Experimental study on flexural strengthening of reinforced concrete beams with U-shaped steel under secondary load

Abstract

This study proposes a new method to strengthen reinforced concrete (RC) beams with U-shaped steel, which can achieve rapid construction and great improvement in the flexural performance of RC beams. To investigate the influence of secondary load defined as newly applied loads after strengthening on the strengthening effect, a total of nine specimens were tested under four-point bending, including a reference beam, a strengthened beam under initial load, and seven strengthened beams under secondary load. The initial loading degree, the thickness of the bottom plate, and the height of the steel box were the main variables considered in this study. Testing results showed that compared with the reference beam, the flexural performance of strengthened beams was significantly enhanced, indicating the good joint performance of the U-shaped steel and the RC beams. Among the three main variables, the initial loading degree was found to have a minimal effect on the flexural performance while the thickness of the bottom plate and the height of the steel box had considerable influence, with the latter having a more pronounced effect. Testing results also showed that most of the strengthened beams experienced flexural failures, which were reflected by steel web peeling and buckling, and concrete crushing. Moreover, a formula was derived for calculating the flexural capacity of strengthened beams under secondary load. The results from the derived formula were found to be in good agreement with those from experiments.

Keywords

reinforced concrete beam strengthened U-shaped steel flexural behavior secondary load height of the steel box

Introduction

Due to material deterioration, natural disasters, and external environmental erosion, reinforced concrete (RC) beams are often subjected to different levels of damage. In comparison to reconstruction, strengthening can not only save cost and greatly shorten the construction period but also avoid generating construction waste. So, it has been always a hot topic to strengthen and transform the damaged existing RC beams to extend their service lives. Further, because of change in usage or emergency rescue, the existing RC structures may face the requirement of a great enhancement in load capacity. Take emergency rescue for example, there are situations in which some large heavy-duty vehicles may have to pass through a small RC bridge with a lower service load, and there will be strict requirements on the range of the load capacity improvement and the construction period. Therefore, it is of great practical significance to develop a strengthening method, which can achieve rapid construction and great improvement in the flexural performance of RC beams.

For the last two decades or so, RC beams strengthened using the steel-encased technique have been widely applied and studied owing to their good mechanical properties, economy, and construction convenience. According to the anchoring method, steel-encased can mainly be classified into three groups (Altin et al., 2005; Aykac et al., 2013; Li et al., 2013, 2018; Lo et al., 2014), namely, bonded steel plates, bolted side-plated, and composite anchorage. The bending tests of rectangular and T-section RC beams strengthened with bonded steel plates indicated that the flexural performance of the member can be significantly enhanced (Arslan et al., 2008; Raoof et al., 2000; Ren et al., 2015). However, bonded steel plates heavily rely on the adhesive quality, and the plates are prone to debonding failure (Hamoush and Ahmad, 1990; Hussain et al., 1995). Besides, they have very poor resistance to high temperature and aging. To avoid the shortcomings discussed above, the bolted side-plated technique was developed (Li et al., 2017; Shan and Su, 2018). This strengthening method can considerably improve both the flexural and shear performance of RC beams. However, it may lead to undesirable brittle failure when the bolted side-plated beams are over-reinforced (Ahmed et al., 2000; Su et al., 2010). The mechanical slipping of the bolts affects the strengthening effect (Li et al., 2016; Nguyen et al., 2001; Shan and Su, 2018; Su and Zhu, 2005). The experimental results showed that the premature debonding of steel plates can be prevented by composite anchorage of adhesive and bolts (Atea, 2019; Gao et al., 2006), while adhesive injection at the interface and bolt holes can reduce the mechanical slipping to some extent. In this way, the joint performance of the steel plates and the RC beams can be distinctly improved, achieving a satisfactory strengthening effect. However, the improvement in flexural capacity is still limited due to the effect of the critical reinforcement ratio (Nguyen et al., 2001; Su et al., 2010). In some cases, traditional steel-encased strengthening methods cannot meet the demand for a great enhancement in the flexural capacity of RC beams.

In recent years, U-shaped steel–concrete composite beams have attracted the attention of many researchers around the world (Heng et al., 2020; Keo et al., 2018; Liu et al., 2017, 2018; Nakamura, 2002; Zhou et al., 2019). It is well recognized that this new composite configuration can be considered as one of the most recent developments in steel–concrete composite beam construction due to its excellent structural performance and reduction of formwork and reinforcement work, as well as reduction of concrete creep and shrinkage. Hence, the existing achievements on U-shaped steel–concrete composite beams have laid a theoretical foundation for the development of the strengthening method using U-shaped steel. Strengthening RC beams with U-shaped steel is equivalent to simultaneously increasing the cross-sectional area and reinforcement ratio of the RC beams. In the strengthened beams, the steel bottom plate carries most of the tensile force originally taken by the steel bars, while the steel web takes the shear force together with the stirrups. More importantly, the steel web extends from the tension zone to the compression zone, so it will serve as both tension and compression reinforcement. Due to the contribution of the steel web in the compression zone, the compressive strain of concrete can be significantly reduced. If the thickness of the steel plate is reasonably designed, this strengthening method will not be affected by the critical reinforcement ratio. Furthermore, U-shaped steel can effectively control the development of cracks in concrete. Therefore, it can be foreseen that this strengthening method will help achieve a rapid and great enhancement in the flexural performance of RC beams, and it will have a broad application prospect.

To date, limited research has been performed on strengthening RC beams with U-shaped steel. The research and development of U-shaped steel strengthening RC beams are significantly lagging compared to the U-shaped steel–concrete composite beams. In practical engineering, the service load on the existing RC beams may not be completely removed before strengthening. As a result, stress and strain already develop in the RC beams. After strengthening, the initial load is still taken only by the RC beam. The U-shaped steel does not start working until the newly load, termed the secondary load, is applied. It means that the U-shaped steel just carries a portion of the newly applied load. Therefore, the stress and strain of the strengthened part always lag behind those in the RC beam, and this is a problem under secondary load (Hu, 2005; Lei et al., 2012). However, the literature review showed that little research had been conducted to study the influence of the secondary load on strengthened beams.

In this study, a new method to strengthen existing RC beams with U-shaped steel is proposed. The advantages of the proposed strengthening method include avoidance of interface debonding between the bottom plate and the RC beam, significant improvement of the flexural performance, and simple construction process. Both experimental and theoretical studies were conducted to evaluate the flexural performance of RC beams strengthened using this method under secondary load. In the experimental tests, a total of nine specimens were prepared and tested in flexure under four-point bending. The failure mode, load-deflection curves, interface slips, and material strains of each specimen were analyzed in detail in this study. The influences of initial loading degree, the thickness of the bottom plate, and the height of the steel box on the strengthening effect were comprehensively studied. Finally, based on the test results and theoretical analysis, a calculation formula for the flexural capacity of strengthened beams under secondary load was derived.

Experimental program

Specimen design and preparation

A total of nine specimens were designed in this study, including a reference RC beam. The dimensions of all the specimens are 150 mm × 250 mm × 2200 mm (with an effective length of 1950 mm and a shear span of 650 mm). The RC beams were cast with C30 commercial concrete. The U-shaped steel was welded with three Q235 B steel plates. For each specimen, two HRB400 (Hot-rolled Ribbed Bar, with a standard value of yield strength of 400 MPa) steel bars with a diameter of 10 mm were selected for the tensile reinforcement, and two HPB300 steel bars with a diameter of 8 mm were taken as the compressive reinforcement. The stirrups were made of HPB300 (Hot-rolled Plain Bar, with a standard value of yield strength of 300 MPa) steel bars with a diameter of 6 mm. The spacing of the stirrups in the pure bending section is either 200 mm or 250 mm while that in the shear bending section is 100 mm. The specific dimensions and reinforcement layout of the RC beams are shown in Figure 1.

Figure 1.

Specific dimensions and reinforcement layout of reinforced concrete beams (mm).

The details of the strengthening scheme for making the specimens are as follows. The U-shaped steel web was attached to the sides of the RC beams by the WSJ structural glue (developed by Jucheng Structural Co., Ltd of Wuhan University) and 8.8-grade bolts with a diameter of 16 mm, and the steel web was extended to the top of the RC beam. A certain distance (i.e., the height of the steel box) between the bottom plate of the U-shaped steel and the bottom surface of the RC beam was not filled with concrete to form a box structure. Based on the GB50367-2013 (2013), GB50017-2003 (2003), and considering the reinforcement position in the RC beams, the longitudinal spacing and edge distance of the bolts were reasonably set. Hence, the bolts were arranged symmetrically, and the longitudinal spacing was as uniform as possible. Because the steel box was not suitable to carry vertical pressure and prevent it from touching the piers due to bending deformation, the distance from the support to the steel box section edge was designed as 75 mm. The schematic diagram of strengthened beams is shown in Figure 2.

Figure 2.

Schematic diagram of strengthened beams (mm).

To explore the effect of the secondary load, four different initial loading degrees were considered including 0%, 30%, 50%, and 70% of the ultimate load of the RC beams. Considering the effects of the thickness of the bottom plate and the height of the steel box on the strengthened beams, the thickness of the bottom plate was designed as 4 mm, 6 mm, and 8 mm, respectively, and the height of the steel box was designed as 50 mm, 100 mm, and 150 mm, respectively. The test parameters and specimen identifications are listed in Table 1. Note that the initial loading degree β is defined as the ratio of the initial load to the ultimate load of the RC beam.

Table 1.

Test parameters and specimen identifications (mm).

Specimen ID^a	Thickness of steel web t₁	Thickness of bottom plate t₂	Height of steel box h_b	Initial loading degree β (%)
RCB1	0	0	0	0
SRCB1	4	4	100	0
SRCB2	4	4	100	30
SRCB3	4	4	100	50
SRCB4	4	4	100	70
SRCB5	4	6	100	50
SRCB6	4	8	100	50
SRCB7	4	4	50	50
SRCB8	4	4	150	50

^aIn the naming system, RCB indicates reinforced concrete beam and, SRCB denotes strengthened reinforced concrete beam.

Strengthening process

The RC beam should be strengthened based on the following construction procedure: (1) Use a rebar detector to accurately detect the stirrup position, and then the bolt holes in Figure 2 are fine-tuned and designed according to the actual position of the stirrups. Drill holes in the designed positions on the RC beam and the U-shaped steel web, respectively. (2) Clean the floating dust on the surface of the RC beam, polish and remove rust on the bonding surface of the U-shaped steel web, and remove oil stains on the steel plate with alcohol. (3) Apply adhesive on the sides of the RC beam, and the thickness should be controlled at 2–4 mm. (4) Install the U-shaped steel to the designed position, and then insert bolt shank and inject structural glue into the bolt holes to avoid undesirable slips due to the presence of gaps between components. Anchor the U-shaped steel by tightening the bolts with hexagonal nuts. Fasten the hexagonal nuts until a little glue is squeezed out. This indicates that the fastening degree is sufficient and then clear away the extruded glue. To prevent loosening, a gasket is placed between the nut and the steel plate.

The specimen SRCB1 was directly strengthened by following the above steps, as shown in Figure 3. However, for the specimens SRCB2 to SRCB8, strengthening was performed after the initial load was imposed on the RC beams, as shown in Figure 4. When strengthening RC beams under the initial load, a forklift was used to assist in positioning the U-shaped steel jacket.

Figure 3.

Processing of specimen SRCB1.

Figure 4.

Strengthening RC beams with U-shaped steel under initial load.

Material properties

Compressive tests of the C30 concrete were conducted on five standard cube blocks (150 mm × 150 mm × 150 mm) (GB50010-2010, 2010), and the average cubic compressive strength and the corresponding axial compressive strength were determined to be 32.86 MPa and 21.98 MPa, respectively. Tensile tests of the steel plates, steel bars, and 8.8-grade-M16 bolts were also performed (GB/T2975-1998, 1998; GB/T228.1-2010, 2010). Three groups of dog bone-shaped specimens were cut and prepared from Q235 steel plates with nominal thicknesses of 4 mm, 6 mm and 8 mm, with each group consisting of five specimens. The mechanical properties of the steel plates, steel bars, and 8.8-grade-M16 bolts measured from the tensile tests are shown in Table 2, respectively.

Table 2.

The mechanical properties of steel plates, steel bars, and 8.8-grade-M16 bolts.

Specimens	Types^a	Measured thickness/diameter (mm)	Yield strength (MPa)	Ultimate strength (MPa)	Elastic modulus (GPa)	Elongation (%)	Shear strength (MPa)
Steel plates	4 mm	3.48	309.2	447.0	205	37.0	-
	6 mm	5.32	308.2	440.8	205	37.0	-
	8 mm	7.46	310.6	446.4	205	39.0	-
Steel bars	D6-HPB300	5.80	394.4	548.2	205	26.7	-
	D8-HPB300	8.10	341.2	490.1	209	25.0	-
	D10-HRB400	9.72	428.4	625.5	201	24.0	-
Bolts	8.8-grade-M16	15.68	601.3	720.4	200	26.7	360.2

^aD6-HPB300 indicates the HPB300 steel bar with a diameter of 6 mm.

Measurement and test procedure

Four-point bending tests were performed on all the specimens listed in Table 1. The load applied by a 1000 kN hydraulic jack was controlled by a force sensor arranged between the hydraulic jack and the reaction force frame. The two-point loads were applied to the top of specimens through a steel distributive girder. The schematic diagrams of the loading mode and the testing set-up are shown in Figures 5 and 6, respectively. Five displacement meters were arranged for each specimen (Figure 5(a)), which was used to record the mid-span deflection, vertical relative slip, support settlement, and end slip (i.e., horizontal relative slip). The layouts of the resistance strain gauges for the concrete, the tensile reinforcement, the U-shaped steel web, and the bottom steel plate are shown in Figure 5(b)–(e), respectively. All the test data including the load, mid-span displacement, relative slip, and material strains were automatically obtained by the DH3816 data acquisition instrument manufactured by Jiangsu Donghua Test Technology Co., Ltd

Figure 5.

The loading mode and measurement (mm).

Figure 6.

Testing set-up.

Before testing, each specimen was pre-loaded to check whether the displacement meters, strain gauges, and the data acquisition instrument were working properly or not. The preload value was considered to be about 20% of the estimated ultimate load. Afterward, all specimens were tested in sequence. In the elastic working stage, every load level was uniformly taken as 10% of the estimated ultimate load. And it remained unchanged for about 2 min, to ensure that the deformation, strain, and stress could be fully developed. After the yielding of the specimens occurred, every load level was reduced to be 1/15 of the estimated ultimate load, and it was kept constant for 3 min–5 min. When the load was added to 70% of the estimated ultimate load, it was further changed to slow continuous loading. The test was terminated until either that the deformation was too large or the load dropped to around 85% of the peak load.

Experimental results and discussions

Failure modes and crack patterns

The failure mode of the specimen RCB1 is a typical flexural failure. When the load was increased to 10 kN, many small cracks appeared at the bottom of the mid-span. When the specimen was loaded to 46.7 kN (the resultant force of the two loading points, the same below), the tensile reinforcement started to yield. Afterward, more cracks appeared and propagated rapidly upward, further reducing the height of the compression zone. The load began to drop when reaching 56 kN. After experiencing obvious deformation, the concrete at the mid-span was crushed as can be seen in Figure 7(a).

Figure 7.

Typical failure modes and crack patterns.

For the specimen SRCB1, the U-shaped steel performed well together with the RC beam in the early stage. Very small deformation was observed, and the strengthened beam was in the elastic stage. When the load was increased to 287.2 kN, the bottom steel plate in the mid-span started to tensile yield. With a further increase of the load, the specimen constantly made a “crackling” sound of degumming. When the load reached 387 kN, with a “bang” muffled sound, the upper part of the steel web on the right side of the pure bending section peeled off. Then a half-wave buckling of the unglued steel web was formed between two adjacent bolts in the top row. There was a slight drop in the load simultaneously. With further loading increase, the peeling-off area gradually expanded and the buckling degree aggravated. But, the steel web without degumming was still tightly bonded to the RC beam. With another muffled sound, the upper part of the steel web near the left loading point in the shear span also peeled off and slightly buckled. Meanwhile, a transverse crack appeared on the top of the RC beam at the right support. When the load was further increased to 429 kN, it began to drop. Subsequently, the mid-span deflection increased rapidly, and the concrete at the junction of the pure bending section and the shear span section on the right side was crushed. Due to the abrupt change of the steel box section at the support, as well as the degumming of the steel web to some extent, the shear contribution of the steel web was greatly reduced. Finally, the RC beam near the right support was cut off under a large shear force, and there was also a thinner transverse crack at the left support, as shown in Figure 7(b).

For the specimens SRCB2 to SRCB8, the RC beams were already loaded with cracks before strengthening. This means that stress and deformation were developed in the RC beam, but the deformation was small. Except for the specimen SRCB2, the other strengthened beams experienced flexural failure, and the failure process is detailed below. Similar to the specimen SRCB1, the U-shaped steel and the RC beam performed well together in the elastic stage, and the corresponding mid-span deflection was insignificant. As the applied load increased, the bottom steel plates started to tensile yield, and then the strengthened beams continuously made crackling sounds of degumming. When loaded to a certain value, accompanied with a “bang” muffled sound, the upper part of the steel web in the pure bending section was peeled off. Then a half-wave buckling of the unglued steel web between two adjacent bolts in the top row occurred, together with a slight drop in load. With further load increase, the steel web at other locations also debonded and buckled, and the previous peeling area gradually expanded downward, while the buckling degree aggravated. Due to the restraint imposed by the bolts, the U-shaped steel in the peeling area still worked together with the RC beam. After the load reached the peak value, it began to drop. Then the mid-span deflection increased rapidly, and the concrete in the pure bending section was crushed, as shown in Figure 7(c).

It should be noted that because of design reasons, there was a sudden decrease in the section area of the steel box near the support, resulting in a significant decrease in the section stiffness. Besides, the strengthened beams might be considerably affected by the strengthening quality during the bolts drilling, gluing, and the U-shaped steel anchoring, etc. Hence, there might be a large stress concentration near the support positions and loading points. For the specimen SRCB2, as the applied load increased to 239.3 kN, the bottom steel plate started to tensile yield. Then, the upper part of the steel web near the left load point degummed and buckled. With the increase of the load, half-wave buckling was formed in the pure bend section and shear span section on both sides of the left loading point, and the buckling degree gradually aggravated. Rather than crushing in the pure bending section, the concrete in the shear span near the left loading point crushed slightly under the combined action of bending moment, shear force, and local pressure. Meanwhile, the RC beam was cut off at the left support position (the failure cause is similar to that of specimen SRCB1). While the RC beam near the right support position was roughly intact as shown in Figure 7(d). Thus to cause the flexural failure, the design principle of “strong shearing and weak bending” must be ensured while the flexural performance of the RC beams is improved. It is very important to minimize stress concentration and enhance strengthening quality. In this way, the mechanical properties of engineering materials can be made full use of.

Load-deflection responses and end slips

Figure 8(a) and (b) show the maximum differences between the top and bottom deflection at the mid-span of SRCB1 to SRCB8 and the maximum end slips, respectively. It can be seen that the interface slip of the specimen SRCB4 is the largest. The relative slip in the vertical and horizontal directions between the U-shaped steel and the RC beam both reached about 16 mm. The interface slips of other strengthened beams were very small, with a maximum value of 6 mm, and most did not exceed 3 mm. This demonstrated the good joint performance of the U-shaped steel and the RC beam.

Figure 8.

The maximum relative slips between the U-shaped steel and the reinforced concrete beam.

Figure 9(a) shows the load-deflection curves (i.e., P-δ curves) before and after the strengthening of the RC beams. It can be seen that the flexural capacity and secant stiffness of the specimen SRCB1 were both greatly enhanced compared to the reference specimen RCB1. The measured ultimate load of SRCB1 was about 6.7 times higher than that of RCB1, while the secant stiffness K_s was 6.5 times higher than that of RCB1. In contrast, the flexural capacity and secant stiffness were enhanced only by no more than 1.3 times and 3.6 times, respectively, when strengthening RC beams using the methods proposed by other researchers (Hawileh et al., 2014; Helal et al., 2020; Lei et al., 2012; Qin et al., 2019; Su et al., 2010). Therefore, a much better flexural performance can be achieved for strengthening RC beams with the U-shaped steel. This strengthening method is especially suitable for RC beams considering emergency rescue. The characteristic values of the load, mid-span deflection, secant stiffness, and ductility coefficient are listed in Table 3.

Figure 9.

P-δ curves of the specimens.

Table 3.

Test data.

Specimen	Yield load P_y (kN)	Yield deflection δ_y (mm)	Ultimate load P_u (kN)	Deflection corresponding to ultimate load δ_u (mm)	Failure load P_0.85 (kN)	Failure deflection δ_0.85 (mm)	Secant stiffness K_s (kN/mm)	Ductility coefficient μ
RCB1	46.7	4.8	56	9.6	47.0	16.7	5.8	3.5
SRCB1	287.2	4.4	429.4	9.9	361.1	18.2	43.4	4.1
SRCB2	239.3	3.1	448.0	9.0	386.5	13.3	49.8	4.3
SRCB3	278.6	4.6	416.6	9.6	340.7	21.3	40.7	4.6
SRCB4	268.6	3.9	411.3	10.3	357.0	17.1	39.9	4.4
SRCB5	277.9	3.8	457.0	8.8	375.0	16.5	51.9	4.3
SRCB6	339.9	4.2	483.3	7.1	410.6	16.9	68.1	4.0
SRCB7	208.7	4.8	345.4	14.5	287.8	20.3	23.8	4.2
SRCB8	337	4.8	468.0	10.0	393.0	21.5	46.8	4.5

To explore the influence of the initial loading degree, the thickness of the bottom plate and the height of the steel box on the flexural performance of strengthened beams under secondary load and the corresponding P-δ curves are plotted in Figure 9(b–d), respectively. Figure 9(b) reveals that the initial loading degree has little effect on the bearing capacity and stiffness, and the P_u and δ_u of SRCB1 to SRCB4 were relatively close. This means that for RC beams under normal working conditions, the original damage caused by the initial load was puny when compared to the excellent strengthening effect.

As the thickness of the bottom plate increased, P_u and P_y were slightly enhanced. Compared with SRCB3, the P_y of SRCB5 was almost the same as that of SRCB3, but the P_y of SRCB6 was increased by 22%. The P_u of SRCB5 and SRCB6 was increased by 9.7% and 16%, respectively. Moreover, the slope of P-δ curves increased significantly, which indicated an improvement in the stiffness. The K_s of SRCB5 and SRCB6 was increased by 27.5% and 67.3%, respectively, when compared to SRCB3. However, the pattern for the change of the corresponding mid-span deflection was not obvious. Increasing the thickness of the bottom plate is equivalent to increasing the reinforcement ratio in the tensile zone, which is easy to understand.

Compared with the initial loading degree and the thickness of the bottom plate, the height of the steel box had the most significant influence on the flexural behaviors. With the increase in the height of the steel box, the values of the P_y of SRCB3 and SRCB8 were increased by 33.5% and 61.5%, respectively, when compared to SRCB7. The values of the P_u of SRCB3 and SRCB8 were increased by 20.6% and 35.5%, respectively. Besides, the values of the K_s of SRCB3 and SRCB8 were increased by 71% and 96.7%, respectively, when compared to SRCB7. The most significant effect of the height of the steel box on the flexural performance can be attributed to the fact that increasing the height of the steel box would be not only equivalent to adding the reinforcement ratio in the tension zone but also increasing the cross-sectional area of the component. Furthermore, as the height of the steel box increased, the neutral axis of the cross-section would move down to some extent. The height of the steel box can be designed flexibly according to the actual demand. With the requirement of a great enhancement in the flexural capacity, it can even make the whole cross-section of the RC beam under compression, and only using the steel box to carry the tension is sufficient. However, when the steel box is too high, the neutral axis may fall within the section of the steel box, and the steel box is prone to local buckling under pressure. Besides, the steel box is too high, it will seriously affect the clear height. Therefore, the height of the steel box should be limited in practical engineering.

Load-strain responses

Figure 10 shows the load-strain curves (i.e., P-ε curves) of tensile reinforcement and bottom plate at the mid-span. In Figure 10, ε_y represents the yield strain of the tensile reinforcement, and ε_py represents the yield strain of the bottom plate. In the early stage of loading, P-ε curves roughly changed linearly, but the growth rate of the strain of the bottom plate was obviously much larger than that of the tensile reinforcement. As the load increased, the bottom plates all yielded. Then, the strengthened beams entered into the plastic deformation stage. As mentioned above, strengthening the RC beam with U-shaped steel would cause a downward movement of the neutral axis. Hence, the tensile reinforcement became closer to the neutral axis, and the tensile force borne by the steel bars was relatively small. At this time, most of the tensile force was borne by the steel box. Under this circumstance, only the tensile reinforcement of SRCB7 yielded when the test was terminated. Since the height of the steel box of SRCB7 is the lowest, the downward movement of the neutral axis is minimal. Consequently, the tensile reinforcement could make full use of its material properties during the experiment.

Figure 10.

P-ε curves of the tensile reinforcement and the bottom steel plate.

As shown in Figure 10, the ultimate strain in the tensile reinforcement of SRCB7 reached about 1000με. However, the strains in the tensile reinforcement of SRCB1 to SRCB4 just exceeded 1000με, while that of SRCB5 was less than 900με. Besides, the strains in the tensile reinforcement of SRCB6 and SRCB8 were not more than 650με. This implies that the contribution of the tensile reinforcement to the tension will be further reduced with the increase in the thickness of the bottom plate and the height of the steel box.

Strain developments at the mid-span

The longitudinal strains at the mid-span of SRCB1 to SRCB8 were found to satisfy the plane cross-section assumption well. Specifically, the longitudinal strains above the steel box developed linearly along with the height of the cross-section. Because the steel box was not filled with concrete, it was prone to a certain degree of buckling under a large external force, resulting in a deviation from the plane cross-section assumption, as shown in Figure 11. It should be noted that if the buckling degree of the U-shaped steel web is large, the longitudinal strains cannot satisfy the plane cross-section assumption.

Figure 11.

Development of longitudinal strain distribution at the mid-span section.

Ductility

As mentioned in the section Load-Strain Responses, the bottom plates all yielded during the experiment, but the tensile reinforcement in all specimens except SRCB7 did not yield. Hence, the corresponding load that caused the bottom plate to yield was taken as the yield load P_y of the strengthened beams, while the corresponding deflection was taken as the yield deflection δ_y. The ductility coefficient μ is defined as the ratio of δ_0.85 to δ_y, which can be expressed as

μ = \frac{δ_{0 . 85}}{δ_{y}}

(1)

Figure 12(a) shows the comparison of ductility coefficients of all specimens. It can be seen that the ductility of SRCB1 to SRCB8 was considerably improved in comparison to RCB1, with the improvement ranging from 14.2%–31.4%. The initial loading degree had a certain effect on the ductility. With the increase in the initial load, the ductility increased first and then decreased, but roughly showed an upward trend. This was closely related to the variation of the secant stiffness under different initial loads. However, there was a decrease in the ductility as the thickness of the bottom plate increased (the same conclusions were drawn by Su et al. (2010) and Nguyen et al. (2001)), while the influence of the height of the steel box on the ductility was not obvious.

Figure 12.

The comparison of ductility coefficients and the comparison of secant stiffness.

Stiffness

Figure 12(b) shows a comparison of the secant stiffness K_s of all specimens. The formula for calculating K_s is as follows

K_{s} = \frac{P_{u}}{δ_{u}}

(2)

Compared with RCB1, the values of the stiffness of the strengthened beams were increased by several times. For the specimen SRCB7 (with a height of the steel box of 50 mm), its stiffness increment was the lowest, which was increased by 3.1 times. For the specimen SRCB6 (with a thickness of the bottom plate of 8 mm), its stiffness increment was the highest, which was increased by 10.7 times. As the initial load increased, K_s roughly showed a downward trend. As expected, with the increase in the thickness of the bottom plate and the height of the steel box, K_s increased steadily.

Theoretical study on the flexural capacity

Basic assumptions

To derive a formula for calculating the flexural capacity of RC beams strengthened with U-shaped steel under secondary load, the following basic assumptions were made: (1) plane sections remain plane; (2) tensile strength of concrete is considered to be negligible; (3) the interface relative slip between the U-shaped steel and the RC beam is ignored; (4) the model suggested by Rüsch (GB50010-2010, 2010) is adopted to represent the compressive stress–strain curve of concrete. In this model, the ascending segment is a quadratic parabola, and the descending segment is a horizontal straight line, as shown in Figure 13(a). This constitutive model of the concrete can be mathematically expressed as follows

σ_{c} = {\begin{cases} f_{c} [2 \frac{ε_{c}}{ε_{0}} - {(\frac{ε_{c}}{ε_{0}})}^{2}] (ε_{c} \leq ε_{0}) \\ f_{c} (ε_{0} \leq ε_{c} \leq ε_{cu}) \end{cases}

(3)

where

σ_{c}

represents concrete compressive stress when concrete compressive strain is

ε_{c}

f_{c}

is the design value of axial compressive strength of concrete,

ε_{0}

is the concrete strain corresponding to

f_{c}

, with a value of 0.002, and

ε_{cu}

is the concrete ultimate compressive strain, with a value of 0.0033.

Figure 13.

The stress–strain relations of materials.

The stress–strain relationships of the steel bars and the U-shaped steel are approximated by the elastic-perfectly plastic model as shown in Figure 13(b), and they can be expressed as follows

σ_{s} = {\begin{cases} E_{s} ε_{s} (0 \leq ε_{s} < ε_{y}) \\ f_{y} (ε_{s} \geq ε_{y}) \end{cases}

(4)

where

σ_{s}

is the stress in tensile reinforcement,

E_{s}

is the elastic modulus of tensile reinforcement,

ε_{s}

is the strain in tensile reinforcement,

ε_{y}

is the yield strain in tensile reinforcement, and

f_{y}

is the design value of the rebar tensile strength

σ_{sp} = {\begin{cases} E_{sp} ε_{sp} (0 \leq ε_{sp} < ε_{py}) \\ f_{py} (ε_{sp} \geq ε_{py}) \end{cases}

(5)

where

σ_{sp}

is the stress in U-shaped steel,

E_{sp}

is the elastic modulus of U-shaped steel,

ε_{sp}

is the strain in U-shaped steel,

ε_{py}

is the yield strain in U-shaped steel, and

f_{py}

is the design value of the tensile strength of the U-shaped steel.

Flexural capacity calculation

For the specimens SRCB2 to SRCB8, the RC beams were already loaded and subjected to cracking before strengthening, and their longitudinal strains accorded well with the plane cross-section assumption. After strengthening, the U-shaped steel began to carry the new additional load together with the RC beam. The strain in the U-shaped steel always lagged behind that in the RC beam, and the plane cross-section assumption was satisfied between the strain increment of the RC beam and the strain in the U-shaped steel. However, the actual total strain of the strengthened beam did not meet the plane cross-section assumption as shown in Figure 14.

Figure 14.

Strain diagram of strengthened beams under secondary load.

According to the plane cross-section assumption, the corresponding strain at the position of the steel box can be obtained from the extension of the initial strain of the cross-section of the RC beam, which is the hysteretic strain $ε_{sp 1}$ of the U-shaped steel. This strain is virtual and does not exist. After the hysteretic strain $ε_{sp 1}$ is added to its actual strain $ε_{sp}$ , the nominal total strain of the composite beam can thus meet the plane cross-section assumption. Therefore, the initial strain of the RC beam section is the key to the calculation of the flexural capacity of the strengthened beams under secondary load.

Initial strain calculation

For the specimens SRCB2 to SRCB8, the longitudinal tensile reinforcement has not yet yielded before strengthening. The initial strain $ε_{c 1}$ of the concrete at the edge of the compression zone did not exceed $ε_{0} = 0.002$ . The stress–strain distribution of the cross-section of the RC beam is shown in Figure 15. Based on the geometric relationship shown in Figure 15, the following expressions can be derived

ε_{s 1} = \frac{h_{0} - x_{c 1}}{x_{c 1}} ε_{c 1}

(6)

{ε^{'}}_{s 1} = \frac{x_{c 1} - {a^{'}}_{s}}{x_{c 1}} ε_{c 1}

(7)

ε_{cx} = \frac{x}{x_{c 1}} ε_{c 1}

(8)

where

{ε^{'}}_{s 1}

is the initial strain in the compressive reinforcement,

ε_{s 1}

is the initial strain in the tensile reinforcement,

ε_{cx}

is the compressive strain at a distance x from the neutral axis in the concrete,

x_{c 1}

is the height of the neutral axis, which is the theoretical height of the concrete compression zone,

h_{0}

is the effective height of the RC beam, defined as the distance from the center of the tensile reinforcement to the top surface of the RC beam, and

a_{s}^{'}

is the distance from the center of the compressive reinforcement to the top surface of the RC beam.

Figure 15.

Initial strain and stress distribution of the cross-section of the reinforced concrete beam.

The resultant force $C_{1}$ of the compressive stress in the compression zone in concrete can be obtained by the following integral

C_{1} = \int_{0}^{x_{c 1}} b f_{c} [2 (\frac{x}{x_{c 1}}) \frac{ε_{c 1}}{ε_{0}} - {(\frac{x}{x_{c 1}})}^{2} {(\frac{ε_{c 1}}{ε_{0}})}^{2}] d x = f_{c} b x_{c 1} \frac{ε_{c 1}}{ε_{0}} (1 - \frac{ε_{c 1}}{3 ε_{0}})

(9)

The distance $y_{c 1}$ from $C_{1}$ to the edge of the concrete compression zone can be given by

y_{c 1} = x_{c 1} - \frac{\int_{0}^{x_{c 1}} x \cdot σ_{cx} (ε_{cx}) b d x}{C_{1}} = \frac{1 - ε_{c 1} / (4 ε_{0})}{3 - ε_{c 1} / ε_{0}} x_{c 1}

(10)

Based on the force and moment equilibriums in Figure 15, the following expressions can be obtained

{\begin{cases} C_{1} + E_{s}^{'} ε_{s 1}^{'} A_{s}^{'} = E_{s} ε_{s 1} A_{s} \\ M_{1} = C_{1} y_{c 1} + E_{s}^{'} ε_{s 1}^{'} A_{s}^{'} a_{s}^{'} - E_{s} ε_{s 1} A_{s} h_{0} \end{cases}

(11)

Where

A_{s}^{'}

and

A_{s}

are the cross-section areas of compressive reinforcement and tensile reinforcement, respectively.

Substituting equations (6), (7), (9), and (10), as well as the initial bending moment $M_{1}$ into equation (11), $ε_{c 1}$ and $x_{c 1}$ can be obtained by solving the nonlinear equations. Then substituting $ε_{c 1}$ and $x_{c 1}$ into equations (6) and (7), $ε_{s 1}$ and $ε_{s 1}^{'}$ can be obtained.

Flexural strength calculation under secondary load

As mentioned in sections Failure Modes and Crack Patterns and Load-Strain Responses, all the bottom plates yielded when the strengthened beams failed. The U-shaped steel web in the compression zone and the tension zone also partially entered into the yielding state. Meantime, the compressive reinforcement yielded, while the tensile reinforcement did not necessarily yield. After strengthening, the neutral axis is still located between the tensile reinforcement and the compressive reinforcement, although it moves down a little bit. The equivalent rectangular stress block can be used to replace the theoretical stress graph of the concrete in the compression zone (GB50010-2010, 2010). Note that $β_{1}$ is the coefficient of the equivalent rectangular stress block, and a value of 0.8 is used for C30 concrete. The stress and strain diagram of the cross-section of a strengthened beam is shown in Figure 16.

Figure 16.

Schematic diagram of stress and strain distribution in the ultimate state.

Based on the plane cross-section assumption and Figure 16, equation (12) can be derived

\frac{x_{c} - x_{1}}{x_{c}} = \frac{ε_{py}}{ε_{cu} - ε_{c 1}}

(12)

where

x_{c}

is the theoretical height of the concrete compression zone under secondary load;

x_{1}

is the distance from the compressive yield point of the steel web to the top surface of the RC beam.

By rearranging equation (12), $x_{1}$ can be determined as

x_{1} = (1 - \frac{ε_{py}}{ε_{cu} - ε_{c 1}}) x_{c}

(13)

Similarly, the strain increment $ε_{s 2}$ of the tensile reinforcement and the strain increment $ε_{s 2}^{'}$ of the compressive reinforcement under secondary load can be determined by equations (14)–(17)

\frac{h_{0} - x_{c}}{x_{c}} = \frac{ε_{s 2}}{ε_{cu} - ε_{c 1}}

(14)

ε_{s 2} = (\frac{h_{0} - x_{c}}{x_{c}}) (ε_{cu} - ε_{c 1})

(15)

\frac{x_{c} - a_{s}^{'}}{x_{c}} = \frac{ε_{s 2}^{'}}{ε_{cu} - ε_{c 1}}

(16)

ε_{s 2}^{'} = (1 - \frac{a_{s}^{'}}{x_{c}}) (ε_{cu} - ε_{c 1})

(17)

Based on the force equilibrium of a strengthened beam under secondary load, equation (18) can be derived

α_{1} f_{c} b β_{1} x_{c} + E_{s}^{'} ε_{s2}^{'} A_{s}^{'} + 2 η f_{py} t_{1} x_{1} = 2 η f_{py} [h_{n} - x_{c} - (x_{c} - x_{1})] t_{1} + η f_{py} b t_{2} + E_{s} ε_{s 2} A_{s}

(18)

Where

α_{1}

and

β_{1}

are the coefficients of the equivalent rectangular stress block, taken values of 1.0 and 0.8 for the C30 concrete, respectively (GB50010-2010, 2010),

η

is a reduction factor, considering steel web buckling and relative slip between the U-shaped steel and the RC beam, with a value of 0.8 (Hu, 2005),

h_{n}

is the total height of the strengthened beam, and

t_{1}

and

t_{2}

are the thicknesses of the steel web and the bottom plate, respectively.

Substituting equations (13), (15), and (17) into equation (18), $x_{c}$ can be obtained by rearranging equation (18). Then substituting $x_{c}$ into equations (15) and (17), $ε_{s 2}$ and $ε_{s 2}^{'}$ can be calculated. If $| ε_{s 1} + ε_{s 2} | > ε_{y}$ or $| ε_{s1}^{'} + ε_{s2}^{'} | > ε_{y}$ , then $| ε_{s 1} + ε_{s 2} | = ε_{y}$ or $| ε_{s1}^{'} + ε_{s2}^{'} | = ε_{y}$ should be substituted into equation (18) for recalculation.

Finally, taking the moments of all the internal forces to the center position of the tensile reinforcement, the formula for calculating the flexural capacity of a strengthened beam under secondary load can be derived as follows

\begin{array}{l} M_{u} = α_{1} f_{c} b β_{1} x_{c} (h_{0} - β_{1} x_{c} / 2) + E_{s}^{'} ε_{s 2}^{'} A_{s}^{'} (h_{0} - a_{s}^{'}) + 4 η f_{py} {(x_{c} - x_{1})}^{2} t_{1} / 3 \\ + 2 η f_{py} t_{1} x_{1} (h_{0} - x_{1} / 2) + η f_{py} (h_{n} - h_{0}) b t_{2} \\ + 2 η f_{py} t_{1} (h_{n} - 2 x_{c} + x_{1}) [h_{n} - h_{0} - (h_{n} - 2 x_{c} + x_{1}) / 2] \end{array}

(19)

Comparison of the flexural strength

The flexural capacity of the specimens SRCB1 to SRCB8 was obtained using the derived equation (19), and a comparison of the results from equation (19) and experiments was conducted as shown in Table 4. SRCB1 is a specimen under initial load, and its initial strain of the RC beam is 0, which can be considered as a special case under secondary load. From Table 4, it can be seen that the calculated values from equation (19) agreed satisfactorily with the experimental ones, with a maximum relative error of 12%. The maximum error found in the specimen SRCB2 was caused by the bending-shear failure, which was different from the flexural failure. Similarly, because the specimen SRCB1 also experienced bending-shear failure, its calculation error is relatively big as well, which is 8.7%. In addition, given that the basic assumptions are not completely consistent with the actual situation, a calculation error of up to 10% is considered to be acceptable.

Table 4.

Comparison of the values of flexural capacity determined from equation (19) and experiments.

Specimen	t₁ (mm)	t₂ (mm)	h_b (mm)	β	a (mm)	P_u (kN)	M_u (kN●m)	M_e^a (kN●m)	Relative error
SRCB1	4	4	100	0	650	429.4	127.4	139.6	−8.7%
SRCB2	4	4	100	30%	650	448.0	128.2	145.6	−12.0%
SRCB3	4	4	100	50%	650	416.6	129.3	135.4	−4.5%
SRBB4	4	4	100	70%	650	411.3	128.6	133.7	−3.8%
SRCB5	4	6	100	50%	650	457.0	136.2	148.5	−8.3%
SRCB6	4	8	100	50%	650	483.3	145.0	157.1	−7.7%
SRCB7	4	4	50	50%	650	345.4	107.8	112.3	−4.0%
SRCB8	4	4	150	50%	650	468.0	146.3	152.1	−3.8%

^aM_e represents the experimental value of flexural capacity and can be calculated by $M_{e} = 0.5 P_{u} a$ .

Conclusions

This study has presented a new strengthening method for existing RC beams with U-shaped steel. To evaluate the flexural performance of RC beams strengthened using this method under secondary load, both experimental and theoretical studies were conducted in this study. The influences of initial loading degree, the height of the steel box, and the thickness of the bottom plate on the strengthening effect were emphatically analyzed. The key conclusions are summarized as follows:

All the strengthened RC beams except the SRCB4 experienced small interface slips between the U-shaped steel and the RC beam, with values of not more than 6 mm. This demonstrated the good joint performance of the U-shaped steel and the RC beam using the proposed strengthening method.

Compared with the reference beam, the flexural capacity of the strengthened beams was improved by 5.2–7.6 times, while the secant stiffness was increased by 3.1–10.7 times. Given that the ease of construction, so the proposed strengthening method can achieve a rapid and great enhancement in the flexural performance of RC beams.

Most of the strengthened beams experienced flexural failure. The failure started from the yielding of the bottom plate and ended with concrete crushing in the pure bending section, accompanied by serious peeling and buckling of the steel web. The strengthened beams mainly relied on the steel box to carry the tensile force, and the tensile reinforcement only made a small contribution to the tensile resistance.

The initial loading degree (β≤70%) was found to only have a negligible role in the flexural performance of the strengthened beams. Compared with the excellent strengthening effect, the original damage of the RC beams caused by the initial load was puny and could be ignored. With the increase in the thickness of the bottom plate and the height of the steel box, the flexural behaviors were considerably improved, with the latter having a more significant effect.

A calculation formula for the flexural capacity of strengthened beams under secondary load was derived. The results from the derived formula were found to have a satisfactory agreement with the experimental ones, with a maximum relative error of 12%.

Due to design reasons, the specimens SRCB1 and SRCB2 experienced undesired bending-shear failure. In addition, when the steel box is too high, the neutral axis may move into the steel box part. At this point, the steel box above the neutral axis is prone to local buckling under compression. Thus, further research is required to estimate the maximum height of the steel box that can lead to local buckling and determine the optimal dimensions of the U-shaped steel box to guarantee “strong shearing and weak bending” and avoid the local buckling.

Footnotes

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 Natural Science Foundation of China (No. 51878520) and Special Project on Technical Innovation of Hubei (No.2019ACA142).

ORCID iD

Yiyan Lu

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