are not changed. Calculation method of computing I_y of beams is the same as its corresponding I_cr.

Calculation of stiffness of U-type strengthened beam

As for U-type reinforcement, situations can be divided into two kinds: (1) neutral axis is out of the TRC layer, as shown in Figure 8(a). (2) Neutral axis is through the TRC layer, as shown in Figure 8(b). And these two kinds of situations seek the same method of calculation section stiffness. Therefore, this article only deduced the calculation formulae of section stiffness about the first situation.

Figure 8.

Neutral axis situation of U-type reinforcement: (a) neutral axis out of the TRC layer and (b) neutral axis through the TRC layer.

Comparing with single-sided strengthened beam, U-type arranged carbon-fiber yarns at lateral side of beam. Therefore, it is needed to convert the area of lateral textile to equal area. When this step is carried out, the height of additional area ((m − 1)A_fc) is equal to the height of lateral TRC, for example, Figure 9(b).

Figure 9.

Conversion cross section of U-type strengthened beam: (a) original section, (b) before cracking, and (c) after cracking.

The height of compression zone x₀ before cracking can be obtained from formula (13)

\frac{1}{2} b x_{0}^{2} = \frac{1}{2} b {(h - x_{0})}^{2} + (n - 1) A_{s} (h_{0} - x_{0}) + (m - 1) A_{f} (h_{f} - x_{0}) + b_{c} h_{c} (h - x_{0} - 0.5 h_{c})

(13)

The conversion moment of inertia and section stiffness before cracking can be gotten from formulae (14) and (15), respectively

I_{0} = \frac{b}{3} [x_{0}^{3} + {(h - x_{0})}^{3}] + (n - 1) A_{s} {(h_{0} - x_{0})}^{2} + (m - 1) A_{f} {(h_{f} - x_{0})}^{2} + \frac{1}{6} b_{c} h_{c}^{3} + 2 b_{c} h_{c} {(h - x_{0} - 0.5 h_{c})}^{2}

(14)

B_{0} = E_{0} I_{0}

(15)

Using the same method, the conversion cross-sectional moment of inertia after cracking and the section stiffness can be calculated by formulae (16) to (18)

\frac{1}{2} b x_{cr}^{2} = n A_{s} (h_{0} - x_{cr}) + m A_{f} (h_{f} - x_{cr}) + m A_{fc} (h - x_{cr} - 0.5 h_{c})

(16)

I_{cr} = \frac{b}{3} x_{cr}^{3} + n A_{s} {(h_{0} - x_{cr})}^{2} + m A_{f} {(h_{f} - x_{cr})}^{2} + \frac{1}{6} \frac{m A_{fc}}{h_{c}} h_{c}^{3} + 2 m A_{fc} {(h - x_{0} - 0.5 h_{c})}^{2}

(17)

B_{cr} = E_{0} I_{cr}

(18)

Checking process

Using above method to calculate the stiffness of strengthened beams, the results are demonstrated in Table 7. From Table 7, it is known that calculated values correspond well to experimental ones, and theoretical ones are slightly bigger than experimental ones, indicating that the formula of stiffness inferred by effective moment of inertia may be suitable for computing the stiffness of TRC-strengthening beams.

Table 7.

Computational and actual stiffness.

Component	Stiffness	Theoretical stiffness (N/mm)	Experimental stiffness (N/mm)	Theoretical stiffness/experimental stiffness
12 single side; two layers	B ₀	4.80 × 10¹²	4.41 × 10¹²	1.09
	B _cr	1.33 × 10¹²	1.30 × 10¹²	1.02
	B _y	3.50 × 10¹¹	3.29 × 10¹¹	1.06
14 single side; two layers	B ₀	4.89 × 10¹²	4.64 × 10¹²	1.05
	B _cr	1.66 × 10¹²	1.51 × 10¹²	1.10
	B _y	4.34 × 10¹¹	4.05 × 10¹¹	1.07
16 single side; two layers	B ₀	4.98 × 10¹²	4.76 × 10¹²	1.05
	B _cr	1.99 × 10¹²	1.93 × 10¹²	1.03
	B _y	5.26 × 10¹¹	4.67 × 10¹¹	1.13
12 U-type; two layers	B ₀	4.83 × 10¹²	4.92 × 10¹²	0.98
	B _cr	1.37 × 10¹²	1.32 × 10¹²	1.04
	B _y	3.8 × 10¹¹	3.16 × 10¹¹	1.20

Conclusion

First, by means of experiments, this article studied crack development and bending performance of TRC-strengthening beams influenced by different strengthening methods and different reinforcement ratio. Then, transformed-section method and effective moment of inertia method were put into use, respectively, to infer the average crack spacing and stiffness of TRC-strengthened beams. Conclusions are given as follows:

No matter single-sided strengthening method or U-type method, the bearing capacity of beams could be enhanced. As for the enhancement of cracking load, U-type one was better than single-sided one; as for the enhancement of the ultimate load, single-sided one was better than U-type one; as for the eventual destruction form, U-type one was better than single-sided one.

Distribution characteristics of cracks in the strengthened beam were presented “multiple and dense” at the bottom and “few and sparse” on the top. The number of cracks of the single-sided strengthened beam was slightly more than that of the U-type, and the spacing of cracks in the single-sided one was slightly less than that of the U-type. The number of cracks in single-sided strengthened beam increases as the reinforcement ratio increases.

Using formulas of the crack spacing provided by concrete structure design GB50010-2010 and combining transformed-section method, the crack spacing formula of TRC-strengthened beam could be achieved. Through checking, values calculated by these formulae corresponded well with experimental values.

The stiffness development of strengthened beam could be divided into three phases: before-cracking phase, after-cracking phase, and after-yield phase. Besides, formulas of the stiffness of single-sided and U-type TRC-strengthened beam in three different phases were achieved through effective moment of inertia methods. Through checking, results calculated by these formulae corresponded well with experimental results.

It is true that plenty of experiments are needed to do to generalize some of the results and check the presented formula. In the future, this research will continue. Moreover, in the subsequent deeper test, specimen with different sizes will be considered to carry out in order to achieve more perfect theory.

Footnotes

Appendix 1 Acknowledgements

The experimental work described in this paper was conducted at State Key Laboratory for Geomechanics & Deep Underground Engineering, Jiangsu Key Laboratory of Environmental impact, and Structural Safety in Civil Engineering in China University of Mining and Technology. Help during the testing from staffs and students at the Laboratory are greatly acknowledged.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Fundamental Research Funds for the Central Universities (2017XKQY053).

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Experimental and theoretical investigations on crack spacing and stiffness of textile-reinforced concrete–strengthened reinforced concrete beams

Abstract

Keywords

Introduction

Experimental program

TRC used in test

Textile

Fine-grain concrete

Concrete used in test

Steel bar

Design and making process of specimen

Load way and experimental content

Experimental results and discussion

Analysis on damage features and bearing capacity

Damage features

Analysis on bearing capacity

Number of cracks and distribution

Analysis on load-deflection

Theoretical analysis

Average spacing of TRC-strengthened beams

Comparing European code with China code

Basic assumption

Calculations of crack spacing

Comparison between experimental and calculative results

Calculations of stiffness of TRC-reinforced beam

Material constitutive model and stiffness model

Calculation of stiffness of single-sided strengthened beam

Calculation of stiffness of U-type strengthened beam

Checking process

Conclusion

Footnotes

Appendix 1

Acknowledgements

Declaration of Conflicting Interests

Funding

References